The Quasi-Riemann Hypothesis (alternate 11/12 proof)

Proof of Theorem 1.1 from Proposition 3.1. Fix 0<\vartheta\le1/10, and put H=D^{1+\vartheta} and Y=H^{1/6}=D^{(1+\vartheta)/6}. For prime ideals Y/2<\NK(\mathfrak p)\le Y, \mathfrak p\notin S, let p be their primary generators, chosen with p\equiv1\pmod3. Then \chi_n(p^6)=\ind_{\mathfrak p\nmid n} and therefore

For this fixed field, Landau’s prime ideal theorem [27] (see [22, Theorem 5.33]) gives J\asymp Y/\log Y such primes. Their sixth powers are distinct rows of norm at most H. Apply (3.1) with loss \varepsilon/2 and use \log Y\ll_\varepsilon D^{\varepsilon/2} to obtain

Thus A_1(D)\ll_{\nu,S,W,\vartheta,\varepsilon} D^{11/12+5\vartheta/12+\varepsilon}. Given any requested exponent loss, choose \vartheta>0 and then the loss in Proposition 3.1 sufficiently small. Renaming the resulting loss \varepsilon, we obtain

We now use (3.3) to rule out a zero of L_K(s,\nu) in \Rea s>11/12. Suppose such a zero \varrho exists. Choose 0\ne\phi\in C_c^\infty((1,2)), \phi\ge0, and W(y)=y^{-\varrho}\phi(y). With the Mellin convention \widehat W(s)=\int_0^\infty W(y)y^s\dd y/y, we have \widehat W(\varrho)=\int\phi(y)\dd y/y>0. The function

is holomorphic on \Rea s>11/12: the estimate (3.3) controls the integral at infinity, and the compact support of W makes A_1(D) vanish for sufficiently small D. Termwise integration for \Rea s>1 gives

where L_K^S is the Euler product outside S:

By Hecke’s meromorphic continuation theorem [21] (see [22, §5.10]) and the identity theorem, the identity L_K^S(s,\nu)\mathcal M_W(s)=\widehat W(s) holds on \Rea s>11/12. The omitted Euler factors are nonzero here, so evaluation at s=\varrho gives 0=\widehat W(\varrho)>0, a contradiction.

To deduce the assertion for Dirichlet L-functions in Theorem 1.1, let \chi_{-3} be the nontrivial character modulo 3. For any Dirichlet character \chi, quadratic base change gives, up to Euler factors nonzero in \Rea s>0,

The Hecke conclusion excludes zeros of either factor away from s=1. At s=1, the only possible pole–zero cancellation is ruled out by Dirichlet’s nonvanishing theorem, which gives L(1,\chi_{-3})>0 (see [6, Chapters 4 and 6]).□

Proof. Inclusion–exclusion followed by k=d\ell gives

Apply lattice Poisson summation [22, Theorem 4.5] in \ell on residue classes modulo \mathfrak m, at scale \mathcal H/\NK(d). The primitive Gauss-sum identity [22, (3.12)] evaluates the finite Fourier transform as \sqrt{\NK(\mathfrak m)}\gamma(\chi)\overline{\chi(h)} for every h, giving (4.8). For a nonprincipal primitive character, \overline{\chi(0)}=0, so the zero-frequency term vanishes. For the principal character, sum \mu(d)/\NK(d) over d\mid\rad \mathfrak r to obtain the displayed zero-frequency contribution.□

Proof. Let C_{\mathfrak r}(X;k,f) denote the inner column sum defining \E_{\mathfrak r}, with \xi,W fixed. Inclusion–exclusion, n=dm, and (4.6) give

Indeed, \chi_m(d)^4 enforces (m,d)=1 and combines with \chi_m(f)^4; terms with (d,f)\ne1 vanish. Each exterior coefficient has modulus at most one. Apply Cauchy–Schwarz in d, then enlarge the injective image f\mapsto df to the squarefree range F\NK(d)\le\NK(df)<2F\NK(d). The normalizing product is unchanged: (X/\NK(d))(F\NK(d))=XF.□

Proof. Write I=[a_0,b_0]. Choose a nonnegative radial Schwartz weight \Phi such that \Phi(x)\ge1 on [0,1] and \operatorname{supp}\widehat\Phi\subset[0,C_\Phi] for some fixed C_\Phi>0. It suffices to prove \mathcal M_D\ll_{\nu,S,I,\vartheta,\varepsilon} \|W\|_{C^{J'}(I)}^2HD^\varepsilon for some J'=J'(\vartheta,\varepsilon), where

Conjugate the inner sum, expand the square, and put g=(n_1,n_2) and n_j=gz_j. The squarefree indices satisfy (z_1,z_2)=(z_1z_2,g)=1, and

Apply Lemma 4.2 with \chi=\overline{\chi_{z_1}}\chi_{z_2} and exclusion g:

The zero frequency occurs only when z_1=z_2=1 and contributes

Retain all h\ne0, including those with z_1=z_2=1.

Expand the fixed ray class function \overline{\nu(t)}G(t^{-1})=\sum_\xi c_\xi\xi(t). The Chinese remainder theorem and (4.5)–(4.7) give

Write N=\NK for the remainder of this proof, and put W_0(x)=x^{-1/2}\overline{W(x)}. Substituting (4.15) into (4.14), including the normalization 1/D in (4.13), gives

since the character sum is fixed and finite. Fix \xi. Using \overline{\chi_z(e)}=\chi_z(e^5), its contribution is

Here and below every starred variable is squarefree, primary, and prime to S, while h,k range over nonzero elements of \OO. We must show |\mathcal S_\xi|\ll_{\nu,S,I,\vartheta,\varepsilon} \|W\|_{C^{J'}(I)}^2HD^\varepsilon.

