The Binary Totient Series

Will Cook

Long paper

The Binary Totient Series

Erdős #249 By Will Cook July 2026; revised 1 October 2026 Authorship, AI use and citation

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Summary. The dyadic totient sections through level $e \geq 1$ have an explicit rational basis of size $2^{e} + 1$ . Truncating a scaled tail difference produces an integer whose residue certifies nonintegrality when its distances from both endpoints of the residue interval exceed an explicit error bound. The record develops the resulting finite denominator exclusions, equivalent residue criteria, and sufficient exponential-sum estimates. Rational comparison sequences identify the limitations of particular size, parity, rank, and finite-prefix arguments. The short paper supplies the all-base basis theorem; here each conditional result carries the arithmetic hypothesis still needed to apply it. Those hypotheses are not established in the form required to prove irrationality of S=∑ $φ (n) 2^{- n}$ .

This paper covers a claim-bounded problem-specific reasoning surface for Erdős #249: selected checked premises, finite certificates, closed routes, surviving obligations, and their exact evidence bands.

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #249, which remains open.

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In this paper

Signed interpolation with entire Dirichlet defects

The small block

z^{2} - 2 z^{4} + z^{6} = z^{2} (1 - z^{2})^{2}

has coefficient sum $1 - 2 + 1 = 0$ and first moment $2 - 8 + 6 = 0$ , but its value at $z = β^{- 1}$ is positive for $β \geq 2$ . It can therefore alter an interior value while preserving these two moments. This is a model for one block, not a construction satisfying all the conditions below; translation, spacing and increasing the difference order supply the additional requirements.

This construction assigns different jobs to the shape and amplitude of a correction. High-order finite differences annihilate low-order moments; their integer amplitudes adjust the series value. As the orders increase, later completed blocks contribute zero to each fixed moment, so its defect stabilises between blocks. The prescribed initial collection of moment defects is zero, not merely constant. Spacing the blocks controls their density and analytic convergence. All corrections occur at even indices, leaving every odd totient fixed. The three numbered remarks below have ordinary proofs; none has yet been formalised in Lean.

For a sequence $a$ , write $F_{a} (z) = \sum_{n \geq 1} a (n) z^{n}$ . For integers $q \geq 1$ , $0 \leq c < q$ and $d \geq 0$ , put

M_{q, c, d} (a; N) = \sum_{\begin{matrix} 1 \leq n < N \\ n \equiv c (\mod q) \end{matrix}} n^{d} a (n) .

Let $μ_{a, X, m}$ be the empirical distribution of the length- $m$ blocks starting at the integers $X \leq N < 2 X$ , with mass $1 / X$ per start. Upper Banach density means the limiting upper density over all intervals of a given length, with their starting points unrestricted.

Remark 10.8 (Sparse interpolation around the totient). Fix integers $β \geq 2$ , $K \geq 1$ , $P \geq 2$ , and any function $f : N_{> 0} \to (0, \infty)$ with $f (n) \to \infty$ . There exist $T > 0$ and target-independent sets $S \subseteq H \subseteq {n > K}$ , with every element of $S$ even, such that every

r \in [F_{φ} (β^{- 1}) - T, F_{φ} (β^{- 1}) + T]

equals $F_{b} (β^{- 1})$ for an integer sequence $b = φ + e$ satisfying:

$0 \leq b (n) \leq n$ , $| e (n) | \leq f (n)$ , and $supp (e) \subseteq S$ . In particular the prefix through $K$ and every odd coefficient are unchanged.
For every fixed $q \geq 1$ , both $e (n)$ and $\sum_{i < n} e (i)$ are divisible by $q$ eventually, with a cutoff independent of $r$ .
For every fixed $(q, c, d)$ , the defect $M_{q, c, d} (b; N) - M_{q, c, d} (φ; N)$ equals an integer $C_{q, c, d, r}$ for all sufficiently large $N \notin H$ . The cutoff is independent of $r$ . If $q \leq P$ and $d < P$ , this defect is zero at every $N \notin H$ .
For each root of unity $ζ$ , the radial function $x \mapsto F_{b} (x ζ) - F_{φ} (x ζ)$ extends smoothly to $x = 1$ . Its Taylor coefficients there lie in $Z [ζ]$ . The coefficients of orders $u < P$ are zero if the order of $ζ$ is at most $P$ .
For every fixed $(q, c)$ the absolutely convergent series
$D_{q, c, r} (s) = \sum_{\begin{matrix} n \geq 1 \\ n \equiv c (\mod q) \end{matrix}} e (n) n^{- s}, Re s > 3 / 2,$
extends to an entire function, and $D_{q, c, r} (- d) = C_{q, c, d, r}$ .

The set $S$ has upper Banach density zero. The set $H$ has ordinary density zero and upper Banach density one. Define

g (n) = min (inf_{k \geq n} f (k), (n / 2)^{1 / 4}), ℓ (X) = 2 ⌊ \frac{\log_{β} g (⌊ X / 4 ⌋)}{8} ⌋ .

For large $X$ , $ℓ (X) > 0$ , and one constant $C$ , independent of $r$ , gives

d_{T V} (μ_{φ, X, m}, μ_{b, X, m}) \leq min (1, C [\frac{m}{ℓ (X)} + ℓ (X)^{- 1 / 2}]), 1 \leq m \leq X .

Thus every block length $m = o (ℓ (X))$ is preserved asymptotically.

Ordinary proof; not formalised in Lean.

