1 Introduction
Let a be an integer and let p be a prime not dividing a. The multiplicative order \ord_p(a) is the least positive integer k such that a^k\equiv1\pmod p. The residue class of a is a primitive root modulo p if \ord_p(a)=p-1. Call a admissible if a\ne-1 and a is not the square of an integer. These are necessary conditions for primitive-root behavior at infinitely many primes: the order of -1 is at most 2, and a nonzero square modulo an odd prime has order dividing (p-1)/2.
Artin’s primitive root conjecture, formulated in 1927, asserts that every admissible integer is a primitive root modulo infinitely many primes; its quantitative form predicts a positive-density asymptotic [1, Section 17.2]. We prove the following lower bound.
Theorem 1.1. For every admissible integer a, there are constants c_a>0 and x_a\ge2 such that, for every real x\ge x_a,
The predicted counting scale is x/\log x; the theorem gives x/(\log x)^2 in every sufficiently large dyadic interval. In particular, it proves the infinitude assertion for each fixed admissible integer, including the prescribed base 2. The constants and the threshold may depend on the base.
The analytic part of the proof establishes a separate theorem. A finite-order Hecke character of a number field F means a continuous finite-image character of the idele class group F^\times\backslash\Adele_F^\times. Write \mu_m for the group of mth roots of unity.
Theorem 1.2. Let F be a cyclotomic number field containing \mu_{12}. For every finite-order Hecke character \eta of F, the meromorphic continuation of L_F(s,\eta) has no zeros in
The principal character is included, with its pole at s=1 permitted. The width is common to all fields and characters in the statement, with no conductor or height cutoff. In the proof, F and \eta are fixed before the auxiliary data are chosen. Constants and the points from which estimates hold may depend on them; the numerical saving in the final Mellin argument does not. This distinction is what allows a uniform splitting estimate for a growing family of fields.
1.1 Historical context and the two obstructions
Hooley proved Artin’s predicted asymptotic under the Riemann hypothesis for the Dedekind zeta functions of the Kummer fields \Q(\mu_n,a^{1/n}), with n squarefree [13]. These fields encode the obstructions to maximal order. For a prime q and p\nmid aq, the conditions
say that p splits completely in \Q(\mu_q,a^{1/q}). They hold precisely when q divides the index (p-1)/\ord_p(a). Hooley’s method counts and sums such splitting conditions. Here a restricted factorization of p-1 confines the possible index divisors to ranges where they can all be removed.
Classical unconditional results allow a finite choice of bases. Gupta and Murty constructed a set of thirteen integers from three distinct primes, at least one of which is a primitive root modulo infinitely many primes [9]. Heath-Brown proved that at most two prime bases fail the infinitude assertion, so at least one of 2,3,5 satisfies it [11]. Averaging over the base gives another approach. Goldfeld and Stephens established early average theorems [7, 26]; Klurman, Shparlinski, and Teräväinen obtained the expected asymptotic for almost all integers in a range as short as 1\le a\le\exp((\log\log x)^2) [17]. The exceptional set may depend on x, so these average results do not by themselves settle a prescribed base.
Two difficulties must be addressed for a fixed base. One is analytic: the splitting fields vary with the possible index divisor. A standard simultaneous zero-free region for Hecke characters associated with an abelian extension has logarithmic width depending on the field, conductors, and height, and permits at most one simple real exceptional zero [28, Theorem 3.1]. The common width in Theorem 1.2 instead gives a common power saving. The other difficulty is the prime construction: we need many primes for which the possible index divisors fall in the range controlled by that saving. We now describe the two parts of the proof and the points where they meet.
1.2 The Hecke zero-free argument
Fix a cyclotomic field F containing \mu_{12} and a finite-order Hecke character \eta. The analytic construction uses cubic theta coefficients under sextic Kummer twists. Patterson’s cubic theta series over \Q(\sqrt{-3}) is the model for the Gauss coefficients and their transformations [24, 25]; the comparison formulas are recorded by Dunn and Radziwiłł [5, Section 5.1 and Appendix A]. The companion manuscript The Quasi-Riemann Hypothesis develops a coefficient reflection and an additive probe over that field while retaining unrestricted cube factors [23, Part I, Sections 5–7]. Its zero-free theorem treats Dirichlet L-functions and finite-order Hecke L-functions over \Q(\sqrt{-3}). We construct the required mechanisms over each fixed F, using the cubic theta representation of Kazhdan and Patterson and a normalized global Whittaker factorization consistent with their correction [15, 16].
Two local calculations govern the comparison. We choose generators for which sextic reciprocity has no supplementary factor. In a reflected interaction between a prime occurring to valuation one in the varying twist and a varying squarefree coefficient prime, the sextic exponents are 1 and -4. Their sum is 3 modulo 6, so the interaction is quadratic. The quadratic large sieve of Heath-Brown, in its number-field form due to Goldmakher and Louvel, then controls the reflected moment [12, 8].
On the Poisson side, a relative-norm formula identifies the cube of the cubic Gauss phase with the negative of a unitary Hecke-character value. A finite-order correction gives the target character \eta. At a prime ideal \mathfrak p outside a fixed excluded set, the first squarefree theta term at the principal frequency is -\eta(\mathfrak p)(\Norm\mathfrak p)^{-\xi}, where \xi is the Mellin variable for the coefficient norm. Together with the constant term, it supplies the local factor of the reciprocal target L-function. The remaining local terms give a holomorphic, nonvanishing correction in the region used at the end of the proof.
Reflection and Poisson summation thus estimate the same additive average in two ways. A conductor large sieve and zero detection control the nonprincipal frequencies. The resulting power saving for the principal inverse Mellin integral continues its Mellin transform across the line \re s=1 and excludes zeros of the target function. The use of ideal norms keeps the numerical exponents in this argument independent of the degree of F. Sections 3–8 carry out this proof; Appendix A provides the theta normalization.
1.3 Controlled predecessors in a fixed progression
The construction produces many primes of the form
where every prime divisor of r lies between \exp((\log x)^{0.1}) and \exp((\log x)^{0.3}). We also prescribe a residue class on which the Legendre symbol (a/p) equals -1. This excludes 2 as an index divisor. A nontrivial remaining index has a prime divisor in rQ: the factor Q leads to a small elementary exceptional set, and the factors of r fall within the uniform splitting estimate.
The construction starts with weighted integers d=crQ+1 of this shape. A sieve removes small prime factors of d. For a remaining composite d=mn, with n its least prime factor, a marked bilinear estimate replaces the restriction on the cofactor m by an explicit density and a weaker condition on its small prime factors. Buchstab’s least-prime-factor identity leaves a positive margin between the initial sifted mass and the mass removed in this comparison. A separate upper bound makes the contribution from two prime factors close to \sqrt x smaller than that margin.
The marked prime estimate is an input from The Poisson–Dirichlet law for the prime factors of p-1 [21, Theorem 3.1]. We extract its scalar coefficient criterion and verify the full character and frequency estimates for the rough-factor comparison. The weights require d-1 to have a prime divisor in each of several prescribed ranges; these choices are the marks. A bounded multiplier in the bilinear estimate may depend on the quotient left after every power of these primes has been removed. A congruence condition on d=mn need not have this invariance. We therefore expand mn\equiv u\pmod M in Dirichlet characters and separate it into factors \lambda(m)\lambda(n) in the two scalar coefficient sequences. Fixed twists preserve the required long character-sum bound.
The resulting fixed-progression construction is stated separately in Proposition 2.1. It applies to any fixed data M,c,u satisfying its local coprimality conditions, independently of the primitive-root application. Its proof uses Bombieri–Vinogradov distribution, a Brun–Hooley block sieve, and the continuous rough-number density [3, 29, 6, 4]. The precise elementary sieve and density forms are adapted from [20, Lemmas 2.9 and 7.4], with proofs included here.
1.4 Organization and notation
Section 2 states the controlled-predecessor and splitting estimates and combines them to prove Theorem 1.1. Sections 3–8 prove Theorem 1.2 for a fixed arbitrary F and \eta, independently of the integer base. The analytic scale parameters in those sections are local to that proof. Section 9 returns to the fixed integer a and derives the uniform splitting bound. Section 10 proves the marked coefficient and rough-factor estimates; Section 11 supplies the elementary sieve facts; Section 12 proves the fixed-progression construction. Appendix A verifies the selected Kazhdan–Patterson specialization and Whittaker normalization.
Throughout, \log is the natural logarithm, and L=\log x in the prime-counting arguments. The symbols \ll and O have their usual meaning; subscripts record permitted dependence when it matters. Auxiliary parameters are fixed before the relevant asymptotic variable tends to infinity. Uniformity in those parameters is stated explicitly. The general representation-theoretic, class-field, Hecke-continuation, and classical sieve theorems are cited at their points of use. The marked Type II argument also uses the companion papers cited in the bibliography.
2 Reduction to controlled predecessors and complete splitting
For a prime p\nmid a, failure of a to generate \mathbb F_p^\times means that some prime divides the index
We construct primes for which the possible divisors of this index belong to two ranges. A large divisor forces p to divide one of a short list of integers a^j-1; the remaining divisors are controlled by complete splitting in Kummer fields. This is the index-divisor reduction underlying Hooley’s work [13, Sections 4–7].
The first proposition supplies the primes, with a fixed congruence that will force quadratic nonresiduosity.
Proposition 2.1 (Primes with controlled predecessors). Let M be a fixed positive multiple of 8, let c\in\{2,4\}, and let u be an integer satisfying
For every sufficiently large real x, there are \gg_{M,c,u}x/L^2 distinct primes p\in(x,2x) satisfying
where r is a positive integer all of whose prime divisors lie in
We prove Proposition 2.1 in Section 12. For the second input, write \mu_m for the group of mth roots of unity. For a nonzero integer a and a prime q, put
The choice of the root a^{1/q} does not affect K_q.
Proposition 2.2 (Uniform bound for complete splitting). Fix an integer a with |a|>1, and put \delta_0=10^{-6}. For all sufficiently large real x and all primes q sufficiently large in terms of a, with q\le\exp(L^{0.3}), one has
The implied constant and the threshold for x are independent of q. For a=2, every prime q\ge5 is allowed, with absolute constants and an absolute threshold for x.
Section 9 proves this proposition from the Hecke zero-free theorem, Theorem 1.2. To apply it, we first record the elementary interpretation of complete splitting.
Lemma 2.3 (Splitting test). Let a\ne0 be an integer, and let p,q be primes with p\nmid aq. Then p splits completely in K_q if and only if
Proof. The field K_q is the splitting field of T^q-a. Fix a prime \mathfrak P of K_q above p. The roots are integral and have distinct reductions modulo \mathfrak P, since p does not divide the polynomial discriminant \pm q^q a^{q-1}. Every inertia element permutes the roots and fixes their reductions, so it fixes every root. Thus p is unramified. Let \sigma be its arithmetic Frobenius at \mathfrak P. On \mu_q, this automorphism acts by \zeta\mapsto\zeta^p, so it fixes E_q exactly when p\equiv1\pmod q. Under this condition, with \alpha=a^{1/q}, we have
The division is valid because p\nmid a. Reduction on \mu_q is injective at p\ne q, so the second congruence in (2.5) is equivalent to \sigma(\alpha)=\alpha. Together the two conditions say that \sigma fixes the generators of K_q. Since K_q/\Q is Galois, this is precisely complete splitting.
We now choose a progression of quadratic nonresidues satisfying all the hypotheses of Proposition 2.1.
Lemma 2.4 (A progression of quadratic nonresidues). Let a\in\Z be neither -1 nor an integer square, and set
There are c\in\{2,4\} and an integer u satisfying (2.1) such that every prime p\equiv u\pmod M satisfies p\nmid a and (a/p)=-1.
Proof. Write a=d_0t^2, where t\ge1 is an integer and d_0\ne1 is signed squarefree. We choose u to be a unit modulo M, with u\not\equiv1\pmod\ell for every odd prime \ell\mid M, and u\bmod8\in\{3,5,7\}. These restrictions already imply p\nmid a.
Suppose first that an odd prime j\ge5 divides d_0. Fix any of the allowed residues modulo 8 and at the odd primes other than j. Quadratic reciprocity expresses (d_0/p) as a fixed sign times (p/j). Among the nonzero residues modulo j other than 1, there are (j-3)/2\ge1 squares and (j-1)/2 nonsquares. We can therefore choose u\bmod j so that (d_0/p)=-1.
If no such j exists, the seven possibilities for d_0 are treated by the following table. Whenever 3\mid M, also take u\equiv2\pmod3.
| d_0 | -1 | 2 | -2 | 3 | -3 | 6 | -6 |
| u\bmod8 | 3 | 3 | 7 | 5 | 5 | 3 | 7 |
The supplementary laws give the first three entries. For the last four, the combined residues modulo 24 are respectively 5,5,11,23; quadratic reciprocity gives (d_0/p)=-1 in each case. Odd primes dividing M but not d_0 occur only in the square factor t^2 and can be assigned any allowed residue. The Chinese remainder theorem combines these choices into a class u\pmod M.
If u\equiv3 or 7\pmod8, take c=2; if u\equiv5\pmod8, take c=4. In each case (u-1)/c is odd. For every odd \ell\mid M, the condition u\not\equiv1\pmod\ell also makes (u-1)/c nonzero modulo \ell. This proves (2.1). Finally, (a/p)=(d_0/p)=-1, since p\nmid t.
Proof of Theorem 1.1. Fix a as in the theorem, so |a|>1. Choose M,c,u by Lemma 2.4, and take the \gg_a x/L^2 primes supplied by Proposition 2.1. For each such prime p, Euler’s criterion gives a^{(p-1)/2}\equiv-1\pmod p. Consequently i_p(a) is odd: an even index would make \ord_p(a) divide (p-1)/2.
If i_p(a)>1, choose a prime divisor q of this index. Since p-1=crQ and c is a power of 2, we have q\mid rQ. Moreover \ord_p(a)\mid(p-1)/q, so (2.5) holds. We bound the number of such primes p in the two possible ranges of q.
If q=Q, then
Thus p divides the positive integer
No factor vanishes because |a|>1, and
Every distinct prime divisor p>x contributes at least L to this logarithm. The number of primes lost in this case is therefore O_a(x^{0.2}/L)=o(x/L^2).
If q\mid r, then \exp(L^{0.1})<q<\exp(L^{0.3}). For large x, all these primes are sufficiently large for Proposition 2.2; also p\nmid aq. Lemma 2.3 and a union bound give at most
primes, where the first sum is over integers. The first estimate uses (2.4) for each possible q; the second uses the telescoping sum of 1/(m(m-1)).
Subtracting these two losses leaves \gg_a x/L^2 primes with i_p(a)=1. All choices depend only on a, and both estimates hold for every sufficiently large real x. This proves the theorem.
For a=2, the theorem gives \gg x/L^2 primes in (x,2x) with primitive root 2, with absolute constants. This case can use the particularly simple choice (M,c,u)=(8,4,5) in Proposition 2.1: the predecessors are 4rQ, and p\equiv5\pmod8 gives (2/p)=-1 directly.
Remark 2.5. Negative squares are covered by the row d_0=-1 in the progression lemma. Odd perfect powers also require no separate exclusion. If a=b^h with fixed odd h>1, every prime divisor of h is eventually smaller than every prime factor of rQ; hence (h,p-1)=1 for the constructed primes. This explains directly how the controlled predecessor removes the obstruction from the fixed exponent.
3 S-integers, residue symbols, and Gauss sums
The analytic argument will sum over ideals while also using their generators in local fields. We first choose generators for which reciprocity has no supplementary factor. We then identify the two Hecke characters which account for the phases of the cubic and sextic Gauss sums.
Fix a cyclotomic field F containing \mu_{12}, and put [F:\Q]=2r. At a finite place v, normalize the absolute value by |\varpi_v|_v=(\Norm v)^{-1}; at a complex place use the square of the ordinary modulus. These are the absolute values in the product formula. Let
be the standard trace additive character. We also fix a finite-order Hecke character \eta. Choose a finite set S containing the complex places, all places above 6, the finite conductor of \eta, and every place at which \mathcal O_v is not self-dual for the pairing \psi_v(xy). The theta normalization in Proposition 5.1 and the local Euler-product estimate in Proposition 7.3 require two additional fixed finite sets of excluded places. Include both sets in S. Their existence is established at those points, independently of the summation variables and of the arithmetic choices below. We may include any further fixed finite set of places. Enlarging S by finitely many primes which generate the ideal class group, we arrange that the ring of S-integers
is principal. This is the final choice of S, and all arithmetic data below are constructed for it before the coefficient family is defined. All assertions in this section concern this fixed F and S; their auxiliary constants may depend on them.
Write S_f for the finite places in S, s_f=|S_f|, and
An exterior ideal is a nonzero integral ideal of R, equivalently an ideal supported at primes outside S. Its norm is q_{\mathfrak a}=\Norm\mathfrak a=|R/\mathfrak a|, extended multiplicatively to fractional ideals. Roman letters will denote generators of the corresponding fraktur ideals. For a\in F^\times, put q_a=|a|_S. Finally, on the additive group F_S set
where dx is the product of the self-dual local additive measures.
Lemma 3.1 (The S-integer lattice). The diagonal copy of R is a discrete cocompact subgroup of F_S, is its own annihilator for the pairing \e(xy), and has covolume 1. For every Bruhat–Schwartz function f on F_S,
If \mathfrak a=aR is an exterior fractional ideal, then
Proof. Put K^S=\prod_{v\notin S}\mathcal O_v, embedded in the adeles with zero S-components. At every place outside S, the defining conductor condition says that \mathcal O_v is its own annihilator. Its self-dual measure is consequently 1, and \widehat{\one_{K^S}}=\one_{K^S}. Adelic additive duality identifies the annihilator of F in \Adele_F with F, and the additive quotient \Adele_F/F has volume 1; see [30, Chapter II, §29]. Strong approximation identifies
The quotient is compact because \Adele_F/F is compact. To check discreteness, choose a compact neighborhood C of zero in F_S. The intersection F\cap(C\times K^S) is finite because diagonal F is discrete and closed in the adeles. Shrinking C excludes the finitely many nonzero S-components in this intersection, so zero is isolated in R. Taking annihilators in the last quotient gives
with R diagonally embedded, so its annihilator in F_S is indeed R. The covolume of a lattice and that of its annihilator are reciprocal under self-dual measure; hence this covolume is 1.
More directly, apply adelic Poisson summation [30, Chapter III, §41, equation (39)] to f\otimes\one_{K^S}. Only the elements of R occur on either side, and the exterior factor is unchanged by Fourier transform. This gives the displayed Poisson identity with no scalar factor. Multiplication by a has additive modulus |a|_S, so scaling this identity gives the formula for aR. The product formula gives
which finishes the proof.
We use Hilbert symbols to remove the supplementary factor in sextic reciprocity. Choose the local reciprocity map \operatorname{Art}_v in which a uniformizer acts as arithmetic Frobenius on an unramified extension. For n\in\{2,3,6\}, set
This orders the two arguments oppositely to the convention in [19, Chapter V, §3]. Thus, at a place of residue characteristic prime to n, for a unit u and a uniformizer \varpi,
The value is read in \mu_n\subset F. On
define H_S(x,y)=\prod_{v\in S}(x_v,y_v)_{6,v}, and let E be the image of U_S in \mathcal V. For T>0, let \ell_T\in F_S^\times have finite components 1 and component T^{1/(2r)} at every complex place. Then |\ell_T|_S=T.
Lemma 3.2 (Generators with fixed power classes). The pairing H_S on \mathcal V is nondegenerate and alternating. The map U_S/U_S^6\to\mathcal V is injective, its image E satisfies E^\perp=E, and there is an isotropic subgroup B such that \mathcal V=E\oplus B.
Every exterior fractional ideal \mathfrak a has a generator a whose class [a] in \mathcal V belongs to B. Any two such generators differ by an element of U_S^6. They can be chosen so that
for one fixed compact subset \mathcal K_{\mathrm{bal}}\subset F_S^1, independent of \mathfrak a.
Proof. For \ell=2,3, put \mathcal V_\ell=F_S^\times/F_S^{\times\ell}. Local Kummer theory and local reciprocity give a nondegenerate Hilbert pairing on this group [19, Chapter V, Proposition (3.2)]. At a finite place v,
Indeed the valuation contributes one dimension. When v\nmid\ell, the residue units contribute one dimension and the principal units are \ell-divisible. When v\mid\ell, the principal units have a \Z_\ell-part of rank [F_v:\Q_\ell] and an \ell-power root-of-unity part contributing one more dimension. The latter is present because F contains \mu_\ell. This is also the local power-class formula [19, Chapter II, Corollary (5.8)]. Complex power-class groups are trivial. All places above \ell lie in S, so their local degrees sum to 2r and
The S-unit theorem [19, Chapter I, Corollary (11.7)] gives U_S\simeq\mu(F)\times\Z^{r+s_f-1}. Since \mu(F) is cyclic and contains \mu_\ell,
We next check that the latter group injects into \mathcal V_\ell. Suppose that u\in U_S is an \ell-th power in every F_v with v\in S. The Kummer extension F(u^{1/\ell})/F is cyclic and split at S. At every exterior prime u is a unit and \ell is invertible, so this extension is unramified there. It is consequently an everywhere unramified abelian extension in which every S-prime splits. The Hilbert class-field correspondence, together with the quotient description of the S-class group, identifies such extensions with quotients of the class group of R, which is trivial [19, Chapter I, Proposition (11.6); Chapter VI, Proposition (6.9) and Theorem (7.3)]. Hence u=t^\ell for t\in F. Its exterior valuations show t\in U_S, proving the injection.
For two S-units, every exterior Hilbert symbol is 1, since both arguments are units and the residue characteristic is prime to 6. Hilbert reciprocity [19, Chapter VI, Theorem (8.1)] therefore makes their S-product 1. The Hilbert pairing is alternating here: -1=\zeta_{12}^6, and the diagonal identity (x,x)_{n,v}=(x,-1)_{n,v} makes its diagonal trivial for n=2,3,6. By Equation (3.1) and Equation (3.2), the injected unit image in \mathcal V_\ell is an isotropic subspace of half the dimension. Nondegeneracy implies that it is self-orthogonal. Symplectic linear algebra supplies an isotropic complementary subspace for each \ell. The Chinese remainder isomorphism \mathcal V\simeq\mathcal V_2\oplus\mathcal V_3, under which the two pairings are the third and second powers of H_S, combines these complements into B. It also gives the asserted injection and self-orthogonality for exponent 6.
Let \mathfrak a=a_0R. Decompose [a_0]=e b in E\oplus B, and choose an S-unit representing e^{-1}. Its product with a_0 is a generator a with [a]\in B. If a' is another, then a'/a is an S-unit whose class lies in E\cap B. It is therefore locally a sixth power at S, and the injection already proved says a'/a\in U_S^6.
Finally a/\ell_{q_{\mathfrak a}} belongs to F_S^1, by Lemma 3.1. To see the relevant compactness explicitly, the norm-one idele class group is compact [19, Chapter VI, Theorem (1.6)]. The subgroup F_S^1\times\prod_{v\notin S}\mathcal O_v^\times is open in the norm-one ideles and meets diagonal F^\times in U_S. Its quotient by U_S is therefore an open, hence closed, subgroup of that compact idele class group. Its projection onto F_S^1/U_S is surjective, so F_S^1/U_S is compact. Since U_S^6 has finite index in U_S, F_S^1/U_S^6 is compact as well. Choosing a compact set which maps onto this quotient and multiplying a by a suitable element of U_S^6 proves the last assertion.
Call a generator with class in B allowed, and call it balanced when it satisfies the compactness condition in Lemma 3.2. We use balanced representatives for estimates involving S-components. In multiplicative identities we may instead use the product of the allowed generators of the prime factors: the product remains allowed and differs from the selected representative by an element of U_S^6.
For an exterior prime \mathfrak p, the reduction of \mu_6 into (R/\mathfrak p)^\times is injective. Define the sextic residue character by
For every integer j, the notation \chi_{\mathfrak p}(x)^j means the usual character power on units and means 0 on nonunits, also when j\equiv0\pmod6. If \mathfrak a=\prod_{\mathfrak p}\mathfrak p^{k_{\mathfrak p}}, put
The same zero convention applies to powers of \chi_{\mathfrak a}.
For distinct exterior primes with allowed generators p,p', global reciprocity gives the exact equality
Indeed the only possibly nontrivial exterior symbols in the product formula for (p,p')_6 are (p,p')_{6,\mathfrak p}=\chi_{\mathfrak p}(p')^{-1} and (p,p')_{6,\mathfrak p'}=\chi_{\mathfrak p'}(p). The S-product is 1 because B is isotropic. If m\in R\setminus\{0\} and \mathfrak b=(m)R, write m=\epsilon b with b allowed and \epsilon\in U_S. Multiplying Equation (3.3) over prime factors gives, for an allowed generator a of \mathfrak a,
This includes the case (\mathfrak a,\mathfrak b)\ne R: both sides are then zero. For an S-unit \epsilon, the same product formula gives
Thus its dependence on \mathfrak a is through the finite class [a].
Lemma 3.3 (Kummer and class characters). For u\in R\setminus\{0\}, the ideal function
is induced away from S and the prime divisors of (u)R by a finite-order Hecke character. Its values at those prime divisors are set equal to zero. If \mathfrak f_u is the conductor of the inducing primitive character, then
The same bound, with a changed constant, holds after twisting by the fixed \eta. The inducing characters associated to u and v agree if and only if u/v\in F^{\times6}.
Every character of the finite group B, evaluated on allowed generators, extends to a finite-order Hecke character unramified outside S.
The following parametrization will index the nonzero Poisson frequencies; its character distinctness and conductor bound will be used in zero detection. Choose a complete set of representatives \mathcal U modulo U_S^6 among the nonzero S-integers whose exterior valuations belong to \{0,1,\ldots,5\}, with 1\in\mathcal U. For allowed generators r, the map
is a bijection. Among u\in\mathcal U, only u=1 induces the principal character, and distinct u’s induce distinct characters, also after the fixed twist by \eta.