Insert the coprimality identity and change variables:

By (4.6), the common factor from the two columns satisfies

where (v,e)=1. Thus the same sum becomes

There is now no restriction (m_1,m_2)=1. Make the bijective change of variables

The resulting b,f are coprime and squarefree, k\ne0 is arbitrary, and

Consequently,

The characters enforce (m_j,f)=1, leaving only the displayed exclusion (m_j,b)=1. The supports of W_0 and \widehat\Phi give

Partition B\le N(b)<2B and F\le N(f)<2F into dyadic ranges. Denote the corresponding contribution to (4.16) by \mathcal S_{\xi;B,F}, and put

These satisfy (4.11). In the variables x_j=N(m_j)/X, the coupled smooth weight is

For every integer q\ge0, these kernels satisfy

For U\in C_c^\infty(I_*), where I_*=[a_0/8,2b_0], put

The shifts (X,F)\mapsto(X/N(d),FN(d)) preserve (4.11). Thus Lemma 4.4 and (4.12), with J=J(\vartheta,\varepsilon/4), give

Here N(b)\ll_I D, so the divisor factors are absorbed in D^{\varepsilon/4}. We use the following smooth-weight principle, stated and proved in Lemma B.2: a mean-square bound valid for every common test function, with a C^J norm, also bounds the corresponding quadratic sum with a kernel depending on the row, at a cost given by the kernel’s C^{2J+4} norm. For each fixed b, apply Lemma B.2 with row index (f,k), coefficient \mu(f)\ind_{(f,b)=1}, and kernel \mathcal K_{b,f,k}. The larger row range 0<N(k)\le\mathcal H adds only terms whose original kernel vanishes. With q=2J+4, this gives

where XF=D/B and ideal counting gives O(B) choices of b. Summing the O((\log D)^2) nonempty dyadic ranges bounds \mathcal S_\xi as required. Sum over the fixed finite set of \xi, include Z, and restore the factor D to obtain (3.1). The required derivative order depends only on \vartheta,\varepsilon.□

Proof. Write N=\NK and I=[u,v]. Complete multiplicativity in (5.3), including at the zeros of \Psi_k, gives

Indeed, after substitution of (5.3), the coefficient of the total cube index b contains \sum_{h\mid b}\mu(h)=\ind_{b=1}. This is Möbius inversion; compare [22, (1.18)] and [7, (8.2)].

Split the right-hand side of (5.8) into P_{\rm short}(k,f)+P_{\rm long}(k,f), where

Define \E_{\rm long} in the same way using P_{\rm long}. Weighted Cauchy–Schwarz for each k,f, followed by Proposition 5.2, gives

Here X/N(h)^3\ge1, and the reciprocal-norm sum contributes only a logarithm. We use Proposition 5.2 with exponent \varepsilon/4 and range C_0+1, since N(f)<2D^{C_0}. If H_c<1, the sum is empty.

Expand T using (5.3) in the remaining terms, with cube index c:

Put b=hc; the pairs giving b are exactly (h,b/h) with h\mid b and N(h)>H_c. Since \Psi_k(b)^3=\xi(b)^3\chi_b(k)^3\ind_{(b,f)=1}, this gives

The divisor function \tau_{\mathrm{div}}(b) counts ideal divisors. All these indices are primary and prime to S; b need not be squarefree. The support of W restricts the sum to N(b)^3\le vX. Weighted Cauchy–Schwarz now gives

The reciprocal divisor sum is \ll_I\log^2(2+X), since

If L_b\le1, the column sum is empty unless L_b\ge1/v; otherwise it has O_I(1) terms, so

Absorb the logarithms in D^\varepsilon and combine the two parts using \E\le2\E_{\rm short}+2\E_{\rm long}. This proves (5.7). If \mathcal H^2\le X, then H_c=X^{1/3}, so N(b)>H_c implies L_b<1.□

Proof of Proposition 5.1. Fix \kappa>0 and C_0\ge1. We prove by induction on j\ge0 that the proposition holds under the additional restriction \mathcal H\le D^{j\kappa}, with the derivative order and implied constant allowed to depend on j. More precisely, for every \varepsilon>0 there is an integer J=J(j,\kappa,C_0,\varepsilon)\ge1 such that

for every compact interval I=[u,v]\subset(0,\infty) and W\in C_c^\infty(I), under the hypotheses of the proposition. The derivative order is independent of I; this allows us to apply the induction hypothesis on the enlarged interval in the transfer estimate. For the base case j=0, we have \mathcal H\le1. In (4.9), the support of W restricts n to the O_I(X) ideals with \NK(n)\in[uX,vX]. The factors a_\xi(n), \chi_n(k), and \chi_n(f)^4 have absolute value at most one, so the triangle inequality bounds each inner sum by O_I(X\|W\|_\infty). There are O(F) choices of f and O(\mathcal H) choices of k. Consequently,

Dividing by \Sigma=XF and using \mathcal H\le1 and F\ge1 proves the assertion for j=0, with J=1.

Suppose the assertion holds for j, and fix \varepsilon>0. Let m be the derivative order supplied by the induction hypothesis with exponent \varepsilon/3. For \mathcal H\le D^{(j+1)\kappa}, apply Lemma 5.3 with exponent \varepsilon/3. We must bound the mean squares in its supremum uniformly in b. If this supremum is nonempty, then \mathcal H^2>X and H_c^3=X^2/\mathcal H^2. For each such b, recall that L_b=X/\NK(b)^3>1. Since L_b\le X and \Sigma\ge\max\{\mathcal H,L_bF\}, Proposition 5.4 applies at column scale L_b, with derivative order m, and exponent \varepsilon/3. Its estimate also bounds \E(\mathcal H,L_b,F;\xi,W) because \E(\mathcal H,L_b,F;\xi,W)\le\mathcal A(W).

Let \mathcal H',X',F' be any parameters in the supremum in (5.11), with \Sigma'=X'F' as in that proposition. By (5.12) and \NK(b)>H_c,

Thus each of these mean squares is covered by the induction hypothesis. Its weight U is supported in [u/16,4v] and has C^m norm at most one. Applying the induction hypothesis on this interval gives

The enlarged interval is determined by I, so the implied constant has only the permitted dependence on the support. Substituting into (5.11) yields, uniformly in b,

Substitution into (5.7) contributes the remaining factor D^{\varepsilon/3}. Choose J at least 4m+12 and at least the derivative order required by Lemma 5.3. We obtain

The choice of J depends only on j,\kappa,C_0,\varepsilon. If the supremum in (5.7) is empty, the same bound follows directly from that lemma. This completes the induction.

Finally, take j=\lceil C_0/\kappa\rceil. The hypothesis \mathcal H\le\Sigma\le D^{C_0} ensures \mathcal H\le D^{j\kappa}, so the induction gives the proposition.□

Proof of Proposition 3.1. Fix 0<\vartheta\le1/10 and put H=D^{1+\vartheta}. The ranges in (4.11) give

For large D, Proposition 5.1 applies with \kappa=\vartheta/2 and C_0=2. If X<1, nonempty support forces X\gg_I1, and counting gives \E\ll_I\mathcal H\|W\|_\infty^2 \ll\Sigma\|W\|_\infty^2; if \mathcal H<1, the sum is empty. Thus (4.12) holds, with derivative order depending only on \vartheta,\varepsilon, and Proposition 4.5 proves (3.1). Bounded D is again covered by counting.□

Proof. The coefficient formula (2.10) gives

Substitution into (5.3), using V_*(y)=\sqrt y\,W(y), proves (6.3). Finite Fourier inversion gives

Multiplication of the \ellth Fourier mode by e(\lambda^3\ell h/q) translates z to z+\lambda^2h/q. The second identity cancels the constant term, proving (6.4).□

Proof. This is Goldmakher and Louvel’s quadratic large sieve [13, Theorem 1.1], after fixing the product of the prime factors of n lying in S, and finitely many ray classes. For completeness, let e_k\in\{0,1\} according as \NK(k)\equiv1,3\pmod4, and let \kappa_\lambda be the nontrivial character modulo \lambda. The character x\mapsto(x/k)_2\kappa_\lambda(x)^{e_k} is trivial on units and has primitive conductor k\lambda^{e_k}. When e_k=0, the factor \kappa_\lambda(x)^{e_k} is omitted. Classes modulo 24\OO fix the supplementary characters and reciprocity factors; for coprime k_1,k_2 in the same class, the product character has conductor k_1k_2. These are the hypotheses in [13, Definition 1 and §2].□

Proof. By homogeneity assume \|W\|_{C^J(I)}\le1, with J chosen below. Write

Fix t; then k_0 is squarefree and primary, with (k_0,g)=1 and (k_0,S)=1. Discard empty ranges, so \mathcal H_0\ge1, and retain these restrictions below. For a coefficient function A(n,b), a positive scale Y, and an integer m\ge-4, put

We shall obtain, after partitioning k_0 into fixed ray classes,

where \mathcal I is independent of k_0, |\mathcal I|\precbd1, |c_{\iota,m}(k_0)|\ll1, and A_{\iota,m} is independent of k_0. For some a_\iota,Y_\iota>0, the coefficients and lengths satisfy

All column restrictions are included by extending A_{\iota,m} by zero. Put k=u_0tk_0. For \mathcal P=\{p\notin S:p\mid kg\}, sextic reciprocity (4.1) gives

Even when j_p=0, the factor \chi_p^0(n)=\ind_{p\nmid n} retains the zero extension at p\mid kg. The fixed factor \Psi_0 contains \xi, the factors at S, the unit factors, and n\mapsto\mathcal R(n,\prod_{p\notin S}p^{v_p(k)}). The fourth power at g contributes no reciprocity sign, and the factors at S depend only on exponents modulo six. Thus \Psi_0 ranges over a fixed finite family. In the notation of Lemma 6.3, take \mathcal P_{\rm fix}=\{p\notin S:p\mid tg\}, so that \mathcal P=\mathcal P_{\rm fix}\sqcup\{p:p\mid k_0\}; the exponents on \mathcal P_{\rm fix} are fixed with t,g. Partitioning k_0 into fixed ray classes fixes \Psi_0 and, by Lemma 6.3, the data d,\psi,c_0 in each transformed term.

Fix one transformed term, with active primes \mathcal A_{\rm fix} away from k_0, and put c_*=c_0\prod_{p\in\mathcal A_{\rm fix}}p. Thus c=c_*k_0 in (6.9). By (6.10), write \ell=u\lambda^mnb^3, with u,m,n,b as there. Every prime dividing k_0 has j_p=1, and hence

Indeed, the transformed exponent is -1-2\equiv3\pmod6 and \chi_p(b)^9=\chi_p(b)^3, also when p\mid b by zero extension. The coefficient bound and rapid decay in Lemma 6.4 give absolute convergence of the transformed series, so we may regroup its terms below. Set

The transformed term is therefore

The representation is unique: away from \lambda, the prime exponents of \ell are 3v_p(b) or 1+3v_p(b). Apart from the quadratic character and weight, the k_0-dependence is a bounded scalar.

Fix u,m and write q=N(p) for an active prime p\nmid k_0. For j_p=4, the local identity is

To display the reindexing in (6.15), let A^{(p)}(n,b) include all the other local factors and set

Then the changes n=pn' and b=pb' give the exact identity

In the second term p\nmid n' preserves squarefreeness; in the third, p\nmid n is retained and b' is unrestricted at p. Both extracted phases have modulus one. The factors q^{1/2} in the local identity cancel against \sqrt{N(pn')} or leave q^{-1/2} after division by N(pb').

Let a track a bound |A(n,b)|\le27a for the coefficient in (6.15), and let Y be its scale. Before inserting the active primes, these are a=1 and Y=N(c_0)^2\mathcal H_0^2/X. Each active prime contributes q^2 to the squared conductor norm. Including this factor, the updates are

Thus a^2 never increases. If p\mid t and p\notin S, then j_p is odd, so a^2Y costs at most N(p)^2. If p\mid g and p\nmid t, the cost is N(p) for j_p=0,4 and N(p)^2 for j_p=2; the latter case requires v_p(g)\ge2. Hence

Reindexing at distinct primes preserves the form and zero extensions. By Lemma 6.3, the transformed terms, units, and at most three choices per prime p\in\mathcal A_{\rm fix} with j_p=4 form an index set of divisor-bounded size in tg. Here and below, divisor-bounded in a means bounded by C\tau_{\mathrm{div}}(a)^A for fixed constants C,A; in particular, this is \ll_\varepsilon\NK(a)^\varepsilon for every \varepsilon>0. For one index \iota, abbreviate a=a_\iota, Y=Y_\iota, and A_m=A_{\iota,m}. The local updates give Y\ll D^{C_1} for some fixed C_1=C_1(C_0). Split b into B\le N(b)<2B, and n by a smooth dyadic partition V(N(n)/U), with U,B\ge1 and uniformly bounded cutoffs. For this part of (6.15), write