Proof. Because \mu_6\subset F, Kummer theory [19, Chapter IV, Theorem (3.3) and Corollary (3.6)] identifies u\in F^\times with the character
whose kernel as a function of u is exactly F^{\times6}. Global class-field theory identifies this finite abelian character with a finite-order Hecke character [19, Chapter VI, Theorem (5.5) and Proposition (5.6)]. At an exterior prime \mathfrak p\nmid(u)R, arithmetic Frobenius satisfies
Reduction is injective on \mu_6, so these are equal as roots of unity. This proves the asserted identification on good primes.
At an exterior place, the principal units are sixth powers by Hensel’s lemma. Hence the associated local Hilbert character is trivial on principal units, giving conductor exponent at most one. The tame Hilbert formula [19, Chapter V, Proposition (3.4)] shows that the pairing of two units is 1, so if u is a local unit the character is unramified. At a place in S, it depends only on the class of u in the finite group F_v^\times/F_v^{\times6}; the finitely many resulting local conductors have a fixed maximum. This proves Equation (3.6). A fixed twist supported in S only changes that maximum. Equality of two inducing Hecke characters is equality of their Kummer characters, since unramified Frobenius elements determine a finite abelian character. The Kummer kernel gives the stated criterion. Setting extra prime values to zero merely omits Euler factors and does not alter the inducing character.
For a character \theta_B of B, extend it to \mathcal V=E\oplus B by \theta_S(eb)=\theta_B(b). This is a continuous finite-order character of F_S^\times trivial on U_S. If \mathfrak a=aR, define \theta(\mathfrak a)=\theta_S(a). It is independent of the generator and multiplicative. At every exterior place take the unramified local character sending a uniformizer to \theta(\mathfrak p), and at S take \theta_S^{-1}. Their product is trivial on diagonal F^\times by construction and is the required Hecke character.
To prove Equation (3.7), divide each exterior valuation of a nonzero h by 6. This determines an ideal \mathfrak r and the exterior valuations of an S-integer u_0=h/r^6 in \{0,\ldots,5\}. Replacing u_0 by its representative in \mathcal U changes u_0r^6 only by an element of U_S^6. Conversely, the exterior valuations of [ur^6] determine \mathfrak r and those of u; equality of two such classes then says that the two representatives u differ by U_S^6. Changing an allowed r by U_S^6 does not affect its image in the quotient. This proves bijectivity.
If u\in\mathcal U induces the principal character, the first part gives u=t^6 in F. Its exterior valuations are divisible by 6 and between 0 and 5, hence are all zero; the same is true for the exterior valuations of t. Thus u\in U_S^6, giving u=1. If u,v\in\mathcal U induce the same character, the differences of their exterior valuations are divisible by 6 and lie in [-5,5], so vanish. Their quotient is then a sixth power of an S-unit, and u=v. The fixed twist by \eta cancels from an equality of twisted characters.
We now determine the phases of the Gauss sums. If \mathfrak c is a squarefree exterior ideal with allowed generator c, define
The additive character on R/\mathfrak c here is primitive by Lemma 3.1. A change c\mapsto\epsilon^6c, with \epsilon\in U_S, changes the additive argument by a sixth power and is absorbed by a change of residue variable; hence the sum is independent of the allowed generator. At an exterior prime, the adelic product formula gives
The reduction of \mu_{12} is injective there, so q_{\mathfrak p}\equiv1\pmod{12} and \chi_{\mathfrak p}(-1)=1. The minus sign can therefore be suppressed in all these prime Gauss sums.
Lemma 3.4 (The Gauss correction characters). There are a unitary Hecke character \Lambda, unramified outside S, and a finite-order Hecke character G, unramified outside S, such that for every exterior prime \mathfrak p,
At each complex place the infinity type of \Lambda is purely angular of exponent 1 or -1, in the ordinary angular coordinate. The values \gamma_3(\mathfrak p) are signs and depend multiplicatively on the class of an allowed p in B.
For coprime squarefree exterior ideals,
Proof. First work over a finite field k of size Q, with a nontrivial additive character \psi_k, a character \chi of exact order 6, and \rho=\chi^2. Write
For nontrivial characters the usual Gauss inversion and Gauss–Jacobi identities give
The first equality uses \rho(-1)=1. Changing \psi_k(x) to \psi_k(ax) multiplies G_j by \overline{\chi(a)}^{\,j}, so the cube in Equation (3.11) is independent of the additive character.
Let E_0=\Q(\omega), where \omega^2+\omega+1=0. Every ideal of \Z[\omega] prime to 3 has a unique generator congruent to 1 modulo 3; denote it by \pi_+(\mathfrak a). This follows from class number one and the bijection of the six units with (\Z[\omega]/3)^\times, and is the convention used in [2, §2.1]. For a prime \mathfrak q\nmid6 with residue size Q, take \rho(x)\equiv x^{(Q-1)/3}\pmod{\mathfrak q}. Reduction of the Jacobi sum gives
since every monomial has degree strictly between 0 and Q-1. The Gauss identities also give |J(\rho,\rho)|^2=Q. To determine the unit, put t=1-\omega; the ideals (t^2) and (3) are equal. Writing \rho(x)=\omega^{k(x)}, with k(x)\in\{0,1,2\}, and expanding modulo t^2 yields
Here each value of \rho occurs (Q-1)/3 times on \Fq_Q^\times, and Q\equiv1\pmod3. Divisibility by \mathfrak q, the norm equality, and this congruence show
This is also the prime case of the classical cubic Gauss identity g(n)^3=\mu(n)\Norm(n)n in [2, §2.2, equation (11) in arXiv:0804.2233v4].
Define on ideals of E_0 prime to 3
The generators congruent to 1\pmod3 multiply, so this is a multiplicative ideal character. On a principal ideal generated by a\equiv1\pmod3 its value is a/|a|; thus it is the unitary Hecke character of finite conductor dividing (3) and angular infinity type of exponent 1, in ideal-character convention. Set \Lambda=\Lambda_{E_0}\circ\Norm_{F/E_0}. If \mathfrak p lies above \mathfrak q with residue degree f, the power definition of the residue character shows that the cubic character at \mathfrak p is the lift of that at \mathfrak q by the residue-field norm. The Davenport–Hasse Jacobi lifting formula says
its hypotheses hold because \rho, \rho, and \rho^2 are all nontrivial [14, Theorem 1.1]. By Equation (3.12), the right side is -\pi_+(\mathfrak q)^f. Since \Norm_{F/E_0}\mathfrak p=\mathfrak q^f, Equation (3.11) proves \gamma_2(\mathfrak p)^3=-\Lambda(\mathfrak p). The local norm at infinity is a product of complex coordinates or their conjugates. Consequently its angular exponent at each complex place of F is 1 or -1, with no radial factor. Pullback by norm is unramified wherever \Lambda_{E_0} is unramified, so its finite conductor is supported in S.
For the quadratic sum we use one global additive character to relate the local signs. For a\in F_v^\times, define the normalized Weil index
It depends only on the square class of a. Weil’s product formula and the Hilbert-symbol identity are
see [30, Chapter II, §28 and Proposition 5 in §30]. At an odd conductor-zero place, the local Gauss integral for the Weil index [30, Chapter II, §27] gives \mathfrak w_v(u)=1 for a unit u and
For the second equality, the number of square roots of x is 1+\chi_{\mathfrak p}(x)^3, and the sum of a nontrivial additive character is zero. The last equality uses \chi_{\mathfrak p}(-1)=1.
Put \mathfrak w_S=\prod_{v\in S}\mathfrak w_v. Applying the first identity of Equation (3.13) to p^{-1}, all exterior places other than \mathfrak p contribute 1, and hence
On B the multiplicative defect in the second identity is
Thus [a]\mapsto\mathfrak w_S(a^{-1})^{-1} is a character of B factoring through square classes. It takes values in \{\pm1\}; equivalently Gauss inversion gives \gamma_3(\mathfrak p)^2=\chi_{\mathfrak p}(-1)^3=1. Extend this B-character by Lemma 3.3. Since 4 is an S-unit, \chi_\bullet(4) is also a finite-order Hecke character unramified outside S. Their product, with \chi_\bullet(4) conjugated, defines G and gives its displayed prime value.
It remains to check the sextic duplication factor. In the finite-field notation above put \phi=\chi^3. Completing the square and counting square roots gives
In the second equality the part without \phi vanishes, since \chi is nontrivial. Applying the Gauss–Jacobi identity to both sides yields
Together with G_2G_4=Q, this gives
Finally use the allowed product c_1c_2 to evaluate the sum at \mathfrak c_1\mathfrak c_2. Chinese remaindering writes its additive character as the product of the characters modulo \mathfrak c_i, with their arguments multiplied respectively by c_2^{-1} and c_1^{-1}. Changing the two residue variables contributes \chi_{\mathfrak c_1}(c_2)^j\chi_{\mathfrak c_2}(c_1)^j. This proves Equation (3.10), including j\equiv0\pmod6 with the specified zero convention.
For later use set
Its finite conductor is supported in S. If \vartheta_v denote its idele-class components, define
The global character is trivial on F^\times, so for every exterior ideal with allowed generator a,
This is a smooth unitary character of F_S^\times. Finite-order characters are trivial on the connected group \C^\times; hence the infinity components contributed by G and \eta are trivial. Those of L_S are therefore purely angular of exponent 1 or -1 at each complex place.
4 The coefficient family and analytic comparison
The two analytic evaluations use the same family of coefficient sums. We define that family and state the second moment proved by the theta and reflection arguments, then explain how the moment enters the zero-free argument.
Retain the field F, the final set S, the ring R=\mathcal O_S, and the generator convention fixed in Section 3. Unless explicitly stated otherwise, all ideals in this section are integral exterior ideals. A roman letter representing an ideal is always an allowed B-generator; it need not be balanced unless this is stated. In particular, products of allowed prime generators are permitted. The row variable m below ranges over all of R.
For \re s>1, write \zeta_F^S(s)=\sum_{\mathfrak a}q_{\mathfrak a}^{-s} for the Dedekind zeta function with its factors at S omitted, and set
The latter notation also denotes the usual meromorphic continuation of this incomplete Hecke L-function; a principal pole is permitted. All primes dividing the conductor of \eta already lie in S, so there are no further omitted exterior factors for this target character. For t>0 put
The transform identity follows by putting v=\log t and completing the square, first for real \xi and then by entire continuation. Write \vartheta=\overline\Lambda\,\overline G\,\eta. For m\in R and Z>0 define
Here c,n,A are allowed generators, and \mathfrak n is unrestricted, so it may share primes with \mathfrak c. Every residue-character power has the zero extension on nonunits fixed in Section 3, including powers with exponent zero modulo six. The sum is independent of the permitted generator choices. It converges absolutely: |\gamma_2(\mathfrak c)|=1 for squarefree \mathfrak c, and for any \sigma>1/2 the bound P_G(t)\ll_\sigma t^{-\sigma} reduces the absolute sum, after a factor O_\sigma(Z^\sigma), to
At m=0, the mask kills every term with \mathfrak A\ne R, so B_0(Z)=P_G(1/Z). The factor \gamma_2(\mathfrak c)/(q_{\mathfrak c}^{1/2}q_{\mathfrak n}) will be identified in Section 5 as the exterior coefficient supplied by the normalized local theta formula.
The comparison uses rows at scale XY, where
Recall the norm section \ell_T fixed in Section 3, for which |\ell_T|_S=T. If \mathscr K\subset F_S^\times is compact, set
Thus every local absolute value of \ell_{Z^{M_0}}^{-1}m is bounded above and away from zero on this set. These are the compact balanced annular row sets used below. For m\ne0 decompose its exterior ideal uniquely as
The ideal \mathfrak d(m) is squarefree and coprime to \mathfrak h(m); every prime exponent of \mathfrak h(m) is at least two. We call such an ideal powerful.
The theta and reflection arguments in Sections 5 and 6 prove the following estimate.
Proposition 4.1 (Reflected second moment). Let F be a fixed cyclotomic field containing \mu_{12}, let S and the arithmetic data be as above, and fix a compact subset \mathscr K\subset F_S^\times. For every \epsilon>0, every Z\geq2, and all dyadic lengths P,D\geq1,
In particular,
The estimate is uniform if \mathscr K is multiplied by an element of any fixed compact subset of the norm-one group of F_S^\times.
Here is the analytic comparison proved in Sections 7 and 8. The first of those sections constructs a scalar I_Z=I(Z^a,Z^b,Z) by pairing B_m(Z), on a fixed compact annular row set of scale XY, with a normalized sum of \chi_{\mathfrak s}(d)\e(-md/s), where q_{\mathfrak s}\asymp Y and d\in(R/\mathfrak s)^\times. It specifies the weights and the average over F_S^1/U_S^6. The whole additive factor has squared row mass O(Y^{-1}), uniformly in that average. Hence Equation (4.6) and Cauchy–Schwarz give, for every \epsilon>0,
The constants here and below may depend on the fixed F,S,\eta and the fixed weights, as well as on any displayed loss.
Poisson summation gives a second evaluation. Its nonzero frequencies are grouped by u\in\mathcal U through Equation (3.7). Only the principal class u=1, the class of sixth powers, contributes both numerator poles in the auxiliary Mellin variables: w=1 and z=1/6. The Euler calculation constructs a specific principal correction \mathcal H_\eta(s) by specializing the correction for u=1 at (w,z)=(1,1/6). It proves that \mathcal H_\eta is holomorphic and nowhere zero on an open neighborhood of the closed half-plane \re s\ge .998, and is bounded on that closed half-plane uniformly in height. The double residue equals c_0f_\eta(Z), where c_0\ne0 and
The exponent left by the two residues is a/2+s-1+h_0/6=s-8/15, whose value at s=1 is 7/15.
The zero-detection argument proves, for sufficiently large Z,
Since 933/2000=7/15-1/6000, this identity and the preceding bound, with the stated losses, give f_\eta(Z)\ll Z^{7/15-1/20000}. Moving the original s-line in the inverse Mellin integral to the right gives f_\eta(Z)=O_A(Z^A) for every A>0 and 0<Z\le1. Consequently the shifted Mellin transform \int_0^\infty f_\eta(Z)Z^{-s+8/15}\,dZ/Z is holomorphic for \re s>1-1/20000. Fourier inversion on \re s=2, followed by meromorphic continuation, identifies it throughout that half-plane with
Its numerator is holomorphic and nowhere zero there, so L^S(s,\eta) has no zero there; a principal pole only gives a zero of the reciprocal. The numerical saving 1/20000 is absolute; the constants and thresholds may depend on F,S,\eta. The later sections supply the residue calculation and the contour and Fourier justifications.
5 The normalized cubic theta input and its radial profiles
To prove the reflected moment, we first realize the exterior coefficient \gamma_2(\mathfrak c)/(q_{\mathfrak c}^{1/2}q_{\mathfrak n}) in the family B_m. The cubic theta representation used here has one-dimensional local Whittaker spaces. Its selected central datum, together with normalized induction, gives the factor q_{\mathfrak p}^{-1} for a triple valuation step. Compatible rational, torus, root, and compact sections fix the unit factor at valuation e as \chi_{\mathfrak p}(u)^{2e}. We state this normalized input first and then identify its exterior coefficient. The verification of the Kazhdan–Patterson specialization is in Appendix A.
Throughout this section F is the fixed cyclotomic field from the preceding section, and \Adele_F is its adele ring. In particular, \mu_{12}\subset F. We use the standard trace character \psi:F\backslash\Adele_F\longrightarrow\C^\times, and choose the self-dual additive Haar measures, for which F\backslash\Adele_F has volume one. Write
The same notation will denote the specified lifts below. A Whittaker functional with character \psi_v means a linear functional \lambda satisfying \lambda(\pi(n(x))\varphi)=\psi_v(x)\lambda(\varphi).
5.1 The automorphic input and its exterior coefficients
We use the determinant-modified cover of Kazhdan and Patterson with parameter c=2, with the ordinary torus section and the canonical root splittings. The embedding \iota:\mu_3(F)\hookrightarrow\C^\times is oriented as in the exterior cocycle formula below. If a_v(y) denotes the chosen torus lift, define the effective scalar \mathcal C_v(x,y) by
in a genuine representation \pi. Thus this notation includes the chosen embedding of the central \mu_3.
Proposition 5.1 (Cubic theta input). For the fixed field F, there is a threefold central cover
which splits over \PGL_2(F), and an irreducible genuine automorphic representation \Theta=\bigotimes'_v\Theta_v on this cover, with the following properties.
- (i)
Global representation and Fourier expansion. The cover is the quotient of the c=2 cover \widetilde{\GL}_2^{(2)} by the ordinary split scalar subgroup. The inducing character of \Theta_v, on the center of the covering torus before this quotient, is
\omega_v\!\left(\zeta\,s_v(\operatorname{diag}(x^3,y^3))s_v(zI)\right) =\iota(\zeta)\left|x^3/y^3\right|_v^{1/6}. \tag{5.1}Here s_v is the ordinary torus section. The normalized inducing norm exponents are 1/6,-1/6. The local exceptional representation is the irreducible quotient for these exponents and the irreducible subrepresentation for their Weyl conjugates. Every \Theta_v has a one-dimensional Whittaker space. The global functional
\lambda(\varphi)= \int_{F\backslash\Adele_F}\theta_\varphi(n(x)) \overline{\psi(x)}\,\dd xon the automorphic realization \varphi\mapsto\theta_\varphi is nonzero. The representation is spherical at all but finitely many finite places. There is a preliminary pure tensor \varphi^\circ on which \lambda is nonzero. Its finitely many nondistinguished finite places lie in the final fixed S, along with the archimedean places, the places over 6, and the places where \psi_v has nonzero conductor.
For a pure tensor \varphi=\varphi_S\otimes\bigotimes_{v\notin S}\varphi_v, with the distinguished spherical vector at almost every v\notin S, the global Whittaker function factors as
\mathcal W_\varphi(g) :=\lambda(\pi(g)\varphi) =\mathcal W_{S,\varphi_S}(g_S) \prod_{v\notin S}W_{v,\varphi_v}(g_v). \tag{5.2}Each exterior spherical functional is normalized at one, so only finitely many factors in this product differ from one for adelic g. The functional at S factors at its complex places; at its finite places it may be retained as a finite sum of products of local Whittaker functionals. For a smooth automorphic vector,
\theta_\varphi(g)=\theta_{\varphi,N}(g) +\sum_{\nu\in F^\times}\mathcal W_\varphi(a(\nu)g),\qquad \theta_{\varphi,N}(g) =\int_{F\backslash\Adele_F}\theta_\varphi(n(x)g)\,\dd x . \tag{5.3}This expansion converges absolutely and locally uniformly on compact sets of g.
- (ii)
Exact lifts and exterior local values. On the quotient the chosen lifts satisfy
\begin{gathered} n(x)a(y)=a(y)n(x/y),\qquad wa(y)w^{-1}=a(y^{-1}),\\ wn(H)=n(H^{-1})a(-H^{-2})\bar n(H^{-1}). \end{gathered} \tag{5.4}In the last identity H\ne0. At an exterior prime, when H^{-1} is integral the last factor is compact, and a compact diagonal may absorb the minus sign.
For every exterior prime \mathfrak p, choose a uniformizer \varpi, put q_{\mathfrak p}=\Norm\mathfrak p, and use the compact splitting on \PGL_2(\mathcal O_{\mathfrak p}). The orientation of \iota, equivalently that of the Hilbert-symbol convention, can be fixed so that the scalar by which the torus cocycle acts in a genuine representation is
\mathcal C_{\mathfrak p}(\varpi,u)=\chi_{\mathfrak p}(u)^{-2} \qquad(u\in\mathcal O_{\mathfrak p}^{\times}). \tag{5.5}The compact splitting agrees with the chosen sections on unit diagonals, Weyl elements, and integral upper and lower root elements. Define
g_{2,\mathfrak p} =q_{\mathfrak p}^{-1/2} \sum_{x\bmod\mathfrak p} \chi_{\mathfrak p}(x)^2\psi_{\mathfrak p}(x/\varpi).Then |g_{2,\mathfrak p}|=1. If W_{\mathfrak p} is the spherical Whittaker function normalized by W_{\mathfrak p}(1)=1, then, for u\in\mathcal O_{\mathfrak p}^{\times},
W_{\mathfrak p}(a(\varpi^e u)) =\chi_{\mathfrak p}(u)^{2e} \begin{cases} 0,&e<0,\\ q_{\mathfrak p}^{-k},&e=3k,\quad k\ge0,\\ q_{\mathfrak p}^{-k-1/2}g_{2,\mathfrak p}, &e=3k+1,\quad k\ge0,\\ 0,&e=3k+2,\quad k\ge0. \end{cases} \tag{5.6}In particular a shift by three nonnegative valuations multiplies the value by q_{\mathfrak p}^{-1}.
- (iii)
Complex local behavior. At a complex place, \Theta_v is the ordinary spherical \PGL_2(\C) principal series with normalized characters (|\cdot|_v^{1/6},|\cdot|_v^{-1/6}), where |y|_v is the square of the ordinary modulus. Its spherical Whittaker function is a nonzero constant times
|y|_v^{1/2}K_{1/3}\bigl(c_v|y|_v^{1/2}\bigr),\qquad c_v>0. \tag{5.7}The constant term of the automorphic theta function belongs to the Weyl-conjugate torus induction. In particular its dependence on a complex diagonal a(y) is radial, also when the vector on its right has been acted on by a fixed compact element.
In (5.2), the condition W_{\mathfrak p}(1)=1 fixes every exterior spherical scalar; the remaining normalization belongs to the S-functional. Appendix A derives this product from the nonzero global functional and local uniqueness, including the correction to the original global formula. The noncompact norm integrations below require the separate absolute bounds in Lemma 5.4.
Corollary 5.2 (The exterior theta coefficient). At every exterior prime use an allowed prime generator as the uniformizer in Proposition 5.1. Let A\in R\setminus\{0\} be an allowed generator of an integral exterior ideal \mathfrak A. Then
In the first case \mathfrak c,\mathfrak n are uniquely determined; they need not be coprime.
Proof. The local support in (5.6) permits precisely the valuations 3k and 3k+1. They give the unique decomposition \mathfrak A=\mathfrak c\mathfrak n^3, with \mathfrak c squarefree, and magnitudes q_{\mathfrak p}^{-k} and q_{\mathfrak p}^{-k-1/2}, respectively. Use allowed product generators A=cn^3; any other allowed A differs by an element of U_S^6, which has no effect on the local values. At a prime \mathfrak p\mid\mathfrak c, the local unit factor is \chi_{\mathfrak p}(c/p)^2. The unit from n^3 has exponent divisible by six in the symbol, including when \mathfrak p\mid\mathfrak n, and disappears. The normalized prime sum g_{2,\mathfrak p} equals \gamma_2(\mathfrak p): the local additive sign from Equation (3.8) has no effect because \chi_{\mathfrak p}(-1)=1. The product of the unit factors and these prime sums is \gamma_2(\mathfrak c) by Equation (3.10). Multiplying the magnitudes gives (5.8).
Thus theta supplies the exterior factor in B_m. The ideal character will enter through an S-average, and finite upper-unipotent translations will insert the moving sextic twist. The chosen S-Whittaker function first supplies a radial profile; an exact radial average will replace it by P_G. We prepare those S-components next.
The arithmetic power classes also describe the torus factor at S. Put \mathcal C_S=\prod_{v\in S}\mathcal C_v. At an exterior prime, Equation (5.5) and the tame Hilbert-symbol formula give \mathcal C_{\mathfrak p}(x,y)=(x,y)_{6,\mathfrak p}^{2} for all local x,y. For rational a,b, the product formulas for the chosen torus lifts and for the Hilbert symbol therefore imply
Both sides are continuous on F_S^\times\times F_S^\times, so weak approximation gives
In particular this factor depends only on the classes in the fixed finite group \mathcal V=F_S^\times/F_S^{\times6}. The local sections here are the same ones used by the rational splitting.
5.2 Bounds and a prescribed finite Whittaker function
For v\mid\infty, define the derivative in the logarithm of the norm by
The factor 1/2 makes |e^{t/2}y|_v=e^t|y|_v.
We give both the Bessel calculation and bounds for the fixed rotations of the vectors that we will use. Choose complex coordinates in which \psi_v(z)=e^{i c_v\operatorname{Re}z}, with c_v>0; a harmless rotation and a change of sign give this form. Put \varrho=|y|_v^{1/2}. In the dominant normalized principal series set \nu_0=1/2+1/6=2/3. Its spherical section, restricted to wn(x)a(y), is a constant times
where |x|_{\mathrm{ord}} is the ordinary complex modulus. The Jacquet integral is absolutely convergent because 2\nu_0>1. Using the Laplace representation of the negative power and the two-dimensional Gaussian Fourier integral gives
The Laplace and Gaussian integrals may be interchanged absolutely: after integrating the Gaussian without its phase the integrand at zero is O(t^{2\nu_0-2}), with exponent greater than -1. This proves (5.7).