Here n,b remain primary. Put z_*=3^mUB^3/Y and fix a decay exponent A>0. With m,U,B,Y fixed, Lemma B.1, applied only in N(n)/U, gives the following representation in a real Mellin variable s:

Indeed, the scale R in (B.2) is 3^mUN(b)^3\mathcal H_0^2/(YN(k_0)^2)\ge z_*. Lemma 6.4 supplies the required derivative bounds, independently of b,k_0,U,B,m. Since \sum_{N(n)\asymp U}N(n)^{-1}\ll1, Lemma 6.5 gives, for any \sigma>0,

Apply weighted Cauchy–Schwarz to the Mellin integral, as recorded in (B.6), with the common majorant in (6.19), then weighted Cauchy–Schwarz in b, using \sum_{B\le N(b)<2B}N(b)^{-1}\ll1:

Since m\ge-4, we have U\le81Yz_* and hence \mathcal H_0U\ll D^{C_0+C_1}(1+z_*). Choose 0<\sigma\le1 small in terms of \varepsilon_1>0,C_0 and take A=3. Taking square roots in (6.20) gives

The infinite dyadic sums converge: for dyadic U,B\ge1 and every Z>0,

These estimates make the sum of \ell^2(k_0) norms finite. The triangle inequality and the geometric sum over m\ge-4 therefore give

Combine the divisor-bounded choices of \iota and use (6.16) to obtain

Sum the fixed ray classes and the divisor-bounded choices t\mid\rad(g\prod_{\mathfrak p\in S}\mathfrak p), choosing \varepsilon_1 and the other small-power losses in terms of \varepsilon. The weight estimates above require a fixed number J=J(\varepsilon,C_0) of derivatives. Homogeneity restores \|W\|_{C^J(I)}^2 and proves (6.14).□

Proof of Proposition 5.2. Write uniquely k=u_0sv^2, with s squarefree and s,v among the chosen ideal generators. No condition (s,v)=1 is imposed. Since 8\equiv2\pmod6, including the zero extensions,

Apply Lemma 6.6 with row bound \mathcal H/N(v)^2 and auxiliary twist g=fv^2. Here N(g)\le\mathcal H N(f)\le D^{2C_0}, so use that lemma with 2C_0. Summing its bounds gives

The last sum is \zeta_K(2)<\infty, where \zeta_K(s)=L_K(s,\mathbf1) is the Dedekind zeta function of K. This proves (5.5).□

Proof. Expand the square defining \mathcal A(W), and write n_i=Cu_i, where C=(n_1,n_2) and (u_1,u_2)=(u_1u_2,C)=1. The row character is \chi_{u_1}\overline{\chi_{u_2}}, primitive modulo u_1u_2, with the extra restriction (k,C)=1. Lemma 4.2 introduces d\mid C and a frequency h. Let Z denote the zero-frequency contribution. It requires u_1=u_2=1, so

For the other frequencies, the Chinese remainder theorem, (4.4), and (4.7) give the paired identity

Expand (\xi G)(z)=\sum_{\xi_1}\ell_{\xi_1}\xi_1(z) on the fixed ray class group. Insert \ind_{(u_1,u_2)=1}=\sum_{t\mid u_1,\ t\mid u_2}\mu(t) and put u_i=tx_i. The inverse is x_i=u_i/t; squarefreeness imposes (x_i,t)=1, but no condition (x_1,x_2)=1 remains. The factors at C,t cancel against their conjugates, leaving \mu(d)\mu(t) and the restrictions (f,Ct)=(h,t)=1. Thus

where, for each integer j_0\ge0,

Here \chi_{x_i}(C)^4 supplies (x_i,C)=1; the other original zeros are supplied by p_{hf^2}(x_i). The displayed frequency range follows from the support of \widehat\Phi.

For fixed C,d,t, the map (f,h)\mapsto y=hf^2 has divisor-bounded multiplicity in y. Since \Sigma/(LF)\ge1, its nonzero image satisfies N(y)\le4C_\Phi v^2L^2F^2N(d)/(\mathcal H N(C)^2)\le Y_{C,d}. Since \Phi\ge1 on [0,1] and is nonnegative,

with the preceding ranges on the left. Apply Lemma B.2 to the full displayed expression for \mathcal A(W)-Z. The kernel bound gives

Together with \mathcal H\le\Sigma, this proves the result after allocating the divisor losses within \varepsilon_0.□

Proof. Expand the square, put g=(n_1,n_2) and n_i=gz_i. The nonzero terms satisfy

Apply Lemma 4.2 to this last character, with exclusion g and divisor e\mid g. Its zero frequency occurs exactly when z_1=z_2=1 and gives the displayed Z; counting g gives its bound. For the nonzero frequencies, (4.5)–(4.7) and the Chinese remainder theorem give

Use also

Thus the full nonzero contribution is

All starred ideal indices are prime to S. Insert

Here (w,gCt)=1. Equation (4.6) and the zero-extended characters give

Consequently the preceding full sum becomes

There is no remaining condition (n_1,n_2)=1. The column support gives N(gw)\le b\ell and N(z_1z_2)\le b^2\ell^2/N(g)^2; the support of \widehat\Phi therefore imposes the stated bound on N(h). Substituting (7.5) gives (7.6), with only finitely many terms.□

Proof. Write N=\NK. Apply Lemma 7.2 with input length \ell, row scale Y_{C,d}, and the same C,d,t, using the character \xi_1. Here h is the new Fourier variable for the sum over y. Its nonzero output has squarefree g,w, e\mid g, and

Thus r,f' are coprime and squarefree, and the coefficient a^\sharp from (7.5) becomes

Write X'_0 for the individual column scale X' in (7.5); below, X' will denote a common dyadic scale. The length, coefficient, and Fourier parameter simplify to

Put U_0(x)=x^{-1/2}U(x). The coupled kernel is therefore

with both columns evaluated at x_j=N(n_j)/X'_0. In particular its nonzero support requires

since C_\Phi(2v)^2/c_I\le1.