For an upper root translate, Whittaker equivariance gives
Suitable complex linear combinations of the two real Lie derivatives in z therefore multiply the spherical function by y and by \bar y, respectively. Write \omega_m(y)=(y/|y|_v^{1/2})^m for m\in\{1,-1\}. After multiplying each derivative vector by a nonzero constant, its Whittaker function is
Here are uniform bounds for every vector in a fixed finite list consisting of these derivative vectors and their fixed compact rotations. A rotation of a first Lie derivative of the spherical vector is another first Lie derivative, since the spherical vector is compact invariant and \pi(k)\dd\pi(X)=\dd\pi(\operatorname{Ad}(k)X)\pi(k). For a fixed real Lie algebra element X, differentiate the spherical section using
On wn(x)a(y), its first Lie derivative and any number of \mathcal D_v-derivatives are the preceding spherical integrand times a finite sum of bounded homogeneous rational functions of (y,x). More precisely, for every norm derivative order b and every x_1-derivative order h, the resulting derivative is bounded by
This follows by differentiating the displayed quotient: each derivative adds a polynomial numerator of the same degree as its denominator, and an x_1-derivative lowers the homogeneous degree by one. The integral of the bound with h=0 is O(\varrho^{2-2\nu_0})=O(\varrho^{2/3}). For \varrho\ge1, integrating by parts h times against e^{-ic_vx_1} instead gives O_{b,h}(\varrho^{2/3-h}). These integrable derivative bounds justify the integrations by parts by cutoff and limit. Consequently, for every integer j\ge0 and every A>0, the fixed list satisfies
At a finite place the corresponding elementary bounds hold for every fixed smooth vector. To be explicit, let W_\varphi(y)=\lambda_v(\pi_v(a(y))\varphi). There is an additive open subgroup J\subset F_v for which n(J) fixes \varphi. The identity
forces W_\varphi(y)=0 when |y|_v is sufficiently large. For the other end, put Q_v=\#(\mathcal O_v/\mathfrak p_v) and z_v=a(\varpi_v^3). This element is central in the covering torus. By the two-cell Jacquet calculation [15, Proposition I.2.1, p. 61, and Theorem I.2.9(e), p. 72], the two constituents of the unnormalized ordinary Jacquet module of the principal series have z_v-eigenvalues of absolute values
These are the exponents 1/2-1/6 and 1/2+1/6 on a triple valuation step. Exactness of the ordinary Jacquet functor gives the same possible eigenvalues on the exceptional subquotient. Hence a polynomial all of whose roots have those absolute values annihilates the action of z_v on this Jacquet module.
For completeness, this gives a recurrence on the Whittaker values themselves. If P is such an annihilating polynomial, then
for finitely many x_i,\varphi_i. Apply \lambda_v\pi_v(a(y)). For all sufficiently small |y|_v the right side is zero because every \psi_v(yx_i) is one. Cube cocycles are trivial, so the left side is the recurrence P(E)W_\varphi(y), where E W_\varphi(y)=W_\varphi(y\varpi_v^3). On each valuation progression its solutions are sums of a polynomial in the valuation times the powers of the roots of P. The unit variable ranges over only finitely many classes after restricting to a sufficiently small unit subgroup which fixes the vector and consists of cubes. It follows that, for every fixed \delta<1/3,
All constants in this paragraph may depend on the place, the vector, and the functional. For a fixed finite list one may choose a common positive \delta. The local twisted Jacquet quotients of \Theta_v are finite dimensional by the quotient argument in Appendix A.2. Thus the same conclusions hold term by term for the finite sum of products at S in Proposition 5.1.
Lemma 5.3 (Finite torus cutoff). Let S_f be a finite set of nonarchimedean places and let \lambda_f\ne0 be a Whittaker functional with character \prod_{v\in S_f}\psi_v on a smooth representation of the product of the local covers. For a vector \varphi, put
Given any open subgroup H_f of the product of the unit groups, there are a compact open product subgroup U\subset H_f and a smooth vector \varphi_U for which
Proof. Choose a vector with \lambda_f(\varphi)\ne0. The map y\mapsto W_\varphi(y) is locally constant near one. Choose a product of sufficiently small groups 1+\mathfrak p_v^{m_v}, contained in H_f, on which it is the constant \lambda_f(\varphi); call that product U. As a subset of the additive product \prod F_v, U is compact and open. By additive Fourier inversion choose a Schwartz–Bruhat function \phi with
The vector \varphi'=\int\phi(t)\pi(n(t))\varphi\,\dd t is a well-defined smooth vector: its integrand is locally constant on the compact support of \phi, so the integral is a finite linear combination of vectors. The split root relation gives
Divide \varphi' by \lambda_f(\varphi).
Here is the choice at S that will be used later. Write S_f=S\setminus S_\infty. Let \mathscr L be any fixed finite family of smooth unitary characters \mathcal L_S of F_S^\times, each trivial on U_S, radially trivial at infinity, and with angular component \omega_{m_v}, m_v\in\{1,-1\}, at each complex place. This permits the arithmetic character L_S and its finitely many finite-order twists used below. Factor the nonzero global functional at its complex places, leaving a nonzero functional at the finite places of S. Apply Lemma 5.3 with an open subgroup contained in the sixth powers and in the intersection of the kernels of the finite components of all \mathcal L_S\in\mathscr L. We obtain one product U=\prod_{v\in S_f}U_v and one finite vector. For each \mathcal L_S, choose at each complex place the derivative of its angular type. The functions in (5.11) are nonzero at one, and their normalization gives an S-Whittaker function
In particular \mathcal L_S(y)^{-1}W_{S,\mathcal L_S}(a(y)) is the product of the same radial Bessel factors and \one_U(y_f) for every member of the family. The radial profile and its Gaussian averaging kernel below may therefore be chosen in common for this finite family.
The complex constant term is radial by Proposition 5.1, whereas \mathcal L_S^{-1} has a nontrivial angular component. We will use this distinction in Section 6, after the actual weighted theta sums have been shown to descend to the quotient on which they are integrated.
5.3 Integration on norm sheets
Recall that r=\#S_\infty=[F:\Q]/2, and define
For d\in F_S^\times, put q_d=|d|_S. Multiplication by d sends X_l to X_{q_dl}, and inversion sends X_l to X_{1/l}. The forward and reflected theta terms will therefore lead to sheet integrals at norms q_dl and q_d/l, respectively. We now bound such one-variable profiles, including absolute envelopes that will justify the ideal sums and the radial average. The two signed orbit substitutions will be proved with their actual integrands in Section 6.
There is at least one complex place. Use the norm section \ell_u from Section 3; it satisfies |\ell_u|_S=u and \ell_u\ell_v=\ell_{uv}. Choose multiplicative Haar measures and Haar measure on X_1 so that
where the measure on X_u is the translate by \ell_u of that on X_1. At a complex place we take angle measure of total mass one, so its multiplicative measure is a fixed constant times \dd t/t times angle measure, with t=|y_v|_v. At a finite place every valuation shell has the same multiplicative measure.
Lemma 5.4 (Norm-sheet profiles). Let \mathcal I be a finite index set. For every \kappa\in\mathcal I, let f_\kappa be a finite sum of products
Assume that for one \delta>0 all these local factors satisfy the following conditions.
- (a)
At a finite v, f_{\kappa,\ell,v} is locally constant, vanishes for |y_v|_v>C_v, and is O(|y_v|_v^\delta) for |y_v|_v\le1.
- (b)
At a complex v, the factor is smooth on F_v^\times and, for every j\ge0 and A>0,
|\mathcal D_v^j f_{\kappa,\ell,v}(y_v)| \le C_{A,j}\min(|y_v|_v^\delta,|y_v|_v^{-A}).
All constants may depend on the finite family. Define
Fix one complex place v_0, and define the nonnegative absolute derivative profile
Then H_\kappa is smooth, and for every j\ge0 and \sigma>-\delta,
In particular, for any 0<\alpha<\delta and every A>0,
The conclusions are unchanged if each f_\kappa is multiplied by a fixed smooth unitary character of F_S^\times and by a bounded function of its class in the finite group \mathcal V.
Let \mathcal K:(0,\infty)\to\C have every absolute moment
Both convolutions
and their norm derivatives obey (5.17), with constants depending also on the indicated moments. For every j, the two absolute majorants \int|\mathcal K(T)|H^{\mathrm{abs}}_{\kappa,j}(uT^{\pm1})\,\dd T/T satisfy the same two-ended size bound. For any x_i>0, complex a_i, indices \kappa_i\in\mathcal I, \tau\in\{1,-1\}, and \sigma>-\delta, one has the absolute bound
In particular the sum and integral may be interchanged if the final series is finite.
Proof. For a finite place, let A_v(Q_v^{-k}) be the supremum of the absolute local factor on its k-th valuation shell. Condition (a) implies, for every \sigma>-\delta,
there are only finitely many negative k, and the positive tail is geometric with ratio Q_v^{-(\delta+\sigma)}. At a complex place condition (b) likewise gives
These statements remain valid for all factors in the fixed finite family.
Use t_{v_0}=|y_{v_0}|_{v_0} as the dependent norm in (5.15). On X_u,
and the sheet measure is the product of the angle measure at v_0 and the multiplicative measures at the other places, up to its fixed normalization. Multiplying the absolute integrand by u^\sigma=\prod_v|y_v|_v^\sigma, take the weighted supremum of the v_0-factor and the weighted integrals of all the others. This gives (5.16), including its absolute version. On this parametrization D differentiates only the v_0-factor and is exactly \mathcal D_{v_0}. The same weighted majorants for each derivative justify differentiation under the integral on compact u-intervals. Taking \sigma=-\alpha and then \sigma=A proves (5.17).
At a complex place the sixth-power quotient is trivial, and at a finite place it is finite. Thus a bounded function of \mathcal V is a finite linear combination of products of local class indicators. A unitary character factors over the finite product S; at a complex place its norm derivatives only introduce fixed constants. Multiplying by these functions preserves the hypotheses after splitting into a finite family.
For the convolutions, apply the bound with \sigma=-\alpha or \sigma=A under the T-integral. For example the two bounds for the reflected convolution have additional factors M_{-\alpha}(\mathcal K) and M_A(\mathcal K). The corresponding absolute derivative profiles dominate differentiation on compact u-intervals. Finally apply (5.16) to each summand in (5.18); Tonelli and the definition of the moment give the displayed bound.
The lemma also gives the smooth separation on norm dyads that we will use. If \rho is a fixed smooth compactly supported function on \R, integration by parts twice in its ordinary Fourier transform gives
Indeed the L^1 norm is bounded by a constant times the suprema of the function and its derivatives of orders one and two; these derivatives are controlled by (5.17). Fourier inversion with t=\log x-\log y, for x,y in fixed ratio intervals, therefore separates the two norm variables with bounded L^1 cost and retains any chosen large-u power saving.
5.4 Exact radial smoothing
For the chosen S-function (5.14), the angular characters cancel against \mathcal L_S^{-1}. Define its radial profile by
It is a positive, nonzero profile covered by Lemma 5.4. In this subsection the Mellin convention is \widehat f(s)=\int_0^\infty f(u)u^s\,\dd u/u. The unreflected theta average will contain V(q_{\mathfrak A}l). At l=T/Z, the next lemma averages this profile to exactly P_G(q_{\mathfrak A}/Z). After the Weyl operation the norm variable is inverted, and the same kernel instead produces the controlled profiles \int\mathcal K(T)H_\kappa(u/T)\,\dd T/T.
Lemma 5.5 (Exact Gaussian smoothing). For V in (5.20), there are constants C_0>0 and A_0>0 such that
There is a smooth function \mathcal K:(0,\infty)\to\C such that, for every N>0 and j\ge0,
and, for every u>0,
Its Mellin transform satisfies the exact sign convention
The quotient on the right is interpreted by its entire continuation. For every fixed finite family of rotated S-profiles H_\kappa covered by Lemma 5.4, the functions \Phi_\kappa(u)=\int\mathcal K(T)H_\kappa(u/T)\,\dd T/T obey the signed and absolute bounds of that lemma.
Proof. The disintegration (5.15) and Tonelli give the Mellin transform of V as the product of its local Mellin integrals. Each finite factor is the positive volume of U_v, because |y_v|_v=1 there. At a complex place put t=|y_v|_v. The classical Bessel integral, or the same Laplace integral used above, gives
The angle integrals only change the positive measure constant. Multiplication over the r complex places proves (5.21).
Set
The reciprocal gamma function is entire, so \mathcal R_V is entire. On each fixed closed vertical strip, Stirling’s formula implies
The same bound with a different polynomial holds after multiplication by any power of s. Hence
converges absolutely and is independent of the real line \sigma: the horizontal sides of a rectangle tend to zero by the Gaussian bound. Differentiation in \log T inserts a power of s. Moving the line to N for 0<T\le1, and to -N for T\ge1, proves (5.22). In particular every M_\beta(\mathcal K) in Lemma 5.4 is finite.
Fourier inversion in \log T on a vertical line in (5.26) gives \widehat{\mathcal K}(-s)=\mathcal R_V(s) on that line. Both sides are entire, so (5.24) holds everywhere. For example on any real line \sigma>0, absolute Mellin Fubini is valid because
It follows that the Mellin transform of the left side of (5.23) is \widehat V(s)\widehat{\mathcal K}(-s)=e^{s^2}. The function on the right of (5.23) has the same transform, by the elementary Gaussian integral. Mellin inversion proves the identity. The final assertion follows from the convolution part of Lemma 5.4.
6 Reflection and the quadratic large sieve
We prove the second-moment estimate in Proposition 4.1. Following the reflection method of [23, Part I, Sections 5–7], we realize B_m as a theta average and apply the rational Weyl element. At a valuation-one row prime the reflected coefficient contributes a first power of the sextic symbol; a squarefree column prime contributes power -4 in the same pairing. Their product has exponent 1-4\equiv3\pmod6, so a quadratic large sieve applies. The reflected norm gives a column length of order D^2P/Z, and the rapid decay of the norm profile controls the remaining terms.
6.1 An automorphic expression for B_m
All implied constants in this section may depend on the fixed field, S, the target character \eta, the compact set \mathscr K, and the chosen finite family of local theta data.
We first replace elements by ideal rows. Choose a balanced allowed generator m_0 of each exterior ideal \mathfrak m. For every nonzero m\in R generating \mathfrak m, write m=\varepsilon_m m_0 with \varepsilon_m\in U_S. When m\in\mathscr B_Z(\mathscr K), the balanced shapes of m_0, the fixed shape set \mathscr K, and q_{\mathfrak m}\asymp_{\mathscr K}Z^{M_0} put \varepsilon_m in a fixed compact subset of the norm-one group. The S-unit theorem makes U_S discrete there, so one ideal is generated by only boundedly many elements of the row set. Put
The group U_6 is finite. The value of B_m depends on the unit \varepsilon_m only through \upsilon_m, which is fixed when the row m is fixed.
For a nonzero row let j_{\mathfrak p}=v_{\mathfrak p}(\mathfrak m)\bmod6 in \{0,\ldots,5\}. For an allowed generator A of an integral exterior ideal \mathfrak A, reciprocity gives
For coprime ideals this is the multiplicative reciprocity law from the arithmetic section. At a shared prime both sides vanish by zero extension, including when j_{\mathfrak p}=0; the zeroth power there is the unit indicator.
Define \lambda_{\varepsilon_m}(y)=H_S(\varepsilon_m,y)^{-1} on F_S^\times. By Equation (3.5), its value at an allowed generator A is \chi_{\mathfrak A}(\varepsilon_m), and isotropy of E makes it trivial on U_S. It depends only on \upsilon_m. With the character L_S from Equation (3.14), put L_{\upsilon_m}=L_S\lambda_{\varepsilon_m}. Then
These characters form a fixed finite family. They are smooth, unitary, and trivial on U_S; their radial components at infinity are trivial, and their angular component at each complex place has power 1 or -1.
We insert the moving twists by finite upper-unipotent translations. At an exterior prime \mathfrak p, write k_{\mathfrak p}=\mathcal O_F/\mathfrak p and use its allowed generator p as a local uniformizer. For 1\le j\le5 put
The character is nontrivial, so \lvert\tau_{\mathfrak p,j}\rvert=q_{\mathfrak p}^{1/2}. Define
For every t\in\mathcal O_{\mathfrak p},
For j\ne0, substitution by t^{-1} proves this for a unit t, and both sides vanish for a nonunit. For j=0, the left side is 1-q_{\mathfrak p}^{-1}\sum_v\psi_{\mathfrak p}(tv/p), the unit indicator. Thus the formula includes the masked zeroth power.
For each \upsilon\in U_6, choose the theta vector in Proposition 5.1 whose finite S-Whittaker function is the indicator of the common small product subgroup contained in sixth powers and in \ker L_\upsilon, and whose complex angular type is that of L_\upsilon. Denote its automorphic function by \theta_\upsilon. The cutoff and the normalizations are those of Equation (5.14); these choices form a fixed finite family. For the remainder of this subsection fix a nonzero m, put \upsilon=\upsilon_m, and write L=L_\upsilon.
For y\in F_S^\times and digits \boldsymbol v=(v_{\mathfrak p})_{\mathfrak p\mid\mathfrak m}, let g(y,\boldsymbol v) have component a(y) at S, component n(v_{\mathfrak p}/p) at \mathfrak p\mid\mathfrak m, and identity at the other exterior places. Use the lifts in Equation (5.4), choose representatives of the residue digits, and set
The combined digit sums that we shall integrate are
Rational automorphy gives \mathscr D_m^+=\mathscr D_m^-. We next show that all four sums descend under \Gamma.
For \gamma\in\Gamma, choose representatives v'_{\mathfrak p}\equiv\gamma v_{\mathfrak p}\pmod{\mathfrak p}, and put
Let k_S=1, k_{\mathfrak p}=n(u_{\mathfrak p})a(\gamma) for \mathfrak p\mid\mathfrak m, and k_{\mathfrak q}=a(\gamma) at the other exterior places. Because \gamma is an exterior unit, these are compact lifts fixing the original exterior spherical vector. At S the torus factors involving \gamma are trivial, since it is a sixth power. At a row prime the exact root relation gives
The rational diagonal and Weyl lifts therefore satisfy
The second identity has the same compact factor at the far right.
The upper constant term has the required invariance at these right arguments as well. For q\in F^\times, any adelic h, and any such exterior compact k, the root relation and the product formula give
Indeed n(x)a(q)=a(q)n(x/q); rational automorphy removes a(q), and right invariance removes k^{-1} at the far right. The substitution x\mapsto x/q preserves Haar measure on F\backslash\Adele_F. This argument applies in particular when h_{\mathfrak p}=wn(v/p). Finally \chi_{\mathfrak p}(\gamma)=1, so \alpha_{\mathfrak p,j}(\gamma v)=\alpha_{\mathfrak p,j}(v), including the zero digit. Reindexing by \boldsymbol v'=\gamma\boldsymbol v in Equations (6.6) and (6.7) proves the asserted descent of the combined sums before and after w.
Recall X_l=\{y\in F_S^\times:|y|_S=l\} and the sheet measures from Equation (5.15). The quotient X_l/\Gamma is compact; choose a measurable fundamental set \Omega_1\subset X_1 and put \Omega_l=\ell_l\Omega_1, using the norm section from that disintegration. Every integral over X_l/\Gamma means the integral over \Omega_l with the restricted sheet measure, equivalently the quotient measure induced using counting measure on \Gamma. The factors \one_B([y]) and L(y)^{-1} also descend, since [\gamma]=1 in \mathcal V and L(\gamma)=1. The resulting integrands are integrable on the quotient: the digit sums are finite and the factors are smooth or locally constant.
Both combined constant terms have zero weighted integral. To see this, rotate y by a complex number z of modulus one at one complex place. In the plus sum this acts by the left diagonal a(z); in the minus sum it acts by a(z^{-1}). The constant term belongs to the torus induction in Proposition 5.1. Its complex inducing law is radial for every right argument, so both sums are unchanged, including at the exterior arguments wn(v/p). The class condition is unchanged because \C^\times is sixth-divisible, while L(y)^{-1} transforms by a nontrivial angular character. Haar invariance therefore gives
We may now define the quotient average
Lemma 6.1 (Theta representation of the smoothed sum). Let V and \mathcal K be the common unrotated profile and Gaussian averaging weight from Lemma 5.5. For l,Z>0,
Here \mathfrak A=\mathfrak c\mathfrak n^3 and A=cn^3 is an allowed product generator. The ideal sums are over integral exterior ideals. Both formulas converge absolutely after the sheet integrations, and the second converges absolutely after the T-integration.
Proof. Use the Fourier expansion and normalized factorization in Equations (5.3) and (5.2) in the plus average. The constant term vanishes by Equation (6.8). At a nonconstant rational frequency \nu, a row-prime factor is
Spherical support forces \nu\in R; summing the digits with Equation (6.4) supplies \prod_{\mathfrak p\mid\mathfrak m}\chi_{\mathfrak p}(\nu)^{j_{\mathfrak p}}.
Write f_\upsilon(v) for the chosen unrotated S-Whittaker function. The S-factor of the frequency is \mathcal C_S(\nu,y)f_\upsilon(\nu y). Its cutoff has [\nu y]=1 in \mathcal V. Together with [y]\in B, this gives [\nu]\in B. By alternation and Equation (5.9),
on that support. If A is an allowed generator of the exterior ideal of \nu, then \nu/A\in U_S. The direct sum E\oplus B and the injection U_6\longrightarrow E now imply \nu=\gamma A for \gamma\in\Gamma. The grouped exterior factor is unchanged by this multiplication, by the local unit rule and \chi_{\mathfrak p}(\gamma)=1.
We justify the orbit substitution with an absolute integral first. The sets \gamma\Omega_l partition X_l up to null sets. Tonelli, multiplicative Haar invariance, and Equation (5.16) give for \sigma>1/2
The signed substitution v=\gamma Ay is therefore legitimate. Since [y]=[A]^{-1}[v] and L(y)^{-1}=L(A)L(v)^{-1}, it gives
For the last equality, [v]=1 on the cutoff, [A]\in B, and the angular characters cancel as in Equation (5.20). Thus the integral depends on A only through its norm. Multiplicative Haar measure introduces no norm Jacobian.
By Corollary 5.2, the exterior spherical coefficient vanishes unless \mathfrak A=\mathfrak c\mathfrak n^3 with \mathfrak c squarefree, and then equals \gamma_2(\mathfrak c)/(q_{\mathfrak c}^{1/2}q_{\mathfrak n}). This includes primes shared by \mathfrak c and \mathfrak n. Together with Equation (6.13) it proves Equation (6.10).
Here are the absolute bounds for the exchanges. Summing Equation (6.12) with the exterior magnitudes is at most l^{-\sigma} times the convergent expression in Equation (4.3). Before evaluating the finite digit sums, their total absolute mass is at most q_{\mathfrak p}^{1/2} for j\ne0 and is less than 2 for j=0. For a fixed row their finite product multiplies the same spherical majorant, which justifies that earlier exchange. Finally Lemma 5.5 gives every absolute moment of \mathcal K. The averaged absolute sum is bounded by a constant times
times the finite sum in Equation (4.3). We may therefore apply Equation (5.23) term by term. Equations (6.1) and (6.2) and the definition of B_m give Equation (6.11).
6.2 The local Weyl calculation
The second expression in Equation (6.9) is now the starting point. At S, wa(y)=a(y^{-1})w replaces the vector by a fixed rotation and the argument by y^{-1}. Its constant term has already been removed in Equation (6.8). At an exterior place with zero or no digit, w is compact. We compute the effect of each nonzero digit.
Lemma 6.2 (Local effect of reflection). Let \mathfrak p be an exterior prime and put W_e=W_{\mathfrak p}(a(p^e)). For j\in\{0,\ldots,5\}, a unit t, and an integer e, define
It vanishes for e<0. For e\ge0,
For j=1, it vanishes for e>0, while
For all j, its magnitude at e=0 is at most 1, and for e>0 is at most q_{\mathfrak p}^{1/2}|W_e|. More precisely, for e\ge0 put s_{\mathfrak p,j,e}=\mathcal R_{\mathfrak p,j,e}(1) and choose the exponents modulo 6 by
For e>0, s_{\mathfrak p,j,e}=0 unless j=4, and it also vanishes for e\equiv2\pmod3. The zero-digit term is available only for j=0; at a frequency \nu it is (1-q_{\mathfrak p}^{-1})W_{\mathfrak p}(a(\nu)), with ordinary spherical support.
Proof. For a unit digit v, the exact lift identity in Equation (5.4), with H=v/p, is
with k_v in the spherical compact group. Only the diagonal minus sign has been absorbed in k_v; its symbols are trivial because F contains \mu_{12}. Commuting the upper root past a(p^{e-2}t) gives the positive additive argument \psi_{\mathfrak p}(p^{e-1}t/v). The effective torus law gives, for integers e_1,e_2 and units u,v,
The unit–unit and uniformizer–uniformizer factors are trivial. Thus torus multiplication contributes
The remaining spherical argument is p^etv^{-2}. It is zero for e<0; otherwise Equation (5.6) contributes \chi_{\mathfrak p}(t)^{2e}\chi_{\mathfrak p}(v)^{-4e}W_e. Their product proves Equation (6.14).
For e>0 the additive character is trivial. Unit-character orthogonality makes the digit sum zero unless j=4, including the case j=0. For every j its absolute value is at most q_{\mathfrak p}^{1/2}: use |\alpha_{\mathfrak p,j}(v)|=q_{\mathfrak p}^{-1/2} for j\ne0 and q_{\mathfrak p}^{-1} for j=0. Its unit dependence is the power 4+2e, and W_e=0 for e\equiv2\pmod3.
For e=0, substitute x=t/v. If j\ne0, this leaves the power \chi_{\mathfrak p}(t)^{8-j}, divided by \tau_{\mathfrak p,j}, and the Gauss sum of \chi_{\mathfrak p}^{j-4} against \psi_{\mathfrak p}(x/p). That sum has magnitude q_{\mathfrak p}^{1/2}, except for j=4, when it has magnitude 1. For j=1 its character is quadratic and 8-j\equiv1\pmod6, giving Equation (6.15). For j=0, the same substitution leaves \chi_{\mathfrak p}(t)^8 times a nontrivial Gauss sum divided by -q_{\mathfrak p}, again of magnitude at most one. These observations prove Equation (6.16) and the remaining bounds. Finally, compactness of w gives the zero-digit assertion by spherical right invariance.
For a row \mathfrak m=\mathfrak h\mathfrak d, call a prime active when its digit is nonzero. Every prime of \mathfrak d is active, because there j_{\mathfrak p}=1. For \mathfrak p\mid\mathfrak h put j_{\mathfrak p}=v_{\mathfrak p}(\mathfrak h)\bmod6. The exact set of possible active products in \mathfrak h is
Only a prime with j_{\mathfrak p}=0 can be inactive. For \mathfrak f\in\mathcal F(\mathfrak h), its exact inactive scalar is
and powerfulness gives
Let \mathcal A(\mathfrak f) be the integral exterior ideals \mathfrak a'_{\mathfrak f} supported on \mathfrak f, with every exponent congruent to 0 or 1 modulo 3; set \mathcal A(1)=\{1\}. This is a convenient common support: the actual coefficient s_{\mathfrak p,j_{\mathfrak p},e} remains zero in the locally vanishing cases of Lemma 6.2.