For fixed r,f',k', all preimages are obtained by choosing

and setting g=e(r/t), h=k'/(de). The factorization f'=Cew is disjoint and squarefree. All these preimages have the same kernel: Ctgw=rf', and the Fourier parameter displayed above depends only on r,k'. Thus the support restrictions discard only zero kernels. The choices t\mid r contribute \tau_{\mathrm{div}}(r). At a prime p\mid f', the choices p\mid C, p\mid e, and p\mid w, respectively, contribute

to the sum of \mu(e)\mu(w) over the preimages; the two choices inside the first term are p\nmid d and p\mid d. Let \mathcal Z be the total zero-frequency contribution: the sum of the terms Z from Lemma 7.2, with the outer weights w_{C,d} and ranges in (7.2). Consequently, regrouping before taking absolute values gives

Fix R\le N(r)<2R, F'\le N(f')<2F' dyadically, and put

Retain only those r for which a nonzero term occurs in these ranges. Since f'\mid k', each such r satisfies

In particular \mathcal H'\ge1. For every j\mid r, using \mathcal H\le\Sigma, we also have

Moreover 1\le X'/X'_0<4. The rescaled kernels

have support in [u/8,2v]^2\subset\operatorname{int}(I'^2) and satisfy

where the supremum is over the retained indices in the dyadic block. This follows from the Schwartz bounds for \widehat\Phi. The common parameters satisfy exactly

Fix \delta>0, to be chosen in terms of \varepsilon_0 at the end. For each retained r, enlarge the f',k' ranges by positivity, keeping this set of r fixed. Lemma 4.4 then gives, for V\in C_c^\infty(I'),

Indeed exclusion removal replaces (X',F') by (X'/N(j),F'N(j)), j\mid r, preserving \mathcal H' and \Sigma'. Equation (7.7) gives X'/N(j)\ge1, and F'N(j)\ge1 as well. Thus every resulting mean square lies in the defining supremum.

There are O(R) ideals r in the dyadic range. Apply Lemma B.2 with row index (r,f',k'), the preceding mean-square bound, and the uniform kernel bound. The absolute contribution of this dyadic block is at most

because (\Sigma R/L^2)R(L/R)^2=\Sigma.

Finally, the zero-frequency bound in Lemma 7.2 gives

This uses only Y_{C,d}>0, not Y_{C,d}\ge1. Sum the O_{I_*,C_0}((\log(2D))^2) dyadic blocks and the fixed finite character set, and take \delta=\varepsilon_0/4. The stated bound follows, with implied constant depending only on I_*,m,C_0,\varepsilon_0, the fixed cutoffs and arithmetic data.□

Proof of Proposition 5.4. Use S_m as in Lemma 7.3, with m as in the proposition. Lemma 7.3, with loss D^{\varepsilon/4}, supplies the hypothesis of Lemma 7.1 with j=2m+4 and M\ll D^{\varepsilon/4}\Sigma(1+S_m). Applying that lemma with the same loss gives

which proves (5.11).□

Proof of Lemma 4.1.

Proof of (4.2). For squarefree n, set G(n)=\overline{\chi_n(4)}\gamma_3(n). Its dependence on a fixed ray class will be verified below. Fix a primary prime p\equiv1\pmod3 outside S. For multiplicative characters A,B of (\OO/(p))^\times, extended by zero at 0, write

For each y\in\OO/(p), the substitution u=2x-1 gives

since p\nmid2 and \chi_p^3 is the quadratic character, extended by zero at 0. Hence

The Gauss–Jacobi identity (cf. [22, (3.18)]) gives

For j\not\equiv0\pmod6, the normalized Gauss sums satisfy

Taking j=2 gives \gamma_2(p)\gamma_4(p)=\chi_p^2(-1)=1, since \chi_p^2 is cubic. Combining this with the Gauss–Jacobi relation above gives, after cancelling \gamma_1(p) and rearranging,

To finish the prime case of (4.2), it remains to evaluate \gamma_2(p)^3. We do so by computing J(\chi_p^2,\chi_p^2). Put q=\NK(p). We have J(\chi_p^2,\chi_p^2)\in\OO and \NK\bigl(J(\chi_p^2,\chi_p^2)\bigr)=q. For 0\le\ell\le(q-1)/3, the exponent (q-1)/3+\ell lies strictly between 0 and q-1, so \sum_{x\bmod p}x^{(q-1)/3+\ell}=0 in \OO/(p). Reducing modulo p and expanding the binomial therefore gives

Thus J(\chi_p^2,\chi_p^2)/p\in\OO has norm one, so J(\chi_p^2,\chi_p^2) is a unit multiple of p. To determine the unit, write \chi_p^2(x(1-x))=\omega^{j_x} with j_x\in\{0,1,2\} for x\in\OO/(p)\setminus\{0,1\}. Then

Since (\omega-1)^2 generates (3), expansion modulo this ideal gives

Together with p\equiv1\pmod3, this fixes the unit:

Consequently

The Chinese remainder theorem extends these prime-modulus identities to (4.2) for squarefree n, with one minus sign for each prime factor accounting for \mu(n).

Proof of (4.6). For coprime squarefree primary a,b prime to S, the Chinese remainder theorem gives

and cubic reciprocity gives \chi_a(b)^2=\chi_b(a)^2. Multiplying by \overline{\alpha(ab)}\xi(ab) therefore proves (4.6).

Proof of (4.1). We now prove (4.1) and the formula for G(n) in (4.3), including their dependence on a fixed ray class. We begin by evaluating the normalized quadratic Gauss sum. For c\in\OO coprime to 2, we will show that

To justify (A.1), apply Poisson summation over z\in\OO to e(z^2/c)e^{-\pi\eta|z|^2}, with \eta>0. The Gaussian makes the sum convergent. The Fourier transform of the product is

Divide both sides by the Gaussian mass

On the Fourier side, replacing r_\eta by 1 has total error

where C_c>0 is fixed. The normalized error is therefore O_c(\eta)\to0. Grouping the original sum modulo c and the Fourier sum modulo 2\OO, then letting \eta\downarrow0, gives (A.1).

For c=a+b\omega, formula (A.1) becomes

Thus \Gamma_{\rm quad}(c) depends only on c\bmod4\OO and is invariant under multiplication by a square in (\OO/4\OO)^\times. Representatives for the four square classes and their values are

Consequently the function

on these square classes satisfies

and is a symmetric bicharacter. For a prime p, each y\in\OO/(p) has 1+\chi_p^3(y) square roots. Grouping by y=x^2 gives

since \sum_{y\bmod p}e(y/p)=0. The Chinese remainder theorem extends this equality to squarefree n. Together with cubic reciprocity, it identifies the above \mathcal R with the quotient \chi_b(a)/\chi_a(b) for coprime primary a,b. Furthermore \chi_c(4)=(-2/c)_3=(c/(-2))_3 is a character modulo 2. Consequently G(c)=\overline{\chi_c(4)}\Gamma_{\rm quad}(c) factors through a fixed ray class group and satisfies (4.1), with \mathcal R(c,c)=\chi_c(-1).