The local support shows that every frequency with a nonzero grouped exterior coefficient occurs among
where \mathfrak a'_{\mathfrak f}\in\mathcal A(\mathfrak f), \mathfrak c' is squarefree, and (\mathfrak c'\mathfrak n',\mathfrak d\mathfrak f)=1. The displayed elements d,f,a'_{\mathfrak f},c',n' are allowed generators. The ideals \mathfrak c' and \mathfrak n' may share primes, and primes of \mathfrak h omitted from \mathfrak f may occur in either. The valuation-one formula forces numerator exponent zero at each prime of \mathfrak d. Away from \mathfrak d\mathfrak f, the valuations 3k and 3k+1 determine \mathfrak c' and \mathfrak n' uniquely. The remaining ambiguity is the displayed S-unit, since allowed product generators are unique modulo \Gamma.
For later absolute estimates define
The local lemma and the spherical magnitudes give |s_{\mathfrak p,j,e}|\le\beta_{\mathfrak p}(e). The other exterior magnitudes together are at most q_{\mathfrak c'}^{-1/2}q_{\mathfrak n'}^{-1}.
6.3 The full reflected expansion
We now assemble these local factors. The S-integrals form a fixed family of one-variable profiles. Let f_{w,\upsilon}(v) be the S-Whittaker function of the chosen vector after the fixed Weyl rotation. Choose once a presentation
The finite S-functional and the fixed cutoff vector provide such presentations by Proposition 5.1 and Equation (5.14); their lengths are bounded over the finite row-class family. All fixed scalar coefficients are included in the displayed local factors. Write H_S also for its induced pairing on \mathcal V. For \kappa\in\mathcal V, define
The class multiplier is bounded and depends only on \mathcal V, by Equation (5.9). The local bounds for the fixed rotated vectors and Lemmas 5.4 and 5.5 therefore give one \alpha>0 such that, for every N>0 and integer j\ge0,
The constants are uniform in \upsilon\in U_6 and \kappa\in\mathcal V. This is a fixed finite family: the functional and tensor terms have been combined in \Phi_{\upsilon,\kappa}, while the bounded class function stays inside its integral. In particular those fixed coefficients will not enter the local majorant \beta_{\mathfrak f}.
Choose representatives \varepsilon_u for the separate frequency-unit classes u\in U_6. For
and class slices s_d,s_c\in B, put
This is a selected class, rather than a further summation index. For \mathfrak p\mid\mathfrak f, abbreviate
With \upsilon,\mathfrak t,s_d,s_c fixed, the next three factors organize the dependence on the two varying ideals: \omega depends on neither, \rho depends on the row \mathfrak d alone, and \lambda depends on the column \mathfrak c' alone. All three are independent of Z. The quadratic symbol and norm profile will contain the remaining joint dependence on \mathfrak d and \mathfrak c'. For L=L_\upsilon and \varepsilon=\varepsilon_u, define
For a squarefree \mathfrak d with (\mathfrak d,\mathfrak h\mathfrak n')=1 and [d]=s_d, put
and put it equal to zero otherwise. For a squarefree \mathfrak c' with (\mathfrak c',\mathfrak f)=1 and [c']=s_c, put
and put it equal to zero otherwise. All character arguments in these formulas are units at their primes on the stated supports. Thus negative character powers are well-defined. These definitions give one coherent row function and one coherent column function for each frozen tuple and class slice.
Lemma 6.3 (The reflected expansion). For every nonzero m\in R and Z>0, let (m)_S=\mathfrak h\mathfrak d and \upsilon=\upsilon_m. Then
All ideal sums are over integral exterior ideals, and the expansion is absolutely convergent. The coefficients satisfy
The row-unit class \upsilon is fixed; u is the independently summed frequency-unit class.
Proof. We use the minus average in Equation (6.9) and its nonconstant Fourier terms. We first supply the absolute bound needed before unfolding any signed unit orbit. Define
This is an absolute profile of Lemma 5.4; in particular H^{\mathrm{abs}}_{w,\upsilon}(x)\ll_\sigma x^{-\sigma} for \sigma>1/2, uniformly over the fixed row-class family. For any \nu_0\in F^\times, the sets \gamma\Omega_l^{-1} partition X_{1/l}. Tonelli and multiplicative Haar invariance give
For a fixed row, this also justifies the exchanges before evaluating the digits. At one nonzero digit, the absolute Whittaker factor in the proof of Lemma 6.2 is |W_e| at numerator valuation e\ge0. Summing the absolute digit weights costs at most q_{\mathfrak p}^{1/2} for j\ne0 and is less than 2 for j=0. For \sigma>1/2, the weighted spherical series is
The denominator shift e-2 multiplies this by q_{\mathfrak p}^{2\sigma}. There are only finitely many row primes, so their resulting product is finite for the fixed row. Here the numerator valuations at the primes of \mathfrak d are still summed. The product of ordinary spherical series at the other primes is bounded by \zeta_F^S(\sigma+1/2)\zeta_F^S(3\sigma+1). Partitioning the units into the finitely many U_6-classes and applying Equation (6.28) bounds the absolute sheet sum. The T-average then uses M_\sigma(\mathcal K)<\infty. Thus these preliminary exchanges are valid for each fixed row. The uniform estimates below use the coefficients after the finite digit sums have been evaluated.
Partition the zero and nonzero digits by \mathfrak f\in\mathcal F(\mathfrak h), apply Lemma 6.2, and parameterize the frequencies by Equation (6.19). For the fixed representative \varepsilon=\varepsilon_u, write
The valuations and then the frequency-unit class give a unique such parameterization on the stated support. At \mathfrak p\mid\mathfrak d, the local unit of \nu_0=p^{-2}t is
At \mathfrak p\mid\mathfrak f, the local unit of \nu_0=p^{e_{\mathfrak p}-2}t is
At \mathfrak q\mid\mathfrak c', put k=v_{\mathfrak q}(\mathfrak n'). The spherical valuation is 1+3k, and its unit exponent is 2 modulo 6. The cube unit from (n'/q^k)^3 disappears from the symbol. A prime of \mathfrak n' outside \mathfrak c' has unit exponent divisible by 6. Consequently the exact grouped exterior factor on (\mathfrak d,\mathfrak h\mathfrak n')= (\mathfrak c',\mathfrak f)=(\mathfrak d,\mathfrak c')=1 is
The factor I_{\mathfrak h,\mathfrak f} is present whether or not an inactive prime occurs in \mathfrak c' or \mathfrak n'. Thus the formula includes their overlap at such a prime as well as every other \mathfrak c'–\mathfrak n' overlap. The local zeros s_{\mathfrak p}=0 keep the broader active numerator support exact.
We identify the only factor involving both row and column variables. At a pair \mathfrak p\mid\mathfrak d, \mathfrak q\mid\mathfrak c', the row factor supplies \chi_{\mathfrak p}(q), and the denominator d^{-2} in the spherical column factor supplies \chi_{\mathfrak q}(p)^{-4}. Allowed-generator reciprocity gives
Multiplying over the pairs gives \chi_{\mathfrak d}(c')^3. There is no further pairing from a prime occurring only in \mathfrak n', since its unit exponent is divisible by 6. The separated S-character is
because L(\varepsilon)=1. Comparing Equation (6.29) with Equations (6.24), (6.25) and (6.26) therefore gives, on the indicated class slices,
The local bounds prove the three coefficient bounds in the lemma. The zero extension of the quadratic symbol makes the right side zero when \mathfrak d and \mathfrak c' meet, as the valuation-one support requires. A prime shared by \mathfrak d and \mathfrak n' must instead be removed by the separate row support of \rho. Together with (\mathfrak n',\mathfrak f)=1 and the column support of \lambda, these conditions reproduce exactly (\mathfrak c'\mathfrak n',\mathfrak d\mathfrak f)=1.
It remains to evaluate the S-integral. The grouped digit sums \mathcal R_{\mathfrak p,j,e}(t) are unchanged under t\mapsto\gamma t for \gamma\in\Gamma, by Equation (6.16). The same is true of each ordinary spherical factor and of the zero-digit factor, since \chi_{\mathfrak p}(\gamma)=1. This includes active j=0 terms and the zero digit. Thus the grouped exterior coefficient may be taken outside the signed \Gamma-orbit. That orbit is absolutely dominated by Equation (6.28).
The reflected S-factor is \mathcal C_S(\gamma\nu_0,y^{-1})f_{w,\upsilon}(\gamma\nu_0/y). In the legitimate substitution v=\gamma\nu_0/y, [y]=[\nu_0][v]^{-1} and L(y)^{-1}=L(\nu_0)^{-1}L(v). Alternation and Equation (5.9) give
We obtain the exact identity
The sheet norm is
there is no norm Jacobian. Inside the integral \nu_0 occurs only through its class in \mathcal V. On the supports of the two class indicators this class is exactly \kappa_6(\mathfrak t;s_d,s_c). Thus no continuous shape of the row or column generator remains inside the profile.
Insert the two class partitions, use Equation (6.31), and split the fixed S-function into its \ell-terms. Putting l=T/Z in Equation (6.11) produces precisely the combined profile in Equation (6.21), and hence Equation (6.27). For a fixed row class and fixed ideal data (\mathfrak h,\mathfrak f,\mathfrak a'_{\mathfrak f},\mathfrak n'), there are only |U_6||B|^2 choices of the frequency unit and two slices. After those choices, the class \kappa_6 is determined and there is one combined profile. The active sets and the ideal sums are the explicit outer sums in the expansion.
We finish by verifying the uniform absolute bound after grouping. For \sigma\ge0,
In particular, for every \delta_1>0, \prod_{\mathfrak p\mid\mathfrak f}E_{\mathfrak p}(\sigma) \ll_{F,\delta_1}q_{\mathfrak f}^{\delta_1}, uniformly for \sigma\ge0. Absorb the finitely many primes of norm below (3+\sqrt2)^{1/\delta_1} into the constant and bound each remaining factor by q_{\mathfrak p}^{\delta_1}.
Put Q=q_{\mathfrak a'_{\mathfrak f}}q_{\mathfrak c'}q_{\mathfrak n'}^3. Dropping squarefreeness and the unfrozen coprimality restrictions in the nonnegative majorant gives
The frozen ideal still ranges over \mathcal A(\mathfrak f). For t_{\mathfrak d}=q_{\mathfrak d}^2q_{\mathfrak f}^2/Z, the absolute profile above dominates every class multiplier and yields
This justifies all exchanges in the grouped expansion. Its constants are uniform in the ideal data, apart from the displayed product of local majorants. Since \mathcal F(\mathfrak h) is finite for each fixed row, the full expansion is absolutely convergent.
The case m=0 remains the separate identity B_0(Z)=P_G(1/Z) from the coefficient-family section.
The expansion also displays the length used in the second moment. On dyads P\le q_{\mathfrak h}<2P and D\le q_{\mathfrak d}<2D, put
where \delta>0 will be chosen small. By Equation (6.18), t_{\mathfrak d}\le\mathcal L. We shall retain Q\le R_0, which in particular gives q_{\mathfrak c'}\le R_0. If R_0<1, this range is empty because Q is the norm of an integral ideal. The tail estimate below justifies this truncation for every dyad, including that case.
6.4 The quadratic large sieve and the moment estimate
We use the quadratic large sieve established over \Q by Heath-Brown and extended to Hecke families by Goldmakher–Louvel [12, 8]. We state the latter result in the form needed for Equation (6.27) and verify its family hypotheses for the present generator convention.
Lemma 6.4 (Quadratic large sieve for the exterior symbols). For C,D\geq1 and complex coefficients z_{\mathfrak c} supported on squarefree exterior ideals with q_{\mathfrak c}\leq C, one has, for every \epsilon>0,
Arbitrary restrictions on the rows and separate restrictions or unitary factors on the columns are allowed by deleting rows and changing the coefficients.
Proof. We verify Definition 1 of Goldmakher–Louvel before applying their Theorem 1.1 [8]. Choose an ordinary integral ideal \mathfrak q_S supported on the finite places of S, with a positive exponent m_v at each such place large enough that 1+\mathfrak p_v^{m_v}\mathcal O_v\subset F_v^{\times2}. Every local quadratic character at v is trivial on this subgroup, so its conductor exponent is at most m_v. Exterior ideals are exactly the ordinary ideals prime to \mathfrak q_S, with the same norms.
For a squarefree exterior ideal \mathfrak c, let \Xi_{\mathfrak c} be the primitive Hecke character of the possibly trivial quadratic Kummer extension F(\sqrt c)/F. It is independent of the allowed generator, since two such generators differ by a sixth power of an S-unit. At an exterior prime its local conductor exponent is one if the prime divides \mathfrak c and zero otherwise: at the odd residue characteristics under consideration an odd valuation gives a tamely ramified quadratic extension, whereas a unit gives an unramified or trivial extension. Thus
Its infinite type is trivial because all infinite places of F are complex. On an exterior ideal \mathfrak d,
This includes the zero at a shared prime, which is in the conductor of \Xi_{\mathfrak c}. These are the primitive global characters, not a character with additional artificially deleted Euler factors.
Let B_2 be the image of B in F_S^\times/F_S^{\times2} and set \mathrm{cl}_2(\mathfrak c)=[c]_{S,2}\in B_2. Allowed generators multiply modulo U_S^6, so this extends to a homomorphism on the group of all fractional exterior ideals. It has finite image. It is also ray-class data for \mathfrak q_S: for a principal exterior ideal (x) its value is the B_2-projection of [x]_{S,2} in the direct sum induced by E\oplus B; if x\equiv1\pmod{\mathfrak q_S}, all its local classes at S are squares, so this projection is trivial.
For coprime squarefree exterior ideals, the exact reciprocity of allowed generators gives
Thus the finite-class reciprocity factor in that definition may be taken identically one. If \mathfrak c_1,\mathfrak c_2 are coprime squarefree exterior ideals and have the same \mathrm{cl}_2, their quotient c_1/c_2 is a square at every place of S. The primitive character inducing \Xi_{\mathfrak c_1}\overline{\Xi_{\mathfrak c_2}} is consequently trivial there. At every prime of \mathfrak c_1\mathfrak c_2 it is tamely ramified with conductor exponent one, and at every other exterior prime it is unramified. Its global conductor is therefore exactly \mathfrak c_1\mathfrak c_2. The product here is understood as the primitive inducing character, with the convention specified in [8, §2]. This verifies the conductor-product axiom, including the part at S.
We have thus obtained a quadratic Hecke family in the precise sense of [8, Definition 1]. Its Theorem 1.1 gives Equation (6.37). Apply that theorem to the full family and then delete unwanted rows or set unwanted coefficients to zero; a single class slice need not itself be a Hecke family.
Proof of Proposition 4.1. The initial element-to-ideal reduction lets us fix one row-unit class \upsilon\in U_6 and sum over ideal rows at a bounded cost. Fix dyads P\leq q_{\mathfrak h}<2P and D\leq q_{\mathfrak d}<2D which meet the row set. They satisfy
There are O_F(P^{1/2}) powerful ideals on the first dyad. In fact every powerful ideal has a unique expression \mathfrak r^2\mathfrak s^3 with \mathfrak s squarefree (choose its exponent as the parity of each prime exponent). The fixed-field bound for the number of ideals of norm at most t is O_F(t), hence
Fix \mathfrak h, and use Q,t_{\mathfrak d},\mathcal L,R_0 from Equation (6.36). For each \mathfrak f\in\mathcal F(\mathfrak h) and \mathbf s=(u,s_d,s_c)\in U_6\times B^2, split its partial sum in Equation (6.27) according to Q\le R_0 or Q>R_0. Denote these sums, which still include \mathfrak a'_{\mathfrak f},\mathfrak n',\mathfrak c', by \operatorname{Ret}_{\mathfrak f,\mathbf s,\mathfrak d} and \operatorname{Tail}_{\mathfrak f,\mathbf s,\mathfrak d}. Sum them over the admitted \mathfrak f and the finite \mathbf s to define \operatorname{Ret}_{\mathfrak h,\mathfrak d} and \operatorname{Tail}_{\mathfrak h,\mathfrak d}. Thus, on a row with class \upsilon,
Set both parts to zero outside the squarefree rows \mathfrak d coprime to \mathfrak h. The threshold R_0 is common to the dyad and independent of the individual row. When R_0<1 the retained part is empty, so no large sieve at a length below one is required.
We first bound the discarded terms, uniformly in this case as well. Put \mathcal J=\max(Z^\delta,\mathcal L^{-1}), and fix N>\sigma>1/2. For Q>R_0 we have both Q/t_{\mathfrak d}>Z^\delta and Q/t_{\mathfrak d}\geq\mathcal L^{-1}, because Q\geq1. The profile estimate therefore gives
Summing absolutely with Equation (6.34), for one active product and one fixed \mathbf s, yields
This one estimate includes all column, cube, and frozen-exponent tails. If R_0<1, then \mathcal J=\mathcal L^{-1} and the corresponding entire partial sum is O(\mathcal L^N\prod E_{\mathfrak p}(\sigma)).
For every \delta_1>0, Equation (6.33) also gives
The same finite-small-prime argument proves this inequality; including all divisors only adds active choices that may actually be forbidden. Sum Equation (6.40) over them and over the fixed finite class list, square, and use O_F(D) for the number of row ideals on the second dyad and Equation (6.39). The total squared tail is
Here Equation (6.38) implies P\ll Z^{M_0} and \mathcal L\ll Z^{2M_0-1}, while \mathcal J\geq Z^\delta. Given any A_0>0, choosing N with
makes Equation (6.42) O(Z^{-A_0}P^{1/2}D). This proves the promised absolute domination of the discarded range, including all dyads for which R_0<1.
It remains to bound the retained terms when R_0\ge1. Fix \mathfrak f,\mathbf s,\mathfrak a'_{\mathfrak f},\mathfrak n'. Then \kappa=\kappa_6(\mathfrak t;s_d,s_c) is fixed, while \rho_{\upsilon,\mathfrak t,s_d} and \lambda_{\upsilon,\mathfrak t,s_c} are coherent functions of all rows and columns in the two slices. Split \mathfrak c' into dyads C\le q_{\mathfrak c'}<2C, C\ge1. Put
Then x\in[-2\log2,\log2], and the profile in Equation (6.27) is \Phi_{\upsilon,\kappa}(v_0e^x). Choose a fixed \varphi\in C_c^\infty(\R) equal to one on that interval and set g_{v_0}(x)=\varphi(x)\Phi_{\upsilon,\kappa}(v_0e^x). Applying Equation (5.19) with this cutoff gives
Fourier inversion now separates the norm factors as (q_{\mathfrak c'}/C)^{i\tau}(q_{\mathfrak d}/D)^{-2i\tau} with the bounded integral cost in Equation (6.43). The retained indicator \one_{q_{\mathfrak a'_{\mathfrak f}}q_{\mathfrak c'}q_{\mathfrak n'}^3\leq R_0} is a column coefficient after the two frozen ideals are fixed.
Use Minkowski’s inequality for this Fourier integral and apply Lemma 6.4 to each \tau. Separate row factors have modulus at most one, and the column square mass is
by ideal counting. For any \delta_2>0, the row \ell^2 norm of this one column dyad, before the factor \beta_{\mathfrak f}(\mathfrak a'_{\mathfrak f})/q_{\mathfrak n'}, is therefore at most
The restrictions (\mathfrak d,\mathfrak h\mathfrak n')=1 merely delete rows. The zero extension of the symbol already treats shared row and column primes, so no joint restriction was removed in applying the sieve.
For the retained range, C\leq R_0 and q_{\mathfrak n'}\leq R_0^{1/3}. There are O(\log(2R_0)) such column dyads, and partial summation of the fixed-field ideal count gives
Drop the final minimum in Equation (6.44) and use Minkowski successively for the column dyads, \mathfrak n', and \mathfrak a'_{\mathfrak f}. Equation (6.33) at \sigma=0 shows that their retained row norm, for fixed \mathfrak h,\mathfrak f,\mathbf s, is
Sum these norms over the admitted \mathfrak f and the finite \mathbf s, using Equation (6.41) for the former. Square the resulting estimate for each fixed \mathfrak h, and only then sum the squares over \mathfrak h. Since each row has a unique powerful part, no triangle inequality across different powerful parts is necessary. Equation (6.39) yields
All the finite unit and S-class choices have only multiplied this by a fixed constant.
We make the power losses explicit. On the range Equation (6.38),
For any \delta_3>0, \log^4(2R_0)\ll_{\delta_3}Z^{\delta_3} because 1\leq R_0\ll Z^{2M_0-1+\delta}. Choose the positive auxiliaries so that
Then Equation (6.45) is bounded by the right side of Equation (4.5). The retained mass is zero for R_0<1, and Equation (6.42) is smaller than that right side in all cases after N is chosen as above. Adding the two parts and restoring the bounded element multiplicities proves Equation (4.5).
Finally, for every nonempty dyad, P\geq1 and Equation (6.38) give
where the first comparison uses M_0>1. There are only O((\log Z)^2) possible dyads. Applying the dyadic estimate with a smaller preliminary loss absorbs this factor and proves Equation (4.6). Multiplying \mathscr K by any element of a fixed compact norm-one set only enlarges the fixed compact shape set used in the initial multiplicity bound, which proves the asserted uniformity.
7 The Poisson probe and its Euler product
We insert the theta sum B_m(Z) of Equation (4.2) into an additive average. The reflected second moment bounds this average from above. Poisson summation will give a second expression whose principal frequency contains the reciprocal of the target L-function. The use of an auxiliary additive polynomial while retaining the unrestricted cube factors follows the method of [23, Part I, §§6–7]; the calculation here is over the fixed field F and uses its sextic characters.
Keep the field F, the set S, the target \eta, and the allowed generators fixed as in Section 3. In particular \vartheta=\overline\Lambda\,\overline G\,\eta, and every character power is zero on nonunits, including the zeroth power. The scale notation a,b,M_0,h_0,X,Y is fixed in Equation (4.4). All implied constants in this and the next section may depend on the fixed F,S,\eta and the fixed weights. Exponents displayed as numerical constants do not have that dependence. An arbitrary positive power loss may always be chosen smaller before an estimate is applied.
7.1 Weights and the estimate from reflection
Choose a nonzero nonnegative function W_1\in C_c^\infty((0,\infty)), and set
Choose a tensor Bruhat–Schwartz function W_0 on F_S, compactly supported in F_S^\times, such that
The condition \widehat W_0(0)=0 will remove the zero additive frequency after Poisson summation. The condition M(1/6)\ne0 will ensure that the weight does not kill the residue at z=1/6 in that expansion. These are separate from the support condition on W_0, which removes the original row m=0 from the probe. Here the Fourier sign is the one fixed in Section 3. We explain the choice and the analytic properties that will be needed. At each place the local zeta functional equation gives, for 0<s<1,
the nonzero constant incorporates the chosen local measures and Fourier sign [27, Theorem 2.4.1]. At one complex place take a smooth annular function whose additive integral is zero but whose pairing with |y|_v^{5/6}d^\times y is nonzero. Such a function is obtained by subtracting suitable multiples of two smooth bumps supported in different radial subannuli: the additive integral is a constant multiple of the pairing with |y|_v d^\times y, and the two radial weights are not proportional. At all other places take nonnegative nonzero annular functions. The local identity then proves both requirements in Equation (7.1).
For every \sigma>0, each local integral of |\widehat W_v(y)||y|_v^\sigma is finite. At a finite place this follows by summing the geometric series of shells near zero and using compact support at infinity; at a complex place it follows from boundedness at zero and Schwartz decay at infinity. The same assertions hold after any number of derivatives in a complex logarithmic norm. Thus M is holomorphic for \re z>0, and integration by parts in that norm gives, for every compact interval J\subset(0,\infty) and every A>0,
The analogous estimate holds for \widehat W_1 on every fixed real strip.
For an exterior ideal \mathfrak s, with an allowed generator s, put
The value is independent of the allowed generator: a change by \epsilon^6, \epsilon\in U_S, is absorbed by the same change of the residue variable. Use product multiplicative Haar measures for which the norm decomposition of F_S^\times has the form
The measure on F_S^1/U_S^6 below is the quotient measure induced by dt. This choice also fixes the normalization in M. Define
All undecorated ideal sums in this section are over nonzero integral exterior ideals. The summand for m=0 vanishes because W_0 is supported in F_S^\times. For \epsilon\in U_S^6, \chi_{\mathfrak A}(\epsilon)=\chi_{\mathfrak s}(\epsilon)=1; changing m to \epsilon m shows that the integrand descends to the quotient. That quotient is compact by Lemma 3.2, and we use representatives in a fixed bounded subset of F_S^1. For each such representative the m-sum has finite support. The Gaussian weight in B_m makes its ideal sum absolutely convergent.
We first record the additive estimate in the form also used later for conductor characters. If \mathfrak s=R, its one residue class is understood to give the single fraction 0.
Lemma 7.1 (Separated additive fractions). Let K be a fixed compact subset of F_S, and let N,V\ge2. Choose balanced generators s for the exterior ideals with q_{\mathfrak s}\le V. Let \mathcal P be any subset of the distinct points
For any complex b_\alpha,
The dual inequality, with the sums over m and \alpha interchanged, has the same constant.
Proof. Reduced fractions with distinct denominator ideals are distinct modulo R. Indeed, at a prime dividing its denominator the valuation of a reduced fraction is the negative of the denominator valuation; at any other exterior prime it is nonnegative. If the denominator valuations differ, their difference therefore has negative valuation there. If all agree, congruence modulo R says the residue classes agree. For distinct fractions and any r\in R, the numerator in
is a nonzero element of R. The norm identity of Lemma 3.1 gives
For r\in R\setminus\{0\} it also gives |r|_S\ge1.
Put d_F=[F:\Q] and \delta=c\min(N^{-1/d_F},V^{-2/d_F}), with c>0 fixed sufficiently small. There is a nonnegative Bruhat–Schwartz function \Phi which is at least one on \ell_NK, has \widehat\Phi(0)\ll\delta^{-d_F}, and whose Fourier transform is supported in a product of fixed finite-place additive subgroups and complex disks of ordinary radius C\delta. Here is a direct construction. At each complex place take the inverse transform of a smooth bump supported in a disk, evaluated at \delta y_v. Its value stays bounded below when \delta|y_v| is sufficiently small. At finite places use the inverse transform of the indicator of a sufficiently small compact open subgroup, whose annihilator contains the finite projection of K. The product, multiplied by a fixed constant and squared in modulus, majorizes the box. Fourier support follows from convolution, and a change of scale gives integral O(\delta^{-d_F}). Shrinking c makes this support contain no translate by R of a difference of distinct points in \mathcal P, by Equation (7.5): the product of the complex squared absolute values on that support is O(\delta^{d_F}), and its finite-place product is bounded by a fixed constant. It contains no nonzero point of R for the same reason.
Expand the square after majorizing it by \Phi, and apply Poisson summation on R. For \alpha,\beta\in\mathcal P,
Since \delta^{-d_F}\ll N+V^2, this proves Equation (7.4). Equality of the operator norms of a finite matrix and its adjoint gives the dual assertion.
Proposition 7.2. For the weights above, for every \epsilon>0 and Z\ge2,
Proof. Write v=q_{\mathfrak s}/Y and y=m/(t_0\ell_{XY}). For v in the support of W_1, the support of W_0(y/\ell_v) puts y in one fixed compact annular set. On this set the smooth function
has a Fourier series in \log v whose coefficients c_j(y) satisfy \sum_j\|c_j\|_\infty<\infty, uniformly in y. To see this, take a compact interval containing the logarithmic support, extend by zero to a slightly larger period, and integrate twice by parts. All the required derivatives are bounded uniformly on the compact y-set. Thus we can separate this weight from the additive sums with a bounded total coefficient cost.
For each Fourier term, expansion of g_{\mathfrak s} gives a coefficient of modulus Y^{-1}q_{\mathfrak s}^{-1/2} at each reduced fraction d/s, with q_{\mathfrak s}\asymp Y. Their total squared mass is at most
using the fixed-field ideal count. Take N=XY and V\ll Y in Lemma 7.1. The bounded set of representatives t_0 only enlarges the fixed compact row box. By the triangle inequality for the Fourier series, the squared \ell^2 norm in m of the whole additive factor in Equation (7.3), including (q_{\mathfrak s}X)^{-1/2}, is
The last inequality uses b<a. On the same enlarged compact annular row box, Proposition 4.1 gives
Cauchy–Schwarz in m, followed by integration over the compact quotient, proves Equation (7.6), after decreasing the preliminary loss.
7.2 Poisson summation and three Mellin variables
For an exterior ideal \mathfrak A with allowed generator A, write
and, for h\in R, define
The argument of the last Gauss sum is in R by the congruence. The definitions for the unit ideal mean a single residue and the value 1 for its character.
The finite Fourier transform of the row coefficient modulo \mathfrak s\mathfrak A=sAR is exactly
In fact expand g_{\mathfrak s}, write m=v+Aj with v\bmod\mathfrak A and j\bmod\mathfrak s, and sum over j. The sum of \e((h-Ad)j/s) is q_{\mathfrak s} precisely when h\equiv Ad\pmod{\mathfrak s}, and is zero otherwise. The remaining sum over v is the one in Equation (7.7). This argument does not require \mathfrak s and \mathfrak A to be coprime.
The dilation t_0\ell_{q_{\mathfrak s}X} has additive modulus q_{\mathfrak s}X, and the lattice sAR has index q_{\mathfrak s}q_{\mathfrak A}. Apply Poisson summation to the m-sum and use Equation (7.8). Its dilation divided by the index is X/q_{\mathfrak A}; after the two square root factors in the probe the multiplier is
The Fourier argument at h is
These norm factors will determine all three Mellin exponents.
For \epsilon\in U_S^6, changing d to \epsilon d and the Gauss residue variable v to \epsilon^{-1}v in Equation (7.7) shows F(s,A,\epsilon h)=F(s,A,h). Sum the nonzero h’s in such orbits and unfold the quotient F_S^1/U_S^6. If
the orbit of h contributes F(s,A,h)V_0(Xq_h/q_{\mathfrak A}). Multiplication by the norm-one part of \ell_{q_{\mathfrak s}X}h/(sA) preserves the sheet measure, so no shape factor remains. This unfolding is absolutely justified: for each \sigma>0, the local integrability used above gives an absolute sheet majorant O_\sigma(T^{-\sigma}), by using one complex weighted supremum and the other weighted local integrals. Taking \sigma>1 sums the q_u^{-\sigma}q_{\mathfrak r}^{-6\sigma} majorant, while the Gaussian handles the ideal \mathfrak A; this permits Tonelli’s theorem in the orbit sum. The zero frequency contributes nothing because \widehat W_0(0)=0.
By the measure disintegration, the Mellin transform of V_0 is M. The same weighted derivative majorants in the complex norm variables justify differentiation of the sheet integral in \log T, so V_0(T) is smooth for T>0. Its absolute weighted integrability and Equation (7.2) then give pointwise Mellin inversion by Fourier inversion in \log T on each line \re z>0. By Equation (3.7), write the orbits uniquely as h=ur^6, where u\in\mathcal U and \mathfrak r is an exterior ideal with allowed generator r. The principal row u=1 consists of the surviving nonzero sixth-power frequencies h=r^6. Hence the precise Poisson expression is
Here P_G is the Gaussian of Equation (4.1).
Use Mellin inversion for P_G,W_1,V_0, with transforms e^{t^2},\widehat W_1(w),M(z), respectively. The factors involving \mathfrak A are q_{\mathfrak A}^{-1}q_{\mathfrak A}^{-t}q_{\mathfrak A}^{z}. Set t=\xi+z-1. We obtain
where the innermost line is \re z=2, the middle line is \re w=4, and the outer line is \re\xi=4, and
All interchanges used for Equation (7.10) are in absolute convergence. For example, |F(s,A,h)|\le q_{\mathfrak s}q_{\mathfrak A}, |\gamma_2(\mathfrak c)|=1, and the ideal count, including the finitely many S-unit classes for u, give absolute convergence on these lines. The factors M(z) and \widehat W_1(w) have arbitrary vertical polynomial decay, while the Gaussian has Gaussian decay in \im(\xi+z). Changing height variables to (\im(\xi+z),\im z,\im w) proves absolute integrability as well.
7.3 The local correction
We extend the notation L^S(s,\eta) from Section 4 to a Hecke character on exterior primes: the factors at S are omitted, and any additional zero prime values are retained in the Euler product. Thus, for example,
As before, \zeta_F^S(s)=\prod_{\mathfrak p\notin S}(1-q_{\mathfrak p}^{-s})^{-1}.
Proposition 7.3. For every u\in\mathcal U, the series in Equation (7.11) has the Euler factorization
initially in absolute convergence. The correction \mathcal H_u is holomorphic in open neighborhoods of both real-part regions
For every \epsilon>0 it satisfies
there, uniformly in all heights. If S contains every prime of norm at most a sufficiently large fixed constant, then \mathcal H_1 is bounded and nowhere zero on both neighborhoods.
The two regions serve different contour movements in the next section. The first permits the small and intermediate nonprincipal rows to place w at .003 and z at .4, while \xi can reach .998 once the reciprocal denominator has been controlled. The second contains the principal numerator poles w=1, z=1/6 and the nearby lines w=.95, z=.16 used to isolate their double residue. The nowhere-zero conclusion for u=1 will ensure that its correction does not cancel a singularity of 1/L^S(\xi,\eta).
Proof. We first factor the finite expression, keeping the units that Chinese remaindering introduces. Use products of allowed prime generators for s,A,r. They differ from any other allowed choices by sixth powers of S-units, which have no effect on the expressions. At an exterior prime \mathfrak p, with generator p and norm Q=q_{\mathfrak p}, write
Write s=p^ks_p, A=p^tA_p, and ur^6=p^Jh_p; the subscripts mean that the full p-power has been omitted. These three subscripted quantities are units at \mathfrak p. Put \psi_{\mathfrak p}^{\circ}(x)=\psi_{\mathfrak p}(-x). For x\in F, the global additive product gives \e(x)=\prod_{\mathfrak p\notin S}\psi_{\mathfrak p}^{\circ}(x), with only finitely many nontrivial factors. Expand the inner Gauss sum in Equation (7.7) and apply the Chinese remainder decomposition to each residue ring. Before changing variables, the local factor of F(s,A,ur^6) is
Here an absent modulus p^0 has one residue and character value 1. For a positive modulus the character retains its unit mask, even when its exponent is divisible by six. Now make the changes
They are bijections of the relevant residue rings. The congruence becomes p^J\equiv p^t d'\pmod{p^k}, and the local additive argument becomes (p^J-p^td')v'/p^{k+t}. The unit multiplier removed by these changes is exactly
There are no hidden coprimality assumptions here: these changes only use the units left after omitting the full p-powers. Moreover \chi_{\mathfrak p}(h_p)=\chi_{\mathfrak p}(u/p^j), because the remaining factors of r^6 are sixth powers.
For completeness define the stripped local sum by
When k=0 the outer modulus is one, its character is one, and its single residue may be taken to be zero. The analogous convention applies to t=0 in the inner sum. When k>0 or t>0, the corresponding character is supported on units even if its exponent is divisible by six. These conventions distinguish the absent modulus from the zeroth power character on a nontrivial modulus.
We check explicitly that the remaining factors in different primes separate. Expand Equation (7.14) using the product generators, and consider an unordered pair of distinct primes \mathfrak p,\mathfrak p'. In their common symbol \chi_{\mathfrak p}(p')=\chi_{\mathfrak p'}(p), the contributions from \chi_{\mathfrak p}(s_p)^t and \chi_{\mathfrak p}(A_p)^{t-k}, together with the analogous factors at \mathfrak p', have exponent
In particular all s-A interactions cancel. By Equation (3.10), the Chinese remainder factors in \gamma_2(\mathfrak c) add 4e_0e_0'. Their total is
Thus all cross-prime terms cancel, including when both moduli have powers of the same primes; those shared powers were retained inside Equation (7.15). The h_p-factors remain precisely as characters of u/p^j. If \theta_{\mathfrak p}=\chi_{\mathfrak p}(u/p^j)\in\mu_6, the exact local factor is consequently
It follows initially by absolute convergence that \mathcal F_u=\prod_{\mathfrak p\notin S}\mathcal F_{u,\mathfrak p}.
Here are explicit evaluations of every C_{t,k}(J). Put
The unit mask is retained for v\equiv0\pmod6. Thus
For t=0 the value is C_{0,0}(J)=1, and C_{0,k}(J)=\one_{J=0} for k>0. For t>0 it is
For the first formula sum the Q^{t-1} lifts of a residue v\bmod p in the inner sum; they cancel unless J\ge t-1. For k=1 the congruence first requires J\ge1. Sum d\bmod p before v. Because v is a unit, the sum is \tau_1\overline{\chi_{\mathfrak p}(v)}; \chi_{\mathfrak p}(-1)=1 removes the sign. The remaining sum over lifts of v is zero unless J\ge t, and then gives the second formula. For k\ge2, varying d by p^{k-1}\lambda, \lambda\bmod p, preserves the congruence and its unit character. For every unit v the phase varies by \psi_{\mathfrak p}^{\circ}(-\lambda v/p), whose sum in \lambda is zero. This proves the last formula. In particular, for t>0,
with the support restrictions visible in Equation (7.17).
Suppose first that \mathfrak p\nmid(u)R, so j=0. Set
The t=0 values give (1-V)^{-1}+W/(1-W): a positive k requires J=0. The first squarefree theta term has e_0=1, l=k=m'=0, hence t=1, J=0, and C_{1,0}(0)=\tau_1. Its contribution is
Indeed \tau_1=\sqrt Q\,\gamma_1(\mathfrak p) in the chosen additive sign, and Equation (3.9) gives \gamma_1\gamma_2\vartheta=-\eta. This negative term is the first nonconstant term of the reciprocal local L-factor 1-D_1; it is the reason that L^S(\xi,\eta\overline{\chi_\bullet(u)}) occurs in the denominator of Equation (7.12). We extract (1-V)^{-1}(1-W)^{-1}(1-D_1), whose first two factors are those of \zeta_F^S(6z) and L^S(w,\chi_\bullet(u)). The mixed terms and the higher theta terms will be included in the correction.
For t>0 and J=6m', the formulas show that only t\equiv0,1\pmod6 occur. For t=6d, d\ge1, the k=0 sum is
and the k=1 term is E_1^dW. For the latter, the only possible valuation is J=t, and \tau_1\tau_5=Q. For t=6d+1, d\ge0, the k=0 term is -D_1E_1^d, extending the first squarefree term just computed. Finally its k=1 sum is
Here J>t and T_0(p^{J-t})=Q-1; the first possible extra m' gives the displayed V. Summing in d proves the closed formula
This derivation also explains the vanishing of the allowed theta exponents t\equiv3,4\pmod6: the possible boundary values of J in Equation (7.17) have the wrong residue modulo six, and the remaining T-sum is a nontrivial character sum at zero.
Define at every prime
where W=D_1=0 at \mathfrak p\mid(u)R, in accordance with the zero extension. We verify normal convergence on explicit open neighborhoods: in each line of Equation (7.13) decrease all three lower endpoints by 10^{-5}, and use the resulting strict inequalities. On these neighborhoods all of V,W,D_1,E_1 have modulus bounded away from one once Q exceeds a fixed constant. Subtracting 1 after multiplying Equation (7.19) by (1-V)(1-W)/(1-D_1) gives
uniformly for \mathfrak p\nmid(u)R. For clarity, before the 10^{-5} enlargement the weakest of the first three terms is |D_1W|\le Q^{-1.001}, in the first region. The exponents in the last two terms are at most
on the union of the two regions. The enlargement leaves respectively the margins 0.00098, 0.90792, and 1.94788 beyond exponent -1, which suffice for the stated \delta. More explicitly, the specialization at E_1=0 satisfies
This accounts for the Q|D_1WV| term as well as the first three terms in Equation (7.20). The difference from this specialization is O(|E_1|) on the stated neighborhoods: the remaining denominators are bounded, and Q|D_1WV|<1 once Q exceeds the fixed cutoff.
At \mathfrak p\mid(u)R, 1\le j\le5. The t=0 contribution is (1-V)^{-1}. The term t=1,k=0 vanishes, since J\ge1 and T_1(p^J)=0. By Equation (7.18), the sum for t=1,k=1 has modulus at most
Its exponent is at most -0.001 on the first region and at most -0.948 on the second. For l\ge1, before the factor V^{m'}, the bound for the summand with e_0\in\{0,1\} is at most a constant times Q to the power
This is at most -0.497 for l=1,e_0=0 on the first region and decreases by at least 0.994 for each further l; the second region has a larger saving. The 10^{-5} enlargement still leaves a saving greater than 0.49 in this sum, and leaves 0.00098 in the t=1,k=1 term. The geometric sums in l,m', followed by multiplication by 1-V, therefore give
on the same neighborhoods. These estimates prove absolute local convergence there as well.
The product of the factors in Equation (7.20) converges normally, since \sum_{\mathfrak p}q_{\mathfrak p}^{-1-\delta}<\infty. The remaining primes are precisely the divisors of (u)R. For every \epsilon>0,
for sufficiently large Q each local factor is at most Q^\epsilon, and the finitely many smaller primes contribute a fixed constant. Consequently \mathcal H_u=\prod_{\mathfrak p\notin S} \mathcal H_{u,\mathfrak p} is holomorphic and obeys the asserted bound. Multiplication of its defining local factors gives Equation (7.12) in the initial region and then as a meromorphic identity on either neighborhood.
For u=1 every prime satisfies Equation (7.20). The local estimate determines a fixed norm cutoff beyond which its uniform bound is smaller than 1/2. Choose S to contain all primes up to that cutoff from the outset, as permitted in Section 3. Every remaining \mathcal H_{1,\mathfrak p} is nonzero, and its logarithm defined near 1 has an absolutely and uniformly convergent sum. Its exponential is \mathcal H_1, which is therefore bounded and nowhere zero on both neighborhoods. This proves the proposition.
For the remainder of the proof, write
This is the principal specialization announced in Section 4. Since (w,z)=(1,1/6) lies in the second region of Equation (7.13), \mathcal H_\eta is holomorphic, bounded, and nowhere zero on an open neighborhood of the closed half-plane \re s\ge .998, uniformly in height.
8 Zero detection and Mellin continuation
We now prove Theorem 1.2 for the fixed but arbitrary F and \eta. The factorization in Proposition 7.3 singles out the frequency u=1: it is the only row whose numerator L-function has a pole. We shall bound the other rows by a conductor large sieve, isolate the two residues of that principal row, and compare them with Proposition 7.2. The resulting bound for one Mellin integral will exclude zeros of the target function.
8.1 A conductor large sieve and a zero detector
Let \omega_u denote the primitive Hecke character inducing \eta\overline{\chi_\bullet(u)}. By Lemma 3.3, on a dyad U\le q_u<2U,
The second bound follows from the ideal count and the finite number of S-unit classes modulo sixth powers. The characters \omega_u in this family are distinct: equality cancels the fixed twist and forces u/v to be a sixth power in F; the selected exterior valuations then agree, and the quotient is in U_S^6. Their local components at S run through a fixed finite set, since the Kummer part depends on F_v^\times/F_v^{\times6}. The principal numerator occurs only for u=1. The primitive denominator \omega_u may be principal for a different row, so it will be treated separately in zero detection.
We give the conductor estimate required for that detection. In the next lemma and in the detector, ideals without the adjective exterior are integral ideals of \mathcal O_F, and q_{\mathfrak n}=\Norm\mathfrak n also for those ideals.
Lemma 8.1 (Primitive conductor large sieve). Fix C_0>0 and a finite set \mathscr L_S of possible local components at S. Let \mathscr X(U) be any family of distinct primitive finite-order Hecke characters of F whose conductor norms are at most C_0U and whose components at S belong to that set. For N,U\ge2, any complex coefficients b_{\mathfrak n}, and every \epsilon>0,
As usual a primitive character is extended by zero on ideals meeting its conductor.
Proof. First restrict the coefficient ideals to one exterior norm dyad and the characters to one choice of their components at S. If \mathfrak q is the exterior part of the conductor, principality of R and the global character relation write the value at \mathfrak n=nR, for its selected balanced generator n, as
Here \lambda_S is a common fixed unitary function of the S-components, and \lambda_{\mathfrak q} is a primitive residue character modulo \mathfrak q. The orientation of the local characters is absorbed in these two symbols. The formula includes zero on nonunits. It follows directly by evaluating the idele character at the rational element n: the unramified exterior components give \omega(\mathfrak n), leaving the ramified exterior components and the components in S. In particular, for this fixed S-choice the residue character determines \omega.
Let q be a balanced generator of \mathfrak q. By Lemma 3.1, (n,d)\mapsto\e(nd/q) is a perfect additive pairing on R/\mathfrak q. Put
For a primitive residue character, this Gauss sum has modulus q_{\mathfrak q}^{1/2}, and
For a unit n this follows by a change of residue variable. For a nonunit the Gauss sum is zero: locally at a prime dividing n the additive character factors through a proper quotient of the conductor, and multiplication by a unit in the kernel of that quotient on which the primitive multiplicative character is nontrivial forces the sum to vanish. Finite additive Parseval now gives
The left side is |(R/\mathfrak q)^\times|\,|\tau(\overline{\lambda_{\mathfrak q}})|^2, which proves the asserted modulus. Put
Sum the square of the preceding Gauss expansion over the characters of conductor \mathfrak q. Extending that sum to all residue characters and using character orthogonality bounds it by
The same assertion holds for the unit conductor with its single residue. Sum now over q_{\mathfrak q}\le C_0U. The fractions are distinct reduced fractions. The balanced generators of the coefficient ideals on a norm dyad lie in \ell_{N'}K for one fixed compact K, with N'\le N, by Lemma 3.2. The dual assertion of Lemma 7.1, with V=\max(2,C_0U), bounds this sum by (N+U^2)\sum|b_{\mathfrak n}|^2.
To pass to all exterior lengths at most N, split into dyads and use Cauchy–Schwarz, at a logarithmic cost. For general \mathcal O_F ideals write \mathfrak n=\mathfrak n_S\mathfrak n^S, with the first factor supported in S_f. There are O_S((1+\log N)^{|S_f|}) possibilities for \mathfrak n_S. Within a fixed S-choice the value \omega(\mathfrak n_S) is fixed (or zero). Cauchy–Schwarz over these possibilities and the preceding exterior estimate introduce only powers of \log N; the square masses over the disjoint pieces sum to the original square mass. The finite partition by S-components costs a constant. Absorbing all logarithms into (NU)^\epsilon proves Equation (8.2).
We use standard Hecke continuation and the functional equation in a deliberately conservative convexity bound. For a nonprincipal primitive finite-order \omega with \Norm\mathfrak f_\omega\ll U, there is an A_F>0, depending on the fixed field, such that on any of the fixed strips below, in particular for .003\le\sigma\le4,
This follows from the functional equation, the absolutely convergent Euler product on a right line, and the Phragmén–Lindelöf principle; the gamma factors contribute the field-dependent height power [27, Main Theorem 4.4.1 and §4.5]. The same estimates give polynomial height bounds on any fixed strip for (s-1)\zeta_F(s). Omitting factors at a fixed set or at primes dividing u changes a positive-real-part estimate by q_u^\epsilon: for a fixed \sigma_*>0, each of
is O_{\epsilon,\sigma_*}(q_u^\epsilon), by bounding a sufficiently large prime’s factor by q_{\mathfrak p}^\epsilon and absorbing the finitely many other primes into the constant.
Proposition 8.2. For the fixed F,S,\eta there are \rho>0 and U_0 with the following property. For every U\ge U_0, there is a set \mathcal E(U)\subset\{u\in\mathcal U:U\le q_u<2U\} of cardinality
which contains every row for which \omega_u is principal or L_F(s,\omega_u) has a zero with \re s\ge .996 and |\im s|\le4U^\rho. For u\notin\mathcal E(U) and every \epsilon>0,
The exponents .996,.998,1/2 are absolute. The height exponent \rho, the constants, and U_0 may depend on the fixed data.
Proof. Set \sigma_0=.996. For a primitive nonprincipal character \omega in the family of Equation (8.1), define the truncated inverse and its product coefficients by
Here \mu_F is the Möbius function of integral \mathcal O_F ideals. In absolute convergence,
Complete multiplicativity with the primitive zero extension justifies this identity also on ideals meeting the conductor. The coefficients are independent of \omega and satisfy
The coefficients with 1<q_{\mathfrak n}\le U^2 have vanished. We shall show that at a zero in the stated rectangle the full exponentially smoothed sum is o(1). After its exponential tail is discarded, the unit coefficient must therefore be cancelled, up to o(1), by terms with U^2<q_{\mathfrak n}\le U^{11}. A dyadic prefix of those terms is consequently large. The conductor sieve will bound the number of characters for which such a prefix can be large at any allowed height.
Suppose L_F(\beta+i\gamma,\omega)=0 with \beta\ge\sigma_0, |\gamma|\le4U^\rho. Absolute convergence of the Euler product implies \beta\le1. Mellin inversion of the exponential gives, with c>1,
Move the v-line to \re v=1/2-\beta, which lies between -1/2 and -.496. The only crossed gamma pole is v=0, and its residue vanishes because L_F(\beta+i\gamma,\omega)=0. The character is nonprincipal, so its L-function has no pole in the shift. The gamma decay and the polynomial height bounds justify the horizontal limits.
On the new line the fixed-field ideal count gives
Use Equation (8.3) and integrate the gamma factor. The right side of Equation (8.6) is at most
Choose once and for all
and then take the preliminary loss small. The last bound is o(1).
The first term on the left of Equation (8.6) is e^{-U^{-10}}=1+o(1), and the intervening terms through U^2 vanish by Equation (8.5). The portion with q_{\mathfrak n}>U^{11} is exponentially small, uniformly in \beta\ge\sigma_0: the divisor bound and ideal count majorize it by a fixed power of U times e^{-U/2}. It follows that the sum over U^2<q_{\mathfrak n}\le U^{11} has modulus bounded below by a positive constant. Decompose this interval into O(\log U) dyads. On any such dyad the remaining real weight is
which is nonnegative, decreasing, and at most one. Partial summation therefore bounds its weighted sum by twice the maximum modulus of the unweighted prefix sums. Consequently for at least one dyadic U^2/2<N\le U^{11} and one y\in[N,2N],
We spell out the two maximal estimates needed to use the conductor sieve on this assertion. Order the ideals on a dyad by norm, with any fixed order for ties. A prefix is the disjoint union of at most O(\log N) binary index intervals. Cauchy–Schwarz and summation over those intervals bound the squared maximum of all prefixes by the sum of their squared interval sums, with O((\log N)^2) total cost after the large sieve is applied; every coefficient occurs in one interval at each level. For a fixed interval its Dirichlet polynomial P_\omega(t) satisfies on a unit interval the elementary bound
The derivative only multiplies its coefficients by \log q_{\mathfrak n}\ll\log N. Apply Lemma 8.1 at each t, and sum these integrals over |t|\le4U^\rho. This costs U^\rho and powers of logarithms. The squared coefficient mass on a dyad is
Since N>U^2/2 and N\le U^{11}, the nonlogarithmic cost in Equation (8.2) is bounded by
Here 2-2\sigma_0=.008, and 11(.008)=.088; the factor (NU)^\epsilon in the large sieve is included in U^{O(\epsilon)}. Chebyshev’s inequality with Equation (8.8), followed by summation over the O(\log U) dyads, gives
The choice \rho<.1 leaves more than 0.31 before exponent 1/2. Take the losses small and include the at most one principal character. Distinctness of the \omega_u’s now gives the asserted bound for \mathcal E(U).