Proof of (4.7). In the fixed ray class group, the multiplicative relation for G gives

Using the symmetry and multiplicativity of \mathcal R, we obtain

which is (4.7).

Proof of (4.4)–(4.5). The second identity in (4.3) follows from the inverse-character Gauss identity: \gamma_1(n)\gamma_{-1}(n)=\chi_n(-1). It remains to prove (4.4)–(4.5). Multiplying the second identity in (4.2) by \overline{\alpha(n)} gives (4.4), and combining it with this inverse-character identity gives (4.5).□

Proof of Proposition 6.2 and Lemmas 6.3 and 6.4.

Finite Fourier expansion. Define \phi:\OO\to\mathbb C by

For \Psi(n)=\Psi_0(n)\prod_{p\in\mathcal P}\chi_p^{j_p}(n), define

When \Psi=\Psi_k, the multiplier of \overline{\tau(-\ell)} is \phi_k(\lambda^3\ell), so this definition recovers \Theta_k in (6.2). We first write \Theta_\Psi as a finite sum of translates of \overline\theta and determine their reduced denominators. Choose once and for all a nonzero L\in\OO, with prime divisors in S, divisible by the conductor of \Psi_0, every prime in S, and sufficiently high powers of the primes above 2 and 3. The supplementary law of cubic reciprocity for \lambda [7, (1.5)], which evaluates \chi_n(\lambda)^2=(\lambda/n)_3, makes \phi periodic modulo L. As before (6.4), define

Since |\phi|\le1, we have |\widehat\phi(h_0)|\le1.

For p\in\mathcal P and 0\le j\le5, we expand \chi_p^j on \OO/(p) in additive characters. Its Fourier coefficients are defined, for h_p\in\OO/(p), by

Finite Fourier inversion then gives, for x\in\OO,

Multiplying (A.3) over the primes in \mathcal P gives, for x\in\OO,

Each summand is indexed by a tuple (h_p)_{p\in\mathcal P}. At x=\lambda^3\ell, its additive character gives the shift \lambda^2\sum_{p\in\mathcal P}h_p/p of \overline\theta. Writing \boldsymbol h=(h_0,(h_p)_{p\in\mathcal P}) with h_0\in\OO/(L), we claim that

where

To verify this identity, expand the fixed factor \phi as well. The finite Fourier expansions identify all nonzero Fourier coefficients, and the constant terms of the translates cancel because

For h_p\ne0, substitute y=h_p^{-1}u in the definition of C_{p,j}(h_p). For h_p=0 and j\ne0, use character orthogonality; for j=0, sum the additive character over nonzero residues directly. These calculations give

For the tuple \boldsymbol h, define its set of active primes by \mathcal A(\boldsymbol h):=\{p\in\mathcal P:h_p\ne0\}. By (A.5), c_{\mathrm F}(\boldsymbol h)\ne0 implies \{p\in\mathcal P:j_p\ne0\}\subseteq\mathcal A(\boldsymbol h). Thus only primes with j_p=0 can be inactive, as asserted in the proposition. Put r=\prod_{p\in\mathcal A(\boldsymbol h)}p, and let c_0 be the reduced denominator of \lambda^2h_0/L. At each active prime p,

so the factor p cannot cancel from the denominator. At primes dividing L, all terms \lambda^2h_p/p are integral. Thus the reduced denominator is c=c_0r up to a unit, including when (h_0,L)\ne1. The finite Fourier expansion has therefore expressed \Theta_\Psi as O_L(2^{|\mathcal P|}) groups of translates \overline{\theta(z+z_{\boldsymbol h},v)}, with shifts z_{\boldsymbol h} and coefficients c_{\mathrm F}(\boldsymbol h) given by (A.4). The groups are indexed by h_0\in\OO/(L) and the active set \mathcal A. Each group has a common reduced denominator c=c_0\prod_{p\in\mathcal A}p up to a unit. For each translate \overline{\theta(z+z_{\boldsymbol h},v)}, we next identify which of the three Fourier expansions in (6.8) will be used after changing coordinates at z_{\boldsymbol h}.

Reduced denominators and cusp coefficients. We next express each translate \overline{\theta(z+z_{\boldsymbol h},v)} using one of the three cusp expansions identified above. Our representatives \gamma_0,\gamma_+,\gamma_- correspond to \gamma_1,\gamma_{10},\gamma_{19}, respectively, in the numbering of [7, §5.1], which follows [38, Table II]. We allow (h_0,L)\ne1 and keep the reduced denominator c_0 of \lambda^2h_0/L in the calculation; this is the extension beyond [7, Corollary 5.1].

Set M=\lambda^{12}L^4. Since (p,M)=1 for p\in\mathcal A(\boldsymbol h), the Chinese remainder theorem lets us choose representatives h_p\in\OO with h_p\equiv0\pmod{M^2}. Changing representatives modulo p changes z_{\boldsymbol h} by an element of \lambda^2\OO=3\OO, under which \theta is periodic. To track dependence on the primes in r, restrict r to a residue class modulo M^2. For fixed h_0, active set, and this residue class, write z_{\boldsymbol h}=a/c in lowest terms, normalizing a\equiv1\pmod3 if \lambda\mid c, and c\equiv1\pmod3 otherwise. Since

these choices fix the residue of a modulo Mc_0. Since (a,c)=1 and (r,M)=1, the Chinese remainder theorem gives \delta'\in\OO satisfying the following congruences, where q ranges over prime divisors of the indicated elements:

Put

The congruences for \delta' give

To choose the cusp expansion at a/c, put

In the last case choose u_0\in\OO from a fixed set of representatives modulo 3. In each case, put g_1=gH^{-1}. Then g=g_1H and g_1\equiv I\pmod3, as in [7, §5.1]. The invariances

reduce \overline\theta(Hw) to one of the three functions \overline{\theta(\gamma_\sigma w)}, \sigma\in\{0,-,+\}, with the matrices \gamma_\sigma from (6.7). These representatives give the infinity expansion and the two additional cusp expansions needed here: translating by \omega or -\omega, then applying inversion, gives the representatives labeled - and +, respectively [7, (5.9)–(5.15), Appendix A]. Write t_\sigma(\ell) for the coefficient of vK_{1/3}(4\pi|\ell|v)\breve e(\ell z) in \theta(\gamma_\sigma w). Let \tau_1,\tau_2:\lambda^{-4}\OO\setminus\{0\}\to\mathbb C be the coefficient sequences defined in [7, (5.13), (5.14)]. Patterson’s cusp calculation [38, Theorem 8.1 and Table III] then gives

Complex conjugation changes the Fourier frequency from \ell to -\ell. Thus the coefficients in the normalization (6.8) are

For indices \ell=u\lambda^mnb^3 as in (6.10), the coefficient magnitudes satisfy

The m=-4 coefficients of t_- and t_+ have magnitude at most 9|b|. For the chosen H, let \sigma\in\{0,+,-\} index the corresponding expansion in (6.8). For each translate \overline{\theta(z+a/c,v)}, we have identified the Fourier expansion in z of \overline{\theta(H(z,v))} as one of the three expansions in (6.8):

These coefficients satisfy the support restriction (6.10) and bound (6.11), proving the coefficient assertions of Lemma 6.4. The next step uses g=g_1H to express the original translate through this Fourier series evaluated at g^{-1}(z+a/c,v), and computes the accompanying multiplier and additive phase. The matrix g maps \infty to a/c, which is why this is called an expansion at the cusp a/c.

The local character transformation. The matrix factorization g=g_1H lets us apply the automorphy law

where \kappa is Kubota’s cubic character [7, (5.4), (5.6)]. We will combine its conjugate with the finite Fourier coefficients C_{p,j}(h_p) in (A.5) to obtain the factors B_{p,j} of (6.5). For the translated theta function, the automorphy law gives

where

At z=0, the term with Fourier index \ell in the expansion of \overline\theta(Hw) therefore acquires the phase \breve e(-\delta'\ell/c). We claim that the multiplier is given by

To verify (A.10), use the determinant equation and cubic reciprocity. For the case v_\lambda(c)=1, the required congruences are

For (c,\lambda)=1, the congruence

removes the supplementary factors; reciprocity at the remaining primes then gives (\delta'/(-b_gc))_3=1. The factors other than (a/r)_3 depend only on the fixed residues at primes in S. We may therefore write

where \kappa_0 is fixed once h_0, the active set, and r\bmod M^2 are fixed.

Fix h_0 and the active set, so that the denominator c=c_0r is fixed while the nonzero residues h_p vary. Put D_0=\lambda^3c_0. For each active prime p, define in \OO/(p)

The inverse in \epsilon_p is taken in the field \OO/(p); it exists because r is squarefree and p\nmid\lambda c_0. The expression for a gives a\equiv\sigma_ph_p\pmod p. For a dual Fourier index \ell\in\lambda^{-4}\OO, the additive form of the Chinese remainder theorem yields

where

The residues \delta'_0 and r^{-1}\bmod D_0 are fixed by the choices above, so \psi ranges over a fixed finite family of additive characters of \OO. We claim the local transformation identity

where B_{p,j} is defined in (6.5), and

The indices of \gamma_j(p) are read modulo six; in the second case, \chi_p(-1)^4=1. In each case |\omega_{p,j}|=1. To prove (A.12), combine (A.5), the conjugate of (A.10), and (A.11). Setting y=h_p^{-1} changes the character exponent as follows:

For j\ne0,4, the character \chi_p^{j+2} is nontrivial, and its Gauss sum gives the first case of B_{p,j}. For j=4, it is trivial on nonzero residues; after rescaling by \epsilon_p\ne0, the sum is

For active j=0, the coefficient -\NK(p)^{-1} combines with a cubic Gauss sum to give

with the unit factors included in \omega_{p,0}. This proves (A.12) in every case. An inactive prime has j=0 and contributes only the scalar 1-\NK(p)^{-1}. Consequently, for each fixed h_0 and active set \mathcal A, combining the Fourier coefficients d_\sigma(\ell) of \overline{\theta(H(z,v))} with the sums over h_p\ne0 gives

up to a scalar independent of \ell. These are the arithmetic factors in the dual sum (6.9). In particular, B_{p,1}(\lambda^4\ell)=\chi_p(\lambda^4\ell)^3 is the quadratic factor used in (6.18).

The archimedean transform. We now derive the transformed weight V_*^\sharp in (6.6) and the scalar multiplying the dual sum. For \Rea s>1, introduce the Dirichlet series associated with the completed sum (5.3):

With V_* from (5.3) and its Mellin transform defined before (6.6), Mellin inversion gives

The identity (6.3), with \Psi in place of \Psi_k, identifies these coefficients with those of \Theta_\Psi after inserting the factor \overline{\alpha(nb^3)}. We now compute the normalization relating \mathcal T(s,\Psi) to the Mellin transform of the derivative of \Theta_\Psi.

For z=x+\mathrm i y, use \partial_{\bar z}=(\partial_x+\mathrm i\partial_y)/2 and \partial_z=(\partial_x-\mathrm i\partial_y)/2. Differentiation in \bar z supplies a factor \bar\ell in each Fourier mode. Combined with the factor |\ell|^{-1} from the Bessel integral below, this produces the required angular factor \overline{\alpha(\ell)}. We therefore define, initially for \Rea s>1, the Mellin transform

For each nonzero Fourier mode, differentiation and the Mellin integral for K_{1/3} (see [35, (10.43.19)]) give the following formulas:

For \Rea s>1, the coefficient formula (2.10) and (A.15) show that the sum of the integrals of the absolute values is finite. We may therefore integrate the differentiated Fourier series term by term.

For \ell=\lambda^{-3}nb^3, the phase and scale simplify to

Combining this with (2.10) gives

We next show that \mathcal J(s) is entire and use (A.16) to continue \mathcal T(s,\Psi).