It remains to obtain the reciprocal bound from the zero-free rectangles just proved. Let T=U^\rho and u\notin\mathcal E(U). For |t|\le2T, consider disks centered at 2+it of radii
For large U the outer disk lies within \re s>.996,\ |\im s|<4T; its left edge is .9965. The nonprincipal L_F(s,\omega_u) is entire and nonzero on the disk. It has a branch of \log L_F there agreeing at the center with the Euler logarithm. The convexity bound gives \re\log L_F=\log|L_F|\le C_F\log U on the outer disk, and the value at the center is O_F(1). Borel–Carathéodory consequently bounds |\log L_F| by O_F(\log U) on radius r_2. On radius r_0 the real part is at least 1.001, so the absolutely convergent Euler logarithm is O_F(1). The three-circles theorem applied to the analytic function \log L_F now gives, on radius r_1,
These disks reach the line \re s=.998; absolute convergence covers the larger real parts outside them. Hence for \re s\ge.998 and |\im s|\le2T,
The estimate for omitted factors preceding the proposition gives the same bound for the incomplete function, proving Equation (8.4).
8.2 The principal component and the other frequency rows
Use the scales in Equation (4.4), and let \sigma_\xi,\sigma_w,\sigma_z denote the real parts of \xi,w,z. Write E_*(\sigma_\xi,\sigma_w,\sigma_z) for the real power of Z outside the Dirichlet factors in Equation (7.10). In the principal row, taking the residues at w=1 and z=1/6 leaves the exponent C_*(\xi) in the second line:
Thus C_*(1)=7/15 is the reference power for comparing the principal remainders and the other frequency rows.
We specify a decay observation that justifies the contour limits and, later, the truncated shifts. On any fixed real strips used below, for every A>0,
The constants can depend on those strips. Away from the poles extracted below, all numerator Hecke and zeta functions on them have polynomial height growth, and \mathcal H_u is uniformly bounded by q_u^\epsilon. After increasing A, their height powers are absorbed in Equation (8.11). In particular for every A_1>0,
with A sufficiently large. Thus integration in z,w gives arbitrary polynomial decay in the remaining \xi-height as well. On lines \re\xi>1 the reciprocal denominator is globally bounded by its absolutely convergent Euler product. These facts justify all infinite contour shifts on such lines by their horizontal limits.
Let I_1(Z) denote the contribution of u=1 to the initially absolutely convergent formula Equation (7.10), on the scales (X,Y)=(Z^a,Z^b). Its numerator is \zeta_F^S(w). Put \delta_1=10^{-7}, \sigma_1=1+\delta_1, and \zeta_1=1/6+\delta_1. Move the initial lines to
These moves cross no poles: both zeta arguments remain to the right of one, and the reciprocal denominator is in absolute convergence. The second region of Equation (7.13) contains all the moves. Now move w to .95, crossing its simple pole at 1, and call the remaining triple integral R_w(Z). In the w=1 residue move z to .16, crossing the pole of \zeta_F^S(6z) at 1/6, and call the remaining residue integral R_z(Z). Their contours are
The other term is the double residue. With its evaluation c_0f_\eta(Z) below, the principal component is exactly
The powers of the two remainders relative to C_*(1) are
The height bounds just given and Proposition 7.3 therefore give
To evaluate the double residue, write R_S=\Res_{s=1}\zeta_F^S(s)>0. It has the form c_0f_\eta(Z), where
Indeed the residue of \zeta_F^S(6z) is R_S/6, and \widehat W_1(1)=\int_0^\infty W_1(t)\,dt>0. The other factor is nonzero by Equation (7.1). The norm measures were fixed before the probe, so this is the exact scalar in its normalization. The line of the double residue was initially \re\xi=\sigma_1; we moved it right to 2 to write Equation (8.13). This uses only absolute convergence of 1/L^S(\xi,\eta) for \re\xi>1.
It remains to bound the rows with u\ne1. Split their sum on the initial lines into dyads U\le q_u<2U, U=1,2,4,\ldots, before shifting any of them. Each dyad contains finitely many rows; the full sum on the initial lines is absolutely convergent. Denote by \mathcal J_{\mathcal C}(Z) the triple integral of a subset \mathcal C of such a dyad, and set
The numerator for each of these rows is entire. From Equation (8.3) and the omitted-factor bound,
On \re\xi=\sigma_1, independently of u and the height,
For each row move the initial lines to
One may first move \xi, then z, then w. The z-line remains above 1/6; the numerator is entire; and the denominator stays in absolute convergence. The first region of Equation (7.13) contains the whole move. If |\mathcal C|\ll U^\alpha, the bounds above give
The factor U^{-.4} is the q_u^{-z} in Equation (7.10). The height integral is bounded by Equation (8.11).
To see how the size of the frequency determines the contours, rewrite Equation (8.15) as
For a general z-line, the corresponding factor is (Z^{h_0}/U)^{\re z}. Its dependence on \re z changes direction at U=Z^{h_0}=Z^{.1}: below that scale a rightward z-shift enlarges this factor, while above it such a shift reduces the factor. We retain the current low-w contour through U\le Z^{h_0+.00001}, and use a rightward z-shift beyond that enlarged threshold.
The common Z-exponent on the current contour, relative to the principal scale, is
For U\le Z^{.09}, use Equation (8.15) with \alpha=1. The U^{1.1} factor adds at most .09(1.1)=.099 to this relative exponent, so these rows are already smaller than the principal scale.
We use Proposition 8.2 only in the remaining interval
where its threshold U_0 is met for all sufficiently large Z. For the rows in \mathcal E(U), use the same contour estimate with \alpha=1/2. The U^{.6} factor then adds at most .10001(.6)=.060006 to the same relative exponent.
For each remaining row truncate its \xi-line at |\im\xi|\le U^\rho, and move that finite portion to \re\xi=.998. The rectangle and its two horizontal sides satisfy Equation (8.4); hence the new central integral satisfies
after summing all the good rows of the dyad. The estimate does not use the reciprocal outside the specified rectangle. Since E_* is linear in \re\xi with coefficient one, this move gains Z^{.998-\sigma_1}=Z^{-.0020001}. Including the U^{1.1} cost up to the upper endpoint of Equation (8.16), the relative exponent is at most
To bound the discarded and horizontal portions, integrate in z,w first in Equation (8.11). For every prescribed A_1>0 the discarded tails on \re\xi=\sigma_1 and the horizontal portions at \im\xi=\pm U^\rho are, in total over a good dyad, at most
On the horizontal portions the real exponent is bounded by its value at \sigma_1; the reciprocal is bounded by Equation (8.4). On the discarded original tails it is bounded by absolute convergence. The convolution estimate following Equation (8.11), used with pointwise exponent A_1+2 before integrating the discarded \xi-tails, supplies U^{-\rho A_1} in both cases. Since U>Z^{.09}, choosing A_1>1/\rho, after \rho is fixed, gives more than enough additional saving for these terms.
All these comparisons leave a fixed margin after the arbitrary small power losses. The sums over the O(\log Z) dyads in these ranges cost only another arbitrarily small power.
Finally consider U>Z^{.10001}. Now Z^{h_0}/U\le Z^{-.00001}, so a fixed rightward z-shift suppresses the whole dyad. Keep \re\xi=\re w=4 and move z to the fixed right line \re z=R_0=350000. The move stays in the Euler region, crosses no pole, and uses the holomorphy of M on \re z>0. Both Hecke factors are bounded by absolute convergence on these lines, so Proposition 7.3 and Equation (8.11) give
The dyads sum geometrically. At their lower endpoint the power of Z is
This is strictly less than C_*(1)-.01 for small \epsilon. The very large z-line is fixed; its weight constants may be large, but are independent of Z and U.
Choose the preliminary losses small enough that their total cost in the finite frequency ranges, including the dyadic count, is less than 10^{-4} in the Z-exponent. This is possible since U\le Z^{.10001} there. The smallest central saving in Equation (8.18) is larger than 0.0012; Equation (8.19) and Equation (8.20) have larger savings. We have therefore proved
Each dyadic integral here is obtained from its initial absolutely convergent integral, and the preceding bounds justify their sum.
8.3 Completion of the Mellin argument
The numerical exponents established above, before the indicated small losses, are collected in Table 1. In particular the exponent from reflection is
| Contribution | exponent relative to C_*(1)=7/15 |
| Probe bound from reflection | -1/6000=-.000166666\ldots |
| Principal remainder R_w | -.00664989 |
| Principal remainder R_z | -.000666566\ldots |
| Nonprincipal rows U\le Z^{.09} | -.010267566\ldots |
| Exceptional rows U\le Z^{.10001} | -.049261566\ldots |
| Good central rows U\le Z^{.10001} | -.001256666\ldots |
| Rows U>Z^{.10001}, R_0=350000 | -.017656666\ldots |
Table 1: The fixed exponent margins in the probe comparison. The row bounds precede arbitrary small power losses.
Combining the principal decomposition I_1=c_0f_\eta+R_w+R_z, Equations (8.12) and (8.21), and Equation (7.10) gives, for sufficiently large Z,
Use Proposition 7.2 with, for example, a loss 10^{-5}. Since 1/6000-10^{-5}>1/20000 and c_0\ne0, the preceding identity implies the safe bound
Set
For \re\xi>1 this is holomorphic. In the definition of f_\eta in Equation (8.13), its \xi-line may be moved right from 2 to any fixed real part \sigma_R>2. The reciprocal there is in absolute convergence, the correction is holomorphic and bounded, and the Gaussian gives rapid vertical decay. It follows that for every A_0>0,
by taking \sigma_R>A_0+8/15. Thus this endpoint estimate has used no zero exclusion to the left of one.
Consider
By Equations (8.22) and (8.23), it converges locally uniformly, and hence defines a holomorphic function, for
Indeed at infinity the real exponent is 1-1/20000-\re s<0, and at zero Equation (8.23) allows an arbitrarily large exponent. The same majorants with powers of \log Z justify differentiation under the integral.
We identify this transform on its original line without a formal contour interchange. Put t=\log Z. From Equation (8.13),
The function \mathscr A(2+i\tau) is continuous and rapidly decreasing by the Gaussian and the bounded Euler factors. The endpoint bounds on f_\eta make the left side integrable in t. Fourier inversion therefore gives
For \re s>1-1/20000, the second neighborhood in Proposition 7.3 makes the numerator
holomorphic and nowhere zero. The Hecke continuation makes \mathscr A(s) meromorphic there. The identity theorem, first on \re s>1 using Equation (8.25) and then for meromorphic functions on the connected larger half-plane, identifies \mathscr A with the holomorphic function \mathcal T. A zero of L^S(s,\eta) in that half-plane would be a pole of \mathscr A, since its numerator is nonzero, which is impossible. A pole of the principal L-function simply makes its reciprocal vanish.
The finite factors removed from L_F(s,\eta) are either one or 1-\eta(\mathfrak p)q_{\mathfrak p}^{-s} with |\eta(\mathfrak p)|=1; their zeros have real part zero. Thus the same zero exclusion holds for the full L_F(s,\eta). The obtained half-plane contains \re s>1-10^{-6}. All onsets and constants in the proof were allowed to depend on F,S,\eta, but the numerical saving 1/20000 is absolute. A field-dependent onset in Equation (8.22) affects only a finite portion of the integral and does not change its half-plane of convergence. Since the initial field and finite-order character were arbitrary in the scope of Theorem 1.2, this proves that theorem.
9 A uniform estimate for complete splitting
We prove Proposition 2.2 for the fields K_q=\Q(\mu_q,a^{1/q}), where a is a fixed integer with |a|>1. Theorem 1.2, proved in Section 8, supplies the common zero-free width \delta_0=10^{-6}. We transfer that region to \zeta_{K_q} and then use a smooth explicit formula whose constants depend on the field only through its degree and discriminant.
9.1 Field bounds and descent of the zero-free region
The field K_q need not be cyclotomic. We first place it in an abelian extension of a cyclotomic field, so that Theorem 1.2 applies, and then descend the resulting zero-free region to \zeta_{K_q}.
Lemma 9.1. Fix an integer a with |a|>1. For every sufficiently large prime q, the degree n_q=[K_q:\Q] and absolute discriminant D_q satisfy
Every rational prime p\nmid aq is unramified in K_q, and
Proof. Fix a rational prime \ell\mid a, and let v_\ell(a)>0 be its valuation. Take q large enough that q\ne\ell and q>v_\ell(a). When a=2, the choice \ell=2 works for every prime q\ge5. The cyclotomic discriminant formula gives |d_{E_q}|=q^{q-2}, so \ell is unramified in E_q and the valuation of a at every prime above \ell is v_\ell(a). This is not divisible by q, and therefore a is not a qth power in E_q.
Since E_q contains \mu_q, the extension K_q/E_q is Galois and the map
embeds its Galois group in \mu_q. Its degree divides the prime q and is not one, so it equals q. This proves the degree formula.
The root a^{1/q} is integral, and the relative field discriminant divides the discriminant of its power basis:
The discriminant tower formula now gives
This also proves unramifiedness away from aq, and taking logarithms proves the last estimate in (9.1). We use the standard cyclotomic and relative discriminant formulas in [19, Chapter I, §10; Chapter III, §2].
To obtain (9.2), put
The extension \widetilde K_q/F_q is cyclic, since it is obtained by adjoining a^{1/q} to a field containing \mu_q. The extension \widetilde K_q/K_q is also abelian: it is Galois, and restriction embeds its Galois group in the abelian group \operatorname{Gal}(F_q/\Q). Abelian Artin factorization, with one-dimensional Artin functions identified with primitive Hecke functions, gives
See [19, Chapter VII, Corollary (10.5), Theorem (10.6), and the following remarks]. Every factor in (9.3) is covered by Theorem 1.2, because F_q is cyclotomic and contains \mu_{12}. Thus \zeta_{\widetilde K_q} is zero-free in the indicated half-plane. Every factor in (9.4) is an entire nonprincipal Hecke L-function. A zero of \zeta_{K_q} there would therefore be a zero of \zeta_{\widetilde K_q}. This proves (9.2).
The width in (9.2) is independent of q. It remains to turn it into a prime-ideal estimate whose constants retain only the explicit dependence on degree and discriminant in (9.1).
9.2 A smooth prime-ideal estimate
Smoothing permits an absolutely convergent sum over zeros. The following form of the explicit formula is useful because its error is linear in \log D+n. We give the calculation with this dependence explicit.
Lemma 9.2. Fix 0<\delta<1/2 and a nonnegative function f\in C_c^\infty((0,\infty)). Let K be a number field of degree n and absolute discriminant D whose Dedekind zeta function has no zero in \operatorname{Re}s>1-\delta. For every real x\ge2,
where \mathfrak p ranges over the nonzero prime ideals of K. The implied constant is independent of K and x.
Proof. Let (r_1,r_2) be the signature of K, so n=r_1+2r_2, and define the Mellin transform
It is entire. Repeated integration by parts shows that, for every fixed vertical strip and every A>0, \Phi(s)\ll_{f,A}(1+|\operatorname{Im}s|)^{-A} on that strip. Mellin inversion and the absolutely convergent Euler series on \operatorname{Re}s=2 give
We shift the contour to \operatorname{Re}s=-1/2. On this line the functional equation gives
with an absolute implied constant. To see the dependence on K, the Euler series at 1-s, whose real part is 3/2, is bounded by
Indeed the sum of the residue degrees of primes above a rational prime is at most n. The remaining terms in the functional equation are \log D and the logarithmic derivatives of
On the lines involved their logarithmic derivatives are O(\log(2+|\operatorname{Im}s|)), by Stirling’s formula and their fixed distance from poles. This proves (9.7); see also [27, Main Theorem 4.4.1 and §4.5] for the functional equation.
The poles crossed in (9.6) are the pole of \zeta_K at 1, its nontrivial zeros \rho counted with multiplicity, and the zero at 0 of order r_1+r_2-1. The last order follows from the gamma factors and the simple pole of the completed zeta function at 0; when the order is zero there is no residue. There are no other trivial zeros between the two lines. The nontrivial zeros lie in 0<\operatorname{Re}\rho<1: the hypothesis excludes the boundary at 1, and the functional equation excludes nontrivial zeros on the boundary at 0.
For completeness, the zero count needed to justify the shift and bound its residues is
uniformly in K and real T. This follows from the uniform Dedekind zero-counting formula of [10, Corollary 1.2, Equation (1.3)]: for |T|\ge2, subtract the formula at heights |T|-1 and |T|+2; for |T|<2, use the count up to height 3. For each fixed K, (9.8) permits horizontal contour edges tending to infinity and staying an inverse polynomial distance from all zeros. The Hadamard partial-fraction expansion of the completed zeta function then bounds its logarithmic derivative on these edges by a polynomial in their heights. The rapid decay of \Phi makes the horizontal integrals tend to zero. The same zero count makes the resulting zero sum absolutely convergent. Hence the contour shift and (9.7) yield
Although the contour edges may be chosen separately for each field, the error here has a uniform constant, since it is bounded directly by the integral on the fixed line in (9.7).
Every nontrivial zero has \operatorname{Re}\rho\le1-\delta. Using (9.8) and the rapid decay on 0\le\operatorname{Re}s\le1, we obtain
The residue at zero in (9.9) is O_f(n). Absorbing it and the left-line error proves (9.5).
Proof of Proposition 2.2. Choose once and for all a nonnegative f\in C_c^\infty((0,\infty)) with f\ge1 on [1,2]. Apply Lemma 9.2 to K_q with \delta=\delta_0, using Lemma 9.1. Each rational prime p\in(x,2x] that splits completely supplies n_q prime ideals of norm p. Their first-power terms contribute at least n_qL to the nonnegative sum \Theta_{K_q,f}(x). It follows that
For q\le\exp(L^{0.3}), the ratio \log(2q)/L is bounded once x is large. This proves (2.4), with constants and an x threshold independent of q. For a=2, the field bounds hold for every q\ge5 with absolute constants, giving the stated specialization.
10 Marked Type II estimates and rough factors
The prime construction will count integers whose predecessor has the form in Equation (2.2), then remove the composite integers from that count. The estimate in this section makes that subtraction possible: inside a marked bilinear sum, it replaces a prime condition on one factor by a condition involving only small prime divisors. We first state the imported prime estimate and isolate the coefficient property used by its proof. We then verify that property for the rough-factor condition needed when a composite’s least prime factor stays below the square-root range.
Return to the real variable x from Section 2, and put
For an integer m>1, write P^-(m) for its least prime factor, and put P^-(1)=\infty. Thus P^-(m)>y means that m is y-rough. All Dirichlet characters below, including principal characters, are extended by zero on nonunits.
Fix an integer K\ge1 and real numbers 0.1<a_1<\cdots<a_K<0.2, all before letting x grow. The prime groups
are disjoint for sufficiently large x. For a positive integer h, let
be its complete part supported on these group primes. A function F:\mathbb Z_{>0}\to\mathbb C satisfies F(ph)=F(h) for every group prime p and every positive h if and only if it depends only on h/h_{\mathcal P}. In particular this invariance removes every power of a group prime, not only a selected divisor.
Define
The prime number theorem and partial summation give V_i=\log 2+o(1). In particular these denominators are positive for large x. The weight is nonnegative and vanishes unless each group supplies a prime divisor. It is bounded by a constant depending only on the fixed K,q, since j q^{j-1} is bounded for positive integers j. In Section 12 we shall take q=1/2.
The following is the one-sided marked Type II estimate of The Poisson–Dirichlet law for the prime factors of p-1 [21, Theorem 3.1].
Theorem 10.1 (One-sided marked Type II estimate). Fix \delta,C,D_*>0 and q\in(0,1). Suppose H_m,H_n\ge x^\delta and X=H_mH_n\asymp x. Let F satisfy |F(h)|\le1 and F(ph)=F(h) for every prime in the groups defined in Equation (10.2). Let \alpha_m be supported on an arbitrary interval in [H_m,2H_m], where it has the value
Let \beta_n be supported on W-rough integers in [H_n,2H_n], with |\beta_n|\le L^C. If K is sufficiently large in terms of \delta,C,D_*,q, then
The required lower bound on K is independent of the particular a_i. The implicit constant and the threshold for x may depend on all the fixed parameters, including the a_i.
The invariance is required for every positive h, including when p\mid h. The estimate is uniform over the permitted F, coefficient intervals, and coefficients. The symbol X in this section denotes only the product H_mH_n.
We need the same marked bound for a rough-factor discrepancy. The following lemma isolates a sufficient condition on a scalar coefficient: cancellation against every character of logarithmic modulus throughout the stated frequency range. Its proof extracts the coefficient dependence from the argument of [21, Theorem 3.1].
Lemma 10.2 (Coefficient criterion for the marked estimate). Fix \delta,C,D_*>0 and q\in(0,1). Retain the marked data and the hypotheses on F,H_m,H_n in Theorem 10.1, with fixed comparison constants in X=H_mH_n\asymp x. Let (\alpha_m) and (\beta_n) be scalar sequences indexed by positive integers, supported on W-rough integers in their respective dyads, with \alpha supported on an arbitrary interval I\subseteq[H_m,2H_m], and with |\alpha_m|,|\beta_n|\le L^C. Suppose that for every fixed A_0,B,A>0, every Dirichlet character \chi modulo k\le L^{A_0}, and every |t|\le2XL^B,
uniformly in the permitted interval, character, and frequency. The constant and threshold in this hypothesis may depend on A_0,B,A, the fixed setup, and all further fixed parameters defining the coefficients; these parameters are chosen before x grows.
Then Equation (10.5) holds when K is sufficiently large in terms of the fixed data, independently of the particular band exponents a_i. Its implicit constant and threshold may depend on all the fixed data, including the a_i.
Proof. We extract the assertion from the proof of [21, Theorem 3.1]. In its marked expansion, write mn-1=ah, where a is the product of the selected primes. Invariance gives F(mn-1)=F(h). Except when a selected prime has its square dividing mn-1, one also has \omega(mn-1)-K=\omega(h). Replacing the damping by q^{\omega(h)} therefore costs O_A(XL^{-A}) for every fixed A. The resulting factor F(h)q^{\omega(h)} has modulus at most one and is discarded in the following Cauchy bound. There are O_K(L) dyads Y for the product of the selected primes. For the contribution S_Y of one dyad, Cauchy’s inequality gives
where the nonnegative expanded square is
Here a,b each select one prime from every group, and \rho is the fixed real smooth dyadic cutoff supported in [1,4]. Pairs a,b sharing a selected prime cost O_A(XYL^{-A}). For the remaining coprime pairs, the common-h condition is equivalent to t=b-a, where t=bmn-am'n'. These reductions use only coefficient bounds and the invariance of F; see [21, Section 3.1].
For a fixed A_0>0, let \mathfrak M\subset\mathbb R/\mathbb Z be the union of the arcs
The selected-product range has \log Y\gg L^{0.1}, so these arcs are disjoint for large x. Choose the companion’s fixed real \psi\in C_c^\infty(\mathbb R), equal to one on [-4,4], and define
The major form is the scalar sum
where a,b now run independently over products of one prime from every group.
The proof of the replacement conclusion in [21, Proposition 4.1] applies to these scalar coefficients. Its operator moment is independent of their values. The passage from that moment to the scalar endpoint pairing uses only W-rough support and norm and supremum bounds with logarithmic exponents depending on C, not on K [21, Sections 3.2–3.3]. This pairing initially has projections onto the good endpoint states of that argument. In removing either projection, only the error from the endpoint outside its good set is majorized absolutely: each whole scalar endpoint factor is replaced by its logarithmic supremum bound, discarding both its coefficient value and its support indicator before the one-edge root replacement. The surviving unprojected pairing retains its exact endpoint vectors. Expanding those vectors and controlling collisions uses only size, rough support, and fixed divisor moments [21, Section 4.9]. Thus the replacement conclusion, including restoration of independent a,b, gives
for every prescribed fixed accuracy, with the choices specified below.
The proof of the major-term estimate [21, Proposition 5.1] separates characters only at moduli k\le L^{A_0}. The group primes and all prime divisors of the rough variables exceed these moduli, so every variable is a unit there. After Mellin separation its retained rectangle, for a fixed B chosen after A_0, has |s'|\le L^B and |t|\le XL^B, so both t and s'-t lie in the range of Equation (10.6). Transform decay and the product mean square on translated intervals control the discarded tails using only scalar coefficient bounds and fixed divisor moments. The further Mellin shift enters only the one-group prime polynomial and the bounded factor from the other groups; the sparse-frequency estimate is uniform in this shift [21, Lemma 5.3]. On the remaining sparse frequency set, the only further input about \alpha is the supremum in Equation (10.6), in the role of [21, Lemma 5.2]. The same hypothesis covers the conjugated endpoint, by conjugating and replacing \chi,t by \overline\chi,-t. The major-term proof therefore gives
for every fixed D>0.
Choose the desired square accuracy D_1 sufficiently large in terms of D_*,C, before choosing K. For Equation (10.9), choose the moment accuracy E_0 after D_1,C, then the arc exponent A_0, and then K, as in [21, Proposition 4.1]. These choices do not depend on the particular a_i. The analytic accuracies in Equation (10.10) are chosen after A_0, as permitted by Equation (10.6) for every fixed accuracy. Taking A=D=D_1 gives \mathcal Q_Y\ll XYL^{-D_1}. Cauchy’s inequality yields |S_Y|\ll XL^{-D_1/2}. The O_K(L) dyads and all fixed coefficient losses are absorbed by the choice of D_1. The logarithmic exponents used in that choice are independent of K; constants depending on K affect only the implicit constant and threshold. This proves Equation (10.5).
We now construct a coefficient satisfying this criterion for x^\gamma-rough integers. Its comparison weight comes from the continuous measure of products of large primes. For w>\gamma>0, define
The term with j\ge2 is zero unless w\ge j\gamma, so the sum is locally finite. This is the density of [20, Lemma 7.4].
Its normalization has an exact continuous meaning. For an interval I whose closure is a compact subset of (x^\gamma,\infty), and each j\ge1,
For j=1, the integrands agree because D_{\gamma,1}(\log y/L)/L=1/\log y. For j\ge2, put y=\prod_i y_i, t_i=\log y_i/L for i<j, and w=\log y/L. The change of variables gives
The strict domain from y_i>x^\gamma differs from the weak simplex in Equation (10.11) only on null boundaries. The factor 1/j! divides the ordered prime-product measure by the number of permutations. This identity concerns the full domain w>\gamma; the upper exponent restriction in the next proposition belongs to the Type II estimate, not to the density.
The functions in Equation (10.11) are jointly continuous on compact subsets of w>\gamma>0, and, for fixed \gamma, are Lipschitz in w on each such compact interval. To check the assertion also at w=j\gamma for j\ge2, write s=w-j\gamma. For s\ge0, the change of variables t_i=\gamma+s y_i gives
Every denominator stays bounded away from zero on the indicated compact sets. The formula extended by zero for s<0 has the claimed regularity; the term 1/w is smooth. In particular wD_\gamma(w) has bounded total variation on each fixed compact interval above \gamma. This variation bound will allow us to weight the high-frequency proxy estimate below, and joint continuity will control the upper Riemann sum in the least-prime-factor subtraction.
The character count proved below gives D_\gamma(\log m/L)/L as the principal local density of the x^\gamma-rough sequence, while the W-rough sequence has density V(W). This explains the normalization of its comparison weight.
Proposition 10.3 (A rough-factor Type II estimate). Retain the marked data and all the hypotheses on F,\beta,H_m,H_n of Theorem 10.1. Fix real numbers 0<\gamma<w_-<w_+<1, and suppose
On an arbitrary interval I\subseteq[H_m,2H_m], set
and set \alpha_m=0 outside I. Then Equation (10.5) holds for every fixed D_*>0 when K is sufficiently large in terms of the fixed data. Its required lower bound is independent of the particular band exponents a_i; the threshold and implicit constant may depend on all the fixed data. In particular, \gamma and the gap w_--\gamma are fixed before x grows.
Proof. We verify Equation (10.6) for this coefficient. At low frequencies the two principal counting terms cancel. At higher frequencies the prime-product sequence and the W-rough comparison sequence are each small. The support and size conditions of Lemma 10.2 will then follow directly from the coefficient formula.
Put M=H_m and write R_\gamma(m)=\one_{P^-(m)>x^\gamma}. On this dyad a number counted by R_\gamma has at most J=\lfloor w_+/\gamma\rfloor prime factors counted with multiplicity. Define
If m is squarefree and x^\gamma-rough, the ordered lists give T_\gamma(m)=1. If the prime multiplicities are e_p, the corresponding value is 1/\prod_p e_p!, which is at most one. Thus the difference is supported on a square of a prime exceeding x^\gamma, and, uniformly for every subinterval of the dyad,
The same bound holds after any character and real power twist. It is smaller than M times every fixed negative power of L.
We next obtain the counting formula needed at low frequencies. For every fixed A_1,A_0>0, every k\le L^{A_0}, every character \chi\bmod k, and every interval I'\subseteq[M,2M], we claim
The constant may depend on the displayed fixed parameters and on \gamma,w_-,w_+, but the estimate is uniform in the interval and character.
To prove it, split every prime variable of a term of T_\gamma into half-open dyads [P_i,2P_i). All their scales satisfy P_i\ge x^\gamma/2 and P_i\le2M. There are O(L^j) possible boxes for the j-prime term. A box that meets the product interval has M/2^j\le\prod_iP_i\le2M. On a prime dyad of scale P, the Siegel–Walfisz estimate, summed against \chi over its reduced classes, gives for every subinterval J'\subseteq[P,2P]
with arbitrary fixed A_2. Indeed \log P\asymp L with fixed comparison constants, and one chooses the Siegel–Walfisz accuracy to absorb the at most L^{A_0} residue classes. This is the classical estimate in [22, Proposition 2.1]. Taking differences of its endpoint estimates gives the absolute error in Equation (10.16) even for an arbitrarily short subinterval. The cutoff p>x^\gamma can be intersected with that interval. For large x, these primes exceed k; hence the principal character is one on them. The same orthogonality argument covers imprimitive characters.
Replace the prime measures in one box successively using Equation (10.16). When all other variables are fixed, whether at prime values or at real values in a previous replacement, the product condition leaves an interval in the remaining prime dyad. Its error is O(P_iL^{-A_2}). The total absolute mass of each other variable, either its prime counting measure or \dd y/\log y, is O(P_h). The error in that box is therefore O(M L^{-A_2}). There are only the fixed j replacements and O(L^j) boxes, so their errors give O(M L^{-A_1}) on choosing A_2 sufficiently large. A nonprincipal character has zero continuous main term. For a principal character the main term of the j-prime summand is exactly
By Equation (10.12), these integrals sum to the principal main term in Equation (10.15). Finally Equation (10.14) absorbs repeated primes and proves the claim.
The matching small-prime count is
for every fixed A_1, with the same interval and character uniformity. This is [21, Lemma 2.5]. Notice that all prime divisors of k\le L^{A_0} are less than W for large x, so its principal character is one on the rough support.
Write u=t+v. Fix temporarily a constant B_2>C+2. If |t|\le L^{B_2}, then |u|\le2L^{B_2}. Partial summation in Equation (10.15) against y^{iu} costs at most 1+O(|u|), since this is its total variation on a dyad. Partial summation in Equation (10.17) against D_\gamma(\log y/L)y^{iu}/(LV(W)) likewise has only a fixed logarithmic cost: D_\gamma is Lipschitz on the fixed interval and Mertens’ theorem gives LV(W)\asymp L^{0.76}. The two resulting principal main terms are both
while both nonprincipal main terms vanish. Choosing A_1 after B_2 proves Equation (10.6) with any fixed accuracy on this low-frequency range.
For the remaining frequencies, use the long prime polynomial estimate [22, Lemma 3.1] and the corresponding bound for the small-prime proxy. For fixed 0<\tau_{\rm lp}<\theta_{\rm lp}<1 and fixed A_0,A_3>0, the prime estimate gives a constant B_0=B_0(\tau_{\rm lp},\theta_{\rm lp},A_0,A_3) such that
for every interval J'\subseteq[P,2P], every character modulo k\le L^{A_0}, and
Choose 0<\tau_{\rm lp}<\gamma and w_+<\theta_{\rm lp}<1. Every prime dyad in the finite convolution lies in this scale range for sufficiently large x. Partial summation changes the reciprocal weight in Equation (10.18) to give \left|\sum_{p\in J'}\chi(p)p^{iu}\right|\ll P L^{-A_3}. In each box of the convolution, fix all but one prime; the remaining product restriction is again an arbitrary interval. This bound in that one slot and trivial bounds in the other slots give O(M L^{-A_3}) per box. Taking A_3 to absorb the O(L^j) boxes and using Equation (10.14) proves any requested logarithmic saving for M^{-1}\sum_{m\in I}R_\gamma(m)\chi(m)m^{iu} throughout this range.
For the proxy, the proof of [21, Lemma 5.2] gives the following uniform estimate. Put T_* =\exp(L/(\log L)^2). For every interval I'\subseteq[M,2M], and with arbitrary fixed A_4, it states
Here C_0,c_0>0 are fixed after A_0,\delta, and C_3>0 is absolute; C_0 does not grow with A_4. The second range uses the logarithmic-phase estimate of [22, Lemma 3.3], valid for arbitrary residue classes and subintervals. The first follows in the cited proof of the long-coefficient lemma by comparison with the integral of y^{iu} on each progression.
For w=\log m/L, the coefficient of the proxy can be written
The multiplier wD_\gamma(w) is bounded and has bounded total variation on our fixed interval. Since Equation (10.19) holds on every subinterval, one more partial summation proves that same bound for the proxy coefficient. Choose the requested accuracy first, then A_3,A_4 and the corresponding B_0, and finally take
When L^{B_2}\le|t|\le2XL^B, one has |u|\asymp|t|, |u|\ge L^{B_0}, and |u|\le2XL^B+L^C<x^2 for large x. The prime convolution and Equation (10.19) therefore give the saving L^{-A} for the two parts separately. The complementary range |t|\le L^{B_2}, including t near -v, was already handled by cancellation of their low-frequency main terms. This covers every |t|\le2XL^B; the conjugated assertion then follows as in Lemma 10.2.
Finally, x^\gamma>W for large x, and D_\gamma is bounded on our fixed interval. Since LV(W)\asymp L^{0.76}, both parts of Equation (10.13) are bounded and supported on W-rough integers. Thus the support, size, and long-coefficient hypotheses of Lemma 10.2 all hold. Applying that lemma proves Equation (10.5) and the proposition.
11 Two elementary sieve facts
The prime construction needs two elementary facts. The first estimates the mass left after excluding local divisibility conditions, with an explicit bound on the moduli in the error term. The second identifies the density removed when a least prime factor is selected. We give proofs, following the forms in [20, Lemmas 2.9 and 7.4].
11.1 A block sieve with bounded coefficients
The advantage of the following form of the Brun–Hooley sieve is that its remainders occur without divisor weights. Its underlying product inequalities are those of Ford and Halberstam [6, Section 1, Lemma 1].
Lemma 11.1 (Block sieve). Let \mathcal A be a finite set with nonnegative weights, and let \mathcal P be a set of primes at most z, where z\ge2. For each p\in\mathcal P, specify a bad condition on \mathcal A. Suppose that, for every squarefree product d of primes in \mathcal P, the weight of the objects satisfying all conditions at p\mid d is
including d=1, where X'\ge0, g(1)=1, and g is multiplicative. Assume that, for fixed constants \eta_0>0 and C_0<\infty,
For every sufficiently large even integer H, the weight S of objects avoiding all the bad conditions satisfies
The constants and the lower bound on H depend only on \eta_0,C_0. In particular, the estimate is uniform when H grows. For H=2 there is the upper bound
Proof. Write b_p\in\{0,1\} for the indicator of the bad condition at p. Use the blocks
ignoring empty blocks and retaining the indicated indices. Put h_j=(j+1)H. Let I_j be the indicator of avoiding the conditions in \mathcal B_j. Define U_j to be the inclusion–exclusion polynomial through degree h_j, and T_j to be the sum of the unsigned degree-h_j+1 inclusion–exclusion terms:
If exactly t conditions in the block hold, then, since h_j is even,
As usual, a binomial coefficient vanishes when its lower index exceeds its nonnegative upper index. Thus 0\le I_j\le U_j and U_j-I_j\le T_j. Telescoping the product, all of whose factors are nonnegative, gives
Consequently the polynomials
satisfy B\le\one_{\text{no bad condition}}\le U.
Every monomial in U has degree at most h_j in block j. A monomial in correction j has degree exactly h_j+1 there and degree at most h_k in every other block. Hence the supports of the corrections are disjoint from one another and from the support of U. All coefficients of both B and U therefore have absolute value at most one. The prime product indexing a monomial of U has logarithm at most
A correction adds at most \log z, so every monomial used is supported on a squarefree d\le z^{4H+2}.
It remains to evaluate the main terms of these polynomials. Give the indicators b_p independent Bernoulli laws with means g(p), and write \mathbb E for expectation under this auxiliary law. By (11.1), each block has total mean at most C_0. Its avoidance probability satisfies
because -\log(1-u)\le u/\eta_0 for 0\le u\le1-\eta_0. Moreover,
The factorial bound implies
for all sufficiently large even H: for sufficiently large n, C_0^n/n!\le\exp(-2n), and h_j=(j+1)H. For H=2, the same sum is bounded in terms of \eta_0,C_0. Independence of the blocks now yields
Thus both bounding polynomials have expectation \prod_ja_j(1+O(\exp(-H))) when H is sufficiently large; for H=2, the upper expectation is O(\prod_ja_j).
Finally evaluate U and B in the original weighted set. Multiplicativity makes their main terms X'\mathbb E U and X'\mathbb E B. The coefficient and support bounds show that each error is at most the sum of |E(d)| in (11.2). Squeezing S between the two weighted polynomial sums proves both assertions.
11.2 Rough-number densities
Recall the density D_\gamma(s) from (10.11). Its jth term describes a product of j primes whose logarithms, in units of \log x, sum to s and are at least \gamma; the factor 1/j! normalizes the ordered prime lists. These are Buchstab’s rough-number densities, expressed in logarithmic coordinates. The identity below is the corresponding least-prime-factor decomposition [4]. The normalization and regularity were established in Section 10; we now prove the two estimates needed to subtract composite mass.
For the product V(y) in (10.1), Mertens’ theorem gives
where \gamma_E is Euler’s constant.
Lemma 11.2 (Rough-number density). For 0<b<1/2,
where the integrand at 1/2 is interpreted by its left limit. There are absolute constants b_0,c_d,C_d>0 such that, for 0<b<b_0,
One may take c_d=1/20.
Proof. To prove (11.6), write a term of D_b(1) with j\ge2 as an integral on
This measure is invariant under permutations of the j coordinates: interchanging a dependent and an independent coordinate has absolute Jacobian one. Except on a set of measure zero, the minimum coordinate is unique. Choose it in j ways and call its value t. The remaining j-1 coordinates sum to 1-t and are at least t; their measure is the (j-1)-factor term of D_t(1-t), multiplied by dt/t, since j/j!=1/(j-1)!. Necessarily t\le1/2. Summing over j\ge2 proves the identity; the omitted one-prime term of D_b(1) equals 1.
For (11.7), we compare D_b(1) with ordinary rough integers and then apply Lemma 11.1. Fix b>0, put L=\log x, and let
We first claim that
For j\ge2 and fixed \eta>0, count ordered prime lists (p_1,\ldots,p_j) with weight 1/j!, subject to
Every product counted is in R_b(x), because p_j>x^{b+\eta}. An integer with prime multiplicities e_1,e_2,\ldots receives total weight at most 1/\prod_i e_i!\le1, even when prime factors repeat. Its total number of prime factors also determines j. Thus summing these counts over j, together with the one-prime contribution, gives a lower bound for R_b(x).
Uniformly in the indicated range of Q, the prime number theorem gives
For fixed 0<\alpha<\beta, Mertens’ theorem gives
Consequently the reciprocal-prime measures on any fixed compact interval of positive logarithmic exponents converge to dt/t. Their finite products converge as well. On the truncated simplex t_i\ge b, \sum_{i<j}t_i<1-b-\eta, the function (1-\sum_{i<j}t_i)^{-1} is bounded, and its boundary has measure zero. The weighted count above, multiplied by L/x, therefore tends to
There are only finitely many relevant j for fixed b. The one-prime contribution tends to 1. First let x tend to infinity with b,\eta fixed, and then let \eta decrease to zero. The boundary hyperplanes are null, so these integrals increase to the terms in D_b(1). This proves (11.8).
For the upper bound on R_b(x), apply Lemma 11.1 to the integers in (x,2x], with bad conditions p\mid n for p\le z=x^b. Here X'=x, g(d)=1/d, and E(d)=O(1), uniformly in d. The hypotheses hold with absolute constants: g(p)\le1/2, and the sums of 1/p over (v,v^2] are bounded. Choose
For sufficiently small b, this is an admissible even integer, H\gg1/b, and b(4H+2)\le1/5+2b<1. Lemma 11.1 gives, with absolute implied constants,
with c=1/20, since \exp(-H)\le\e^2\exp(-1/(20b)). Multiplying by L/x, applying (11.5), and letting x tend to infinity proves (11.7) by (11.8).
The term 1 in (11.6) is the one-prime contribution. The integral accounts for the composite contributions by selecting their least prime factor. This is the cancellation that will leave a positive prime mass in the construction below.
12 Primes with a controlled predecessor in a fixed progression
We prove Proposition 2.1. Fix its data M,c,u, so
We first assign nonnegative weights to integers d=crQ+1 in the specified progression. A lower sieve bound supplies integers with no small prime factor. The marked Type II estimate then allows us to bound the mass of composites according to their least prime factor. Factors close to the square root require a separate upper sieve bound. Keeping the constant in that last bound independent of the number of marks will permit the final parameter choices.
12.1 The weights and the Type II estimate
Use the parameters L,W,V, the prime groups \mathcal P_i, and the marks \mathcal W of (10.1), (10.2), and (10.4), with q=1/2. All prime-indexed sums and products below are over rational primes. The integer K\ge1 and exponents 0.1<a_1<\cdots<a_K<0.2 are fixed before x grows; their eventual choice will follow the other sieve parameters. The groups are disjoint for large x, and Mertens’ theorem gives V_i=\log2+o(1). Their primes are then coprime to M and lie in (\exp(L^{0.1}),\exp(L^{0.3})).
The factor \omega_i(h) counts choices of a prime divisor of h from the ith group. Thus \mathcal W(h) vanishes unless every group contributes, while \omega_i(h)2^{1-\omega_i(h)} remains bounded when several primes from one group divide h. These choices are the marks in the Type II estimate below.
Call a positive integer a group integer if all its prime divisors belong to \bigcup_i\mathcal P_i, and include 1. Use the complete group-prime part h_{\mathcal P} defined in (10.3). Fix a smooth function \Psi compactly supported in (1,2), with 0\le\Psi\le1 and \int\Psi>0, and define
Set w(1)=0. Since (ph)_{\mathcal P}=p h_{\mathcal P} for every group prime p, including when p\mid h, we have
On the support of w, the representation d=crQ+1 is unique: r=(d-1)_{\mathcal P}, since neither c nor Q has a group prime factor. In particular,
for a constant B_K. The last assertion follows from j2^{1-j}\le1 for each integer j\ge1; in fact 0\le\mathcal W(h)\le B_K for every h\ge1.
Write X_0=\sum_d w(d) for the total mass. Our aim is to prove that the prime mass is \gg X_0/L, while X_0\gg x/L. The bound w\le B_K will then give the required count of primes. The marked estimate supplies the comparison needed to subtract composite mass.
Use the rough-number density D_\gamma(s) from (10.11). For fixed \gamma>0 and integers m>x^\gamma, put
We will replace R_\gamma by B_\gamma inside sums weighted by w(mn): the factor D_\gamma/(LV(W)) compensates for retaining only W-roughness. Lemma 10.2 and the character-transform bound proved in Proposition 10.3 permit this replacement after the fixed congruence is expanded in characters. We verify that application next. All Dirichlet characters retain the zero extension off the units used in Section 10.
Lemma 12.1 (Type II estimate in the progression). Fix \delta>0 and 0<\gamma<s_-<s_+<1. Let H_m,H_n\ge x^\delta, with X=H_mH_n\asymp x and fixed comparison constants, and let I\subseteq[H_m,2H_m] be an interval. Suppose
Let |b_n|\le1 be supported on W-rough integers in [H_n,2H_n]. For sufficiently large K,
This is uniform in the dyads, interval, and coefficients. The lower bound on K is independent of the a_i. Constants and thresholds may depend on all fixed parameters, including the band exponents.
Proof. The congruence in w cannot be incorporated into F while preserving (12.3). Instead, character orthogonality and (u,M)=1 give
By the proof of Proposition 10.3, the sequence \alpha_m=\one_I(m)m^{iv}(R_\gamma(m)-B_\gamma(m)) satisfies (10.6) for every fixed A_0,B,A,C>0, every character of modulus at most L^{A_0}, and every |t|\le2XL^B, uniformly for |v|\le L^C and every interval I under the present fixed exponent hypotheses. If \alpha satisfies that bound, so does \alpha_m\lambda(m): the product \lambda\chi, with its zero extension, is a character modulo \operatorname{lcm}(M,k), which is at most ML^{A_0}\le L^{A_0+1} for large x.
To separate the smooth factor, write
Fourier inversion yields
For |v|\le L, apply Lemma 10.2 to the invariant function F in (12.2), with C=2, D_*=6, and
Here the formula for \alpha_m applies on I, and \alpha_m=0 elsewhere. Both sequences are W-rough and bounded by L^2 for large x: x^\gamma>W, D_\gamma is bounded on the fixed exponent interval, and LV(W)\asymp L^{0.76}. The finite number of characters is fixed with M. The integral of |\widehat\psi| is finite, whereas its tail beyond L is O_A(L^{-A}) for every fixed A. On that tail the bilinear sum is O_K(X) by (12.4). Integration proves (12.7).
All parameters in the remainder of the section are fixed before x tends to infinity. Unless an independence is explicitly asserted, constants, thresholds, and rates of convergence may depend on those fixed parameters. The estimates are valid for arbitrary fixed marks, subject to the lower bounds on K in the Type II applications. We will choose the parameters together at the end.
12.2 Mass and distribution in progressions
Let r range over group integers, and define
We shall show that X_0 is of order xJ_0/L, and that the family at each small r has enough distribution to apply the block sieve. The harmonic sum makes this possible without constants depending on the pointwise bound B_K.
Lemma 12.2 (Harmonic mass). For each group put
The harmonic mass has the exact Euler-product expression
For every fixed \eps>0,
Consequently the part of X_0 with r>x^\eps is O_K(x\exp(-\eps L^{1-a_K})).
Proof. At a prime p, summing over its positive exponents contributes \sum_{j\ge1}p^{-j}=1/(p-1). Consequently E_i(t) is the harmonic generating function for integers supported on the ith group, with t recording their number of distinct prime factors. Differentiating its finite product and evaluating at t=1/2 inserts the factor \omega_i(r)(1/2)^{\omega_i(r)-1}. The groups are disjoint, so multiplying these marked sums and dividing by \prod_iV_i proves (12.9).
Integers with one prime, to exponent one, from each group contribute
to J_0. For the upper bound and tail put s=L^{-a_K}. Every group prime satisfies p^s\le\e^2, so, for large x,
Bounding the mark by B_K and multiplying the geometric series over group primes gives
since \sum_iV_i\ll K. Multiplication by x^{-\eps s} on r>x^\eps proves the tail. Finally, A_r\ll x/r by counting integers in the interval for Q whenever A_r\ne0; otherwise the bound is immediate.
Take temporarily
For an odd squarefree integer \ell, let A_r(\ell) denote the sum defining A_r with the additional condition \ell\mid crQ+1, and set
In particular, g_r(1)=1 and E_r(1)=0.
Lemma 12.3 (Distribution of the weighted family). For every fixed A>0,
Moreover, writing A_\Psi=\int_1^2\Psi(y)\,dy, we have
with all construction parameters fixed and J_0=J_0(x). In particular,
with comparison constants independent of K,\kappa,\eps,b.
Proof. Put N=M/c. For r\le x^\eps, the support of \Psi makes Q>x^{0.9} automatic. The progression for d is equivalent to the reduced class
If (\ell,Mr)>1, then A_r(\ell)=g_r(\ell)=0. Otherwise the additional condition is Q\equiv-(cr)^{-1}\pmod\ell, and the two classes combine to one reduced class modulo N\ell.
We use the prime-counting form of the Bombieri–Vinogradov theorem [3, 29]: for fixed A',\eta>0,
Here \Li(Y)=\int_2^Ydt/\log t. Uniformly for r\le x^\eps and Y\asymp x/(cr), the moduli N\ell, \ell\le D, lie in this range. Indeed, with \eta=(\kappa-\eps)/4,
which also absorbs the fixed factor N.
Define
For every fixed 0<\theta<0.1, the support of \Psi forces Q>x^{0.9} whenever r\le x^\theta. The prime number theorem in the fixed reduced progressions modulo N, followed by partial summation, therefore gives, for every fixed A>0,
The class varies with r, but N is fixed and every class is reduced. The implied constant is independent of \theta, although the threshold for the automatic cutoff may depend on \theta. This single-modulus estimate thus holds throughout 0<\theta<0.1; the restriction \eps<\kappa is needed for the joint modulus estimate that follows.
Partial summation in (12.16) yields
To check the smoothing uniformly, integrate the progression remainder against the derivative of \Psi((crt+1)/x). Its total absolute integral is \int|\Psi'(y)|\,dy, independently of r, and all integration points have t\asymp x/(cr). The moduli N\ell are distinct, and the maximum in (12.16) covers their varying residue classes.
The term \ell=1 of (12.18) evaluates A_r. Replacing I_r/\varphi(N) by A_r in the other terms costs at most one logarithmic factor, because
For example, expand n/\varphi(n)=\sum_{d\mid n}\mu^2(d)/\varphi(d) and use \sum_d\mu^2(d)/(d\varphi(d))<\infty. Choosing A' sufficiently large gives \sum_{\ell\le D}|E_r(\ell)|\ll(x/r)L^{-A}; summing with the marks proves (12.13).
On the support of \Psi, for r\le x^\eps the logarithm in I_r is comparable to L, with absolute comparison constants since \eps<0.01 and c\in\{2,4\}. Thus I_r\asymp_\Psi x/(rL). Summing the evaluation of A_r and using Lemma 12.2 proves (12.15). The tail divided by xJ_0/L tends to zero since J_0\ge1. Although its threshold may depend on K, the resulting comparison constants do not.
To obtain the precise asymptotic, fix an auxiliary exponent 0<\theta<\eps, without changing the sieve parameters. Uniformly for r\le x^\theta and y in the support of \Psi,
and hence
The constants in these relative errors are independent of \theta. Use the \ell=1 term of (12.18) with A'>2, multiply by \mathcal W(r), and sum over r\le x^\theta. The summed progression error is O(xJ_0L^{-A'}). Lemma 12.2 bounds the omitted parts of both X_0 and J_0; after the harmonic tail is multiplied by x/L, both are o_\theta(xJ_0/L), since J_0\ge1. Thus
First let x tend to infinity with \theta fixed, and then let \theta decrease to zero. This proves (12.14). For (M,c,u)=(8,4,5), it becomes X_0\sim xJ_0A_\Psi/(4L), since \varphi(M/c)=\varphi(2)=1.
To express the sieve products, define
Division by V(y) and Mertens’ theorem give
where \gamma_E is Euler’s constant. The primes dividing any supported r are large enough that their omission changes these products negligibly. More precisely, (12.4) gives
uniformly for r\le x^{0.11} and all y. The bound on the relative error is independent of K.