Write (z',v')=g^{-1}(z+a/c,v). Differentiating (A.9) at z=0 gives

Thus the derivative in cusp coordinates is -(cv)^{-2}\partial_{z'}, with no height-derivative term. The defining Fourier series (A.2) for \Theta_\Psi controls v\to\infty. For v\to0, use (A.8) and the cusp expansions (6.8). In each expansion the horizontal derivative removes the constant mode. The remaining modes decay exponentially as v\to\infty; at v\to0, the transformed height 1/(\NK(c)v) tends to infinity, giving exponential decay in 1/v. Thus the integral defining \mathcal J(s) converges for every s and defines an entire function. Equation (A.16), after division by its gamma factors, also continues \mathcal T(s,\Psi) to an entire function.

For the term indexed by \boldsymbol h in (A.4), write c_{\boldsymbol h},\delta'_{\boldsymbol h},g_{1,\boldsymbol h},H_{\boldsymbol h} for the corresponding choices above. Let \sigma_{\boldsymbol h} be the cusp index determined by H_{\boldsymbol h}. Define its cusp Mellin transform by

Substituting v\mapsto(\NK(c_{\boldsymbol h})v)^{-1} in each translated term of \mathcal J(s), using (A.8), gives

Here c_{\mathrm F}(\boldsymbol h) is the coefficient in (A.4). The cusp coefficients d_{\sigma_{\boldsymbol h}}(\ell) and (A.15) give, for \Rea s>1,

The Dirichlet series \mathcal T(s,\Psi) converges absolutely for \Rea s>1, while the series in (A.19) at 1-s converges absolutely for \Rea s<0. The functional equation (A.18) and Stirling’s formula [22, §5.A.4] therefore give polynomial bounds in |\operatorname{Im}s| for \mathcal T(s,\Psi) on both sides of the strip 0\le\Rea s\le1. Splitting the integral defining \mathcal J(s) at v=1 gives finite order; Phragmén–Lindelöf [22, Theorem 5.53] then gives the same type of bound inside the strip (compare the arguments of Dunn and Radziwiłł in [7, Propositions 5.1–5.2]). Together with the rapid decay of \widehat V_* on vertical lines, these bounds justify moving the s-contour in (A.13) to \Rea s<0. No poles are crossed, since \mathcal T(s,\Psi) is entire. Setting t=\tfrac12-s then gives a line \Rea t>1/2, where the dual coefficient series converges absolutely.

For fixed h_0 and active set, insert (A.19) into (A.18) and use (A.12) to sum over the nonzero h_p. After the normalization in (A.16), the scalar C in (6.9) for this group of translates is

Dividing (A.16) by its gamma factors and setting t=\tfrac12-s produces the gamma quotient in (6.6). Its numerator gamma factors have their first pole at t=-5/6, and its reciprocal denominator gamma factors are entire. Thus no poles lie between the current contour \Rea t>1/2 and \Rea t=0. The rapid decay of \widehat V_*, together with Stirling’s formula, allows us to shift each kernel contour to \Rea t=0. This gives the weight V_*^\sharp defined in (6.6), evaluated at \NK(\ell)X/\NK(c)^2. Together with (A.12), this gives the dual sum (6.9).

For k_0 as in Lemma 6.3, fix h_0, the factors of \Psi other than \chi_{k_0}, and the active/inactive choices at their primes. Every prime dividing k_0 has exponent j_p=1 and is therefore active, so

Thus r/k_0 is fixed, and fixing k_0\bmod M^2 fixes r\bmod M^2. The choices of H,c_0 and the residues in (A.11) are therefore fixed in each such class. Thus d=d_\sigma, \psi, and c_0 have precisely the asserted dependence on k_0.

It remains to bound the transformed weight. The first numerator pole of the gamma quotient in (6.6) is at t=-5/6. We may therefore shift the kernel contour to \Rea t=-1/4 for 0<x\le1, obtaining the factor x^{1/4}, and to \Rea t=A for x\ge1, obtaining x^{-A}. Each application of x\partial_x introduces a factor -t. Stirling’s formula on -1/4\le\Rea t\le A then gives, for every A>0 and j\ge0,

When V_* is supported in a fixed compact interval I\subset(0,\infty), repeated integration by parts in its Mellin transform gives rapid decay in |u|, uniformly for -A\le\eta\le1/4. Thus, for a sufficiently large J=J(A,j), the supremum of integrals in (A.20) is \ll_{A,j,I}\|V_*\|_{C^J(I)}. This proves (6.12). Thus (6.9) expresses T(X;\Psi) as O_{\Psi_0,S}(2^{|\mathcal P|}) dual sums with |C|\le1/81, the asserted ray class dependence of (d,\psi,c_0), and the weight decay just established. Together with the support and coefficient bounds proved above, this completes the proof of Proposition 6.2 and Lemmas 6.3 and 6.4.□

Proof. Take the Fourier transform in logarithmic coordinates:

Applying Fourier inversion to this gives (B.1). Since the intervals I_j are fixed and bounded away from zero, the C^q norm in logarithmic coordinates is bounded by a constant times \|\mathcal K\|_{C^q(\mathcal I)}. Applying (1-\Delta_{\mathbf y})^{q/2} under the integral therefore gives

Multiplication by (1+|\mathbf t|)^J and integration proves the bound when q>J+d. See also the multivariable Mellin formulation in [39, §10.1, (130)–(131)].□

Proof. Choose a real V\in C_c^\infty(I_*) equal to one on I, and set U_t(x)=V(x)x^{\mathrm i t}. Apply (B.1) of Lemma B.1 to each \mathcal K_r with d=2 and \mathcal I=I^2. Replacing the second Mellin variable by its negative and multiplying by V(x)V(y), which equals one on the support of \mathcal K_r, gives

with the following uniform bound, obtained from the pointwise coefficient estimate in the proof of Lemma B.1 with d=2 and \mathcal I=I^2:

The hypothesis applies to each U_t, and \|U_t\|_{C^m(I_*)}\ll_{I,I_*,m}(1+|t|)^m. Taking the common bound for b_r before integrating and applying Cauchy–Schwarz in r therefore bounds the absolute value in (B.4) by a constant depending on I,I_*,m,q times

Taking q=2m+4 makes the integral converge and proves the claim.□

Proof. For each fixed r, weighted Cauchy–Schwarz gives

Multiply by w_r and sum over r to obtain (B.6). The same argument with sums in place of integrals proves the finite version.□