12.3 Two sieve estimates
The block sieve of Lemma 11.1, from [20, Lemma 2.9], now gives the initial mass without small prime factors and a conditional mass for a specified divisor. Both estimates use the distribution already proved, but at different sieve depths.
First sieve the prime objects Q in A_r, for r\le x^\eps, by the conditions p\mid crQ+1 at odd primes p\le z=x^b. The local densities are g_r(p), with g_r(p)\le1/2 and \sum_{v<p\le v^2}g_r(p)\ll1 with absolute constants. Choose the even depth
For sufficiently small fixed b, this is large enough for the lower bound in Lemma 11.1; moreover,
All remainders are therefore covered by (12.13). Every supported d is odd. Summing the sieve bound with the marks, using (12.20)–(12.21) and the negligible tail in Lemma 12.2, gives
where C_s,c_s>0 are absolute; the displayed bound on \e^{-H} permits c_s=1/20, after enlarging C_s. The remainders are o(X_0/L) by taking A>2 in (12.13).
Lemma 12.4 (Mass conditioned on a divisor). For any fixed b\le\gamma<\gamma'\le1/2-\kappa,
Proof. For large x, the primes n in this sum exceed W and all group primes, and do not divide M. Fix r\le x^\eps and condition the family in A_r on n\mid crQ+1. For a squarefree product \ell of odd primes at most W, the intersection count is
This includes \ell=1, as allowed in Lemma 11.1. Apply that lemma with main mass A_r/(n-1) and H=2\lceil(\log L)^2\rceil. Its relative error tends to zero, and its remainder moduli satisfy
The map (n,\ell)\mapsto n\ell is injective, since n is the unique prime factor exceeding W. Thus the total remainder after summing n and the marked r’s is bounded by a single copy of (12.13).
The sieve product is (1+o(1))V_M(W) uniformly in r, and Mertens’ theorem gives
By (12.8), Lemma 12.2, and (12.15), for every fixed A>0 we have
This evaluates the contribution of r\le x^\eps with the exact main mass X_0; taking A large in (12.13) makes the remainders o(X_0V(W)). Finally, each supported d has at most \log(2x)/(bL)=O(1/b) distinct prime divisors above x^b. Consequently the harmonic tail bounds the omitted r>x^\eps contribution even after summing over n. This proves (12.23).
12.4 Removing composites by their least prime factor
We next pass from S_b to
For a bin (\gamma,\gamma']\subseteq(b,1/2-\kappa], with the first left endpoint allowed to equal b, define
If x^\gamma<P^-(d)\le x^{\gamma'}, taking n=P^-(d) and m=d/n shows that its weight is included in this sum. This remains true when n^2\mid d.
Partition both factors into disjoint half-open dyads with endpoints powers of two, retaining pairs that meet the smooth product support and the range of n. On that support, with t=\log n/L,
Every retained full m-dyad therefore satisfies (12.6) with
for large x; in particular s_->\gamma. Both dyad scales are at least x^{b/2}, and their product is comparable to x. There are O(L) such pairs, since each n-dyad allows only a bounded number of m-dyads.
Apply Lemma 12.1 in every pair with \delta=b/2 and with b_n the prime indicator restricted to its dyad and bin. These coefficients are W-rough. The total error is O(xL^{-5})=o(X_0/L), by (12.15) and J_0\ge1. Hence
Joint continuity of the density on the fixed compact exponent range gives, on the support of the summand,
Since n>W, the remaining roughness condition is precisely P^-(mn)>W. Lemma 12.4 thus gives
The density estimate in Lemma 11.2 is normalized to match this expression:
for sufficiently small fixed b. On b\le\gamma\le t\le1/2-\kappa, the point (\gamma,1-t) stays a fixed distance from s=\gamma. Uniform continuity and the finite mass of dt/t therefore give a sufficiently fine fixed partition
such that
Subtracting (12.24) for these finitely many bins from (12.22) proves
The unit term in (12.25) is what leaves a positive mass after this subtraction. It remains to remove the composites whose two prime factors are both close to \sqrt x.
12.5 Composites with two nearly equal prime factors
A composite counted by U_\kappa has exactly two prime factors with multiplicity: three factors greater than x^{1/2-\kappa} would have product exceeding 2x. Each factor lies in [x^{1/2-\kappa},2x^{1/2+\kappa}]. Cover the ordered factor pairs by dyadic boxes
that meet this range in each coordinate and admit a product in [x,2x]. They satisfy x/4\le M_1M_2\le2x, and their full coordinate intervals lie in [x^{0.46},x^{0.54}] for large x. There are at most C_{\rm dyad}(\kappa L+1) boxes, where C_{\rm dyad} is absolute: this bounds the number of first coordinate dyads, and the product condition leaves only a bounded number of second coordinate dyads for each.
The rough-pair bound holds on a slightly wider fixed range of boxes, which includes all the boxes just selected.
Lemma 12.5 (An upper bound for rough pairs). Let \mathcal B=[M_1,2M_1)\times[M_2,2M_2) be any dyadic box such that
Then
where C_1 depends at most on M,\Psi, and is independent of K,\kappa,b,\eps.
Proof. Fix a contributing group integer r\le x^{0.11}, and set k_r=cr, z'=x^{0.002}. The corresponding pairs satisfy mn\equiv1\pmod{k_r}. At every odd prime j\le z' with j\nmid r, they avoid
Indeed, mn-1=k_rQ with Q>x^{0.9}>z', while both factors are W-rough. We impose no further condition at primes dividing k_r and may omit the fixed progression for this upper bound.
For a squarefree product \ell of the designated odd primes, put
The conditions mn=1 and mn=0 modulo j are disjoint and have respectively j-1 and 2j-1 residue pairs. There are \varphi(k_r) pairs with product one modulo k_r. The Chinese remainder theorem therefore gives exactly \varphi(k_r)\rho(\ell) pairs modulo k_r\ell satisfying the base congruence and all the forbidden conditions at primes dividing \ell. Their count in the box is
where
This formula also holds at \ell=1.
The local bounds g(j)\le7/9 and g(j)\le3/j verify the gap and block-sum hypotheses of Lemma 11.1, with absolute constants. Its upper bound with H=2 uses \ell\le(z')^{10}=x^{0.02}. For squarefree \ell,
where \tau is the divisor function. Since \varphi(k_r)\le k_r and k_r=cr\le4r, (12.31) gives
The needed divisor sum follows elementarily. Let \tau_4(n) count ordered decompositions of n into four positive factors. Prime by prime, \tau(n)^2\le\tau_4(n); summing over the first three factors therefore yields, for Y\ge2,
Consequently, taking Y=x^{0.02} and using M_1+M_2\le4x^{0.54} from (12.28), we obtain
uniformly for r\le x^{0.11}. The ratios of the two displayed power terms to x/r are at most x^{-0.33} and x^{-0.74}, respectively, and absorb the logarithmic factor.
For j\le W, the surviving sieve factor is
and for W<j\le z' it is
The correction products are bounded. Omitting the primes dividing r costs a bounded factor by (12.21), and omitting two costs a fixed factor. Thus
As X'\ll x/r, the upper sieve and (12.32) bound the pair count at this r by O((x/r)V(W)^2V(z')), uniformly in all the marks. Bound \Psi by one, multiply by \mathcal W(r), and sum. The result is
Finally V(z')\asymp L^{-1}, with fixed comparison constants, and (12.15) proves (12.29). Only J_0, rather than the pointwise bound B_K, entered this calculation, so C_1 has the claimed independence.
Return now to the boxes selected in (12.27), whose full coordinate intervals lie in [x^{0.46},x^{0.54}]. We replace the prime indicators in each of these boxes. Apply Lemma 12.1 with
In this range R_{0.4}(m) is exactly the prime indicator, since 2(0.4)>0.54, and D_{0.4}(s)=1/s. Its comparison coefficient is therefore
The identity
allows two successive applications, interchanging the variables in the second. The companion coefficient is W-rough and bounded by one for large x in both cases. All coefficients retain their box restrictions. The total error over the boxes is O(xL^{-5})=o(X_0/L).
Since \log m,\log n\ge0.46L, Lemma 12.5 bounds the resulting sum in one box by (0.46)^{-2}C_1X_0/L^2. The ordered prime-pair sum includes prime squares and may count other semiprimes twice, which is harmless for an upper bound. Summing the boxes gives
where C_2=(0.46)^{-2}C_1C_{\rm dyad} depends at most on M,\Psi, and is independent of K,\kappa,b,\eps.
12.6 Choice of parameters and completion
Completion of the proof of Proposition 2.1. Fix M,c,u,\Psi first, and hence the constants \mathfrak S_M and C_2. Choose 0<\kappa<0.01 such that
and then choose 0<\eps<\kappa. Choose b>0 sufficiently small for the sieve and density estimates, and so that
Fix the partition used in the least-factor subtraction. Only finitely many Type II parameter sets now occur. Choose K sufficiently large for all of them, and choose the exponents afterward, for example
This order is permitted by Lemma 10.2 and Proposition 10.3, hence by Lemma 12.1.
Every supported prime is counted by U_\kappa. Subtracting (12.33) from (12.26), and then taking x sufficiently large for the fixed parameters, gives
By (12.15), J_0\ge1, and w\le B_K, the number of distinct supported primes is therefore \gg_{M,c,u}x/L^2. Their support in (12.2) ensures x<d<2x, d\equiv u\pmod M, and d=crQ+1 with Q>x^{0.9} prime. Every prime factor of r belongs to a group in (10.2), hence lies in (\exp(L^{0.1}),\exp(L^{0.3})) for all sufficiently large x. These are precisely the assertions of the proposition.
A The cubic theta normalization
This appendix supplies the Kazhdan–Patterson specialization and normalization behind Proposition 5.1. We first specialize their cover, inducing datum, and global representation, and derive the normalized Whittaker factorization. We then verify the exact lift identities and compute the spherical Jacquet integral in the sections used in the paper. The local calculation fixes the Gauss phase and the unit factor together, and includes the modular factor in the triple-valuation step. The main argument uses the resulting global Fourier expansion with normalized local factors and the exact local rules for the Weyl calculation.
A.1 The selected cover and global representation
Denote the cubic Hilbert symbol at a place v by (\,\cdot\,,\,\cdot\,)_{3,v}. In the convention of Kazhdan and Patterson, the restriction of their determinant-modified cocycle to the diagonal torus is
The root groups have their canonical splittings. These conventions are those of [15, Section 0.1, pp. 39–44].
For the n-fold cover of \GL_d, where d is the rank parameter, the center criterion of [15, Proposition 0.1.1, p. 42] requires z^{d-1+2dc}\in F_v^{\times n} for a scalar zI. For d=2,n=3 the exponent is 1+4c, which is zero modulo three exactly when c=2. Moreover, (A.1) on two scalars is
Thus their full preimage is central and the ordinary scalar section splits. Quotienting its image gives the asserted local covers of \PGL_2. The adelic cover and its rational splitting are obtained by the reciprocity construction in [15, Section 0.2]. The ordinary scalar section agrees with that splitting on rational scalars, so the quotient retains a rational splitting.
We check the datum in (5.1). A Hilbert symbol with a cube argument is one; the cocycle on scalar pairs is also one. Consequently the cube and scalar sections commute and multiply without any remaining cocycle. If an element has two presentations in (5.1), their two cube coordinates differ by a common scalar cube, so the displayed absolute-value ratio agrees. The formula is therefore a genuine character, trivial on the split scalar subgroup. On a coroot cube it gives
This is exactly the exceptional condition of [15, Section I.2, p. 71]. The exceptional condition alone would only specify the square of the value on a(\varpi^3); the datum in the main text chooses the pure absolute-value value.
Use the idele norm in (5.1) to define the global character. A rational diagonal lying in the adelic cube torus times scalars has a rational ratio which is a cube at every completion. If that ratio were not a cube in F, its nontrivial Kummer extension would have a nonsplit unramified prime by Chebotarev. Thus the ratio is a rational cube. The product formula now shows that our character is trivial on the intersection with the rational lift. This is the compatibility hypothesis in [15, Section II.1, p. 107, immediately after Lemma II.1.1]. Extending by the trivial character on rational lifts gives the character on the rational maximal abelian subgroup of that lemma. As in [15, Section II.1, pp. 107–109], one chooses an extension on the product of local maximal abelian subgroups which agrees with the former character on their intersection. The character on the scalar center is unitary.
At finite places, the local quotient and subrepresentation assertions are [15, Theorem I.2.9(a),(b), p. 72]. At tame finite places, part (f) of that theorem supplies the spherical vector for the normalized unramified datum. For a tame place, the uniqueness criterion is
by [15, Corollary I.3.6, p. 79]. It applies to (d,n,c)=(2,3,2). The residue construction and restricted tensor product are [15, Theorem II.2.1, p. 118]. Since the tame local Whittaker spaces have dimension one, [15, Theorem II.2.5(a), p. 122] gives a nonzero global Whittaker functional, and part (b) gives local uniqueness at every place. At the complex places this also follows directly from [15, Theorem I.6.5(c), p. 105]. The inducing spaces, their exceptional images, and the automorphic residue all inherit the scalar action of \omega. That action is trivial, so these representations descend to the scalar quotients.
On the quotient torus the section of a(y) has cocycle (x,y)_{3,v}^{2}. Since -1 is a cube in F, this is an alternating bimultiplicative law. At a tame place it is trivial on pairs of units, and its value on a uniformizer and a unit is the cubic residue symbol or its inverse, with a common orientation at all places. Choose the embedding \iota with the orientation giving (5.5). The canonical compact splitting agrees with the sections of unit diagonals, Weyl elements, and integral upper root elements by [15, Proposition 0.1.3, p. 44]. Conjugating by the Weyl element gives the same agreement for the lower root group. [15, Corollary I.3.4, p. 78] gives relations for the coefficient functions defining local Whittaker functionals. [15, Theorems I.4.2–I.4.3, p. 87] convert them to spherical Whittaker values with the modular factor and give the central step. The direct local calculation below proves (5.6) and the lift identities (5.4) in these conventions.
At a complex place the Hilbert symbol and the cover are trivial. [15, Theorem I.6.4(c), p. 104] identifies the exceptional representation with the ordinary principal series when d\le n. Our cube datum fixes the characters on all of \C^\times, because every complex number has a cube root. Triviality on scalars removes the determinant character; equivalently any possible remaining finite-order continuous character of the connected group \C^\times is trivial. This gives the characters in part (iii) of Proposition 5.1. Their Bessel function is computed in the main text from the Jacquet integral. Finally, the two Bruhat terms in the constant term of the Eisenstein series are described by [15, Proposition II.1.2, p. 110]. At the exceptional residue the surviving term belongs to the Weyl-conjugate induction, as in [15, Section II.1, p. 114]. Its complex inducing character, including the modular factor, is |y|_v^{1/3}; in particular it has no angular part.
A.2 The normalized global Whittaker expansion
Let \lambda be the nonzero global functional just obtained. The restricted tensor product is spanned by pure tensors, so there is a pure tensor \varphi^\circ with \lambda(\varphi^\circ)\ne0. Its nondistinguished finite places form a finite set. The final S contains this set; the arithmetic data are constructed only after that final choice. For v\notin S, varying only the v-component of \varphi^\circ gives a nonzero local Whittaker functional. Normalize it at \varphi_v^\circ. If other finitely many components have already been varied, the resulting partial functional at v is still a Whittaker functional, and local uniqueness makes it a scalar multiple of the normalized one. Evaluating at \varphi_v^\circ identifies that scalar. Induction on the number of varied components proves (5.2). The same argument factors the complex components.
At a finite product of remaining places the functional descends to the tensor product of the local twisted Jacquet quotients. The twisted Jacquet quotient of the principal series is finite dimensional by [15, Lemma I.3.2, p. 75]. Since \Theta_v is its quotient by Theorem I.2.9(b), its twisted Jacquet quotient is a quotient of that finite-dimensional space. A functional on the finite tensor product is therefore a finite sum of products. This is the finite-S assertion in Proposition 5.1.
This normalization accounts for the correction to the original global formula. The erratum withdraws an auxiliary spherical-functional vanishing assertion on pp. 119–120. It replaces stabilization of the auxiliary coefficient C_S by stabilization of C_ST(S), where in rank two
The sums in Theorems II.2.2–II.2.3 become finite-S limits with the factor T(S)^{-1}, and the same change applies on p. 130 [16, correction to Theorems II.2.2–II.2.3]. The preceding argument begins with the nonzero automorphic functional and normalizes each exterior local functional at the distinguished spherical vector. It proves the product (5.2) without the withdrawn assertion or an unnormalized infinite product.
The rational splitting identifies every nonconstant Fourier coefficient with this Whittaker function. Conjugate n(x) by the rational lift of a(\nu) and change x to \nu x; the adelic Jacobian is one. This gives (5.3). On a fixed compact set of g, smoothness at infinity and invariance under a fixed finite open subgroup reduce the expansion along F\backslash\Adele_F to the smooth Fourier expansion on a compact real torus. Integration by parts in sufficiently many archimedean directions gives absolute local uniform convergence there.
A.3 Sections and the spherical induced vector
Fix an exterior finite place \mathfrak p. In this appendix write \mathcal O=\mathcal O_{\mathfrak p}, let \varpi be the chosen uniformizer, put Q=q_{\mathfrak p}, and abbreviate \psi=\psi_{\mathfrak p}, \chi=\chi_{\mathfrak p}, and \mathcal C=\mathcal C_{\mathfrak p}. Thus \psi has conductor \mathcal O, additive measure gives \mathcal O volume one, and \chi is the sextic residue character on \mathcal O^\times. In all finite sums it is extended by zero to the zero residue, including when its exponent is zero. The embedding of the cubic symbol has been chosen so that the effective torus cocycle is
The last two identities follow from tameness and alternation. In particular pure uniformizer powers multiply without a cocycle. All symbols with -1 are one because -1\in F^{\times6}.
We first record explicitly why the ordinary rank-one decompositions are valid for these lifts. See Kubota [18, p. 114] for the local rank-one construction. In the convention of [15, Section 0.1, p. 41], the determinant-modified formula is
We take c=2 and then quotient the split scalar subgroup. Substitution in this formula gives
on the quotient. Before quotienting, the second equality has the extra scalar yI. All its cocycles are symbols between powers of y and -1, hence are one. For H\ne0, the matrix identity
also lifts exactly. The product of the first two factors has cocycle one, and multiplication by the last factor has cocycle (H^{-1},-H)_3=1. The middle diagonal is the product of the split scalar HI and a(-H^{-2}), again without a cocycle. Thus
on the quotient. If H^{-1}\in\mathcal O, the last factor is in the compact splitting. The latter agrees with the sections on unit diagonals, w, and integral upper root elements by [15, Proposition 0.1.3, p. 44]; conjugation by w gives agreement on the lower root group. The diagonal a(-1) can therefore be absorbed in a compact factor in any spherical calculation.
Let \sigma^- be the unramified representation of the covering torus with central character the Weyl conjugate of (5.1). Its norm exponent on a(y) is -1/6. Modulo scalars, the subgroup of torus elements of valuation divisible by three is maximal abelian: the tame group F_{\mathfrak p}^\times/F_{\mathfrak p}^{\times3} has one valuation and one unit coordinate over \Z/3\Z, and the commutator pairing pairs those coordinates nondegenerately. The torus construction in [15, Section 0.3, pp. 54–55, and Section I.1, p. 58] realizes \sigma^- by induction from a character of this subgroup. Its index is three, so the unramified Heisenberg representation has dimension three. The canonical unramified extension is trivial on unit lifts [15, Section I.1, p. 60]; the induced basis vector supported on that subgroup is therefore unit fixed. Call it e_0, and set
The central element a(\varpi^3) acts by Q^{1/2}, so e_{j+3}=e_j. These three vectors form a basis. From (A.2) and unit invariance one obtains
Use the vector-valued realization of normalized induction
Its spherical vector f^\circ is determined by f^\circ(k)=e_0 on the compact subgroup. The total norm exponent in this transformation law is 1/2-1/6=1/3. The compact-invariant line of this normalized unramified induction is one dimensional by [15, Lemma I.1.3, p. 60]. By [15, Theorem I.2.9(a),(f), p. 72], the local exceptional representation is the irreducible subrepresentation of I^- and has a nonzero compact-invariant vector. That vector is a scalar multiple of f^\circ, so the subrepresentation contains f^\circ.
A.4 The shells of the Jacquet integral
For a smooth f\in I^-, define the vector Jacquet functional by
This limit stabilizes. Indeed, when |x| is sufficiently large, the right invariance of f makes \bar n(x^{-1}) fix f. The lifted Bruhat formula then expresses f(wn(x)) as
Choose m so that 1+\varpi^m\mathcal O consists of cubes and its unit lifts fix the vector f(1) in the smooth torus representation. On a shell x=\varpi^{-l}v, the displayed expression is unchanged when v is replaced by vu, u\in1+\varpi^m\mathcal O: the norm is unchanged, all cocycles with u are trivial, and a(u^{-2}) fixes f(1). Each such unit coset is an additive coset of \varpi^m\mathcal O. For l>m, the character \psi(\varpi^{-l}v) has zero integral on it. All sufficiently large shells therefore vanish. This argument applies to every smooth f, including each of its right translates.
If t\in F_{\mathfrak p}, choose a ball large enough to contain t and to realize the stable values for both f and \pi(n(t))f. Translation x\mapsto x+t preserves that ball and gives \mathscr J(\pi(n(t))f)=\psi(t)\mathscr J(f). Thus each scalar coordinate of \mathscr J is a Whittaker functional. This is a direct stable version, at the present inducing parameter, of the Jacquet functionals introduced in [15, Section I.3, pp. 74–75].
For e\ge0 put J_e=\mathscr J(\pi(a(\varpi^e))f^\circ); explicitly,
The integral retains the stable interpretation just established. We now evaluate it shell by shell.
Put x=\varpi^eH and use the first two lift identities above. The measure contributes Q^{-e}, the modular factor of a(\varpi^{-e}) contributes Q^{e/2}, and its torus norm factor contributes Q^{-e/6}. Thus every shell has the common factor
If H\in\mathcal O, then wn(H) belongs to the compact subgroup, and \psi(\varpi^eH)=1. This region contributes e_{-e} inside the common factor.
Suppose instead that H=\varpi^{-l}v, with l\ge1 and v\in\mathcal O^\times. In (A.3) the last factor is compact, while the first acts trivially in the inducing transformation law. The modular factor of a(-H^{-2}) is Q^{-l}, and its torus norm factor is Q^{l/3}. Furthermore,
The minus sign in -H^{-2} has no effect. Multiplication by a(\varpi^{-e}) changes the basis index by -e, without a cocycle between the pure uniformizer powers. Finally \dd H=Q^l\,\dd v. It follows that this shell contributes, inside (A.7),
This accounts separately for the measure, modular, norm, and Hilbert-symbol factors.
Define the normalized prime Gauss sums
The powers in these sums retain the unit mask. Since \chi(-1)=1, replacing \psi(x/\varpi) by its complex conjugate has no effect on g_j. For j\not\equiv0\pmod6, the usual finite-field Gauss identity gives
In particular g_2g_4=1. These identities follow, for example, by expanding the product of the two sums, substituting x=ty, and using the additive sum in y; the result is Q\chi(-1)^j before division by Q.
Write I_{e,l} for the integral in (A.8). The character there is \chi^{-2l}, since -8l\equiv-2l\pmod6. Its exact values are
Indeed, for l\le e the additive character is trivial, so unit orthogonality gives the first two cases. For l=e+1, integration over the residue classes gives Q^{-1} times the corresponding prime Gauss sum; the trivial character with its unit mask gives \sum_{x\ne0}\psi(x/\varpi)=-1. For l>e+1, split each unit class modulo \varpi^{l-e-1} into its Q lifts modulo \varpi^{l-e}. The multiplicative character is unchanged on these lifts, whereas the additive factors sum to \sum_{t\bmod\mathfrak p}\psi(t/\varpi)=0. This also proves the asserted stabilization of (A.6).
We can now sum the shells. For k\ge0, the integral unit ball and the shells l=3,6,\ldots,3k have the same basis index and combined coefficient
For e=3k, the terminal shell l=3k+1 has coefficient Q^{k-1/6}g_4 and basis vector e_2. For e=3k+1, the terminal shell l=3k+2 has coefficient Q^{k+1/6}g_2 and basis vector e_0. For e=3k+2, the terminal shell l=3k+3 has coefficient -Q^k and the same basis vector e_1 as the preceding combined contribution. Multiplying by (A.7) therefore gives
The second equality uses g_2g_4=1.
In particular J_0=e_0+Q^{-1/6}g_4e_2\ne0. Choose a scalar coordinate nonzero on J_0. Its Jacquet functional is nonzero on the exceptional subrepresentation, since that subrepresentation contains f^\circ. The local Whittaker space is one dimensional by [15, Corollary I.3.6, p. 79]. Hence every spherical Whittaker function normalized at one has the ratios in (A.10):
The middle term is Q^{-k-1}\sum_x\chi(x)^2\psi(x/\varpi), as claimed. This computation agrees with the central-step statement of [15, Theorem I.4.3, p. 87]: on a(\varpi^3) the Weyl-conjugate inducing character is Q^{1/2} and the modular half-character is Q^{-3/2}, so their product is Q^{-1}.
A.5 Negative valuations and the unit normalization
For e<0, choose x\in\mathcal O for which \psi(\varpi^e x)\ne1. The spherical vector is fixed by n(x), while the split root relation and Whittaker equivariance give
Thus the value is zero. For a unit u, the compact section of a(u) fixes the spherical vector. The torus law gives
where \mathcal C includes the scalar action on a genuine vector. Consequently
Together with (A.11), this proves (5.6).
There is a useful check that fixes the orientation without referring to a convention for the name of a cubic Gauss sum. At a tame place the residue character is unchanged when \varpi is replaced by \varpi u. Substitution x=uv in the residue sum yields
This is exactly the e=1 transformation in (A.12). Thus the Gauss phase and the unit factor are a single compatible normalization. Changing the sign of an additive Gauss argument is harmless because \chi(-1)=1.
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