The allowance can grow as slowly as desired and need not be monotone. For $f (n) = (\log (n + 3))^{ϵ}$ , $ϵ > 0$ , the allowed block scale is $o (\log \log X)$ ; for $f (n) = (n + 3)^{α}$ , $α > 0$ , it is $o (\log X)$ . The exact Taylor series, growing moduli, zeros and regular parts of Dirichlet functions are not preserved. The correction has both signs, so the earlier one-sided perturbation constraint is not retained. The construction makes no rationality assertion about $F_{φ} (1 / 2)$ . The two exceptional sets also have different roles. The support $S$ contains the individual altered indices. The larger set $H$ contains the intervals spanned by correction blocks, where a prefix may have included only part of a cancelling block. Its upper Banach density is one because it contains arbitrarily long intervals, even though its ordinary density is zero. Thus the moment conclusion outside $H$ does not assert exact cancellation at every sufficiently large cutoff.

Proof. We construct the blocks first, independently of the target value. Their increasing order enforces the moment and boundary conditions; only their amplitudes will be chosen afterwards to obtain that value. All blocks lie beyond the prescribed prefix.

The tail infimum in $g$ is positive, nondecreasing and tends to infinity. The cap gives $g (n) \leq (n / 2)^{1 / 4}$ , while $g (n) \leq f (k)$ for every $k \geq n$ . Choose a large even $n_{- 1} > K$ so that $g \geq 1$ thereafter and the first spacing below is at least $max (16, (2 P P!)^{2})$ . Define

\begin{aligned} s_{j} & = 2 ⌊ \frac{1}{8} \log_{β} g (n_{j - 1}) ⌋, & n_{j} & = n_{j - 1} + s_{j}, \\ k_{j} & = max {k \geq 1 : (2 k k!)^{2} \leq s_{j}}, & M_{j} & = k_{j}!, t_{j} = 2 k_{j}!, W_{j} = k_{j} t_{j}, \\ A_{j} & = β^{s_{j} + 2} . \end{aligned}

These parameters serve two different constraints. The inequality $W_{j} \leq \sqrt{s_{j}} < s_{j + 1}$ keeps one block clear of the next. The larger digit range $| d_{j} | \leq A_{j}$ will make the possible values overlap, but its coefficient cost must still fit within $g$ . The spacings and orders are nondecreasing and tend to infinity, with $s_{j} = O (\log n_{j - 1})$ . For digits $d_{j} \in {- A_{j}, \dots, A_{j}}$ , form

P_{j} (z) = d_{j} M_{j} z^{n_{j}} (1 - z^{t_{j}})^{k_{j}}, E (z) = \sum_{j \geq 0} P_{j} (z) .

The factor $(1 - z^{t_{j}})^{k_{j}}$ supplies cancellation of moments; $M_{j}$ supplies divisibility; and the even $n_{j}, t_{j}$ leave every odd coefficient unchanged. The digit $d_{j}$ remains free to adjust the value at $z = 1 / β$ . Let $S$ consist of all $n_{j} + l t_{j}$ , $0 \leq l \leq k_{j}$ , and let $H$ be the union of their full integer hulls $[n_{j}, n_{j} + W_{j}]$ . These hulls are disjoint and all points of $S$ are even.

Bounds for the coefficients. If $s \geq 16$ and $2 k k! \leq \sqrt{s}$ , then $4 k! 2^{k} \leq 2 \sqrt{s} 2^{\sqrt{s} / 2} \leq 2^{s}$ . It follows for $β \geq 2$ that $β^{2} k! 2^{k} \leq β^{s}$ . Therefore

\begin{matrix} (1) & ‖ P_{j} ‖_{1} \leq M_{j} β^{s_{j} + 2} 2^{k_{j}} \leq β^{2 s_{j}} \leq \sqrt{g (n_{j - 1})} . \end{matrix}

At every modified index $n$ this is at most $f (n)$ and at most $(n_{j - 1} / 2)^{1 / 8} \leq \sqrt{n / 2}$ . Both signs of the correction are permitted at even indices, because the product formula gives

\frac{φ (n)^{2}}{n} = \prod_{p^{a} ∥ n} p^{a - 2} (p - 1)^{2} \geq \frac{1}{2},

because all odd-prime factors are at least one and the smallest possible factor at two is $1 / 2$ . Also $φ (n) \leq n / 2$ for even $n$ . Thus $b = φ + e$ , where $e$ is the coefficient sequence of $E$ , stays between zero and $n$ . The coefficient bound ensures absolute convergence of $E$ inside the unit disk.

An interval of values. At $z = 1 / β$ , the contribution of block $j$ is $d_{j} w_{j}$ , where $a_{j} = (1 - β^{- t_{j}})^{k_{j}}$ and $w_{j} = M_{j} β^{- n_{j}} a_{j}$ . We must check that the later blocks can fill the gaps between successive choices of $d_{j}$ . Bernoulli’s inequality gives $3 / 4 \leq a_{j} \leq 1$ . Since $M_{j + 1} \geq M_{j}$ ,

A_{j + 1} w_{j + 1} = β^{2} M_{j + 1} β^{- n_{j}} a_{j + 1} \geq 3 M_{j} β^{- n_{j}} \geq 3 w_{j} .

The sum $T = \sum_{j} A_{j} w_{j}$ is positive and finite, by (1). After choosing $d_{j}$ , the remaining digits have total allowance $T_{j} = \sum_{i > j} A_{i} w_{i}$ . Consecutive choices of $d_{j}$ change the current contribution by $w_{j}$ , while $T_{j} \geq A_{j + 1} w_{j + 1} \geq 3 w_{j}$ . Hence the intervals $[d w_{j} - T_{j}, d w_{j} + T_{j}]$ , $- A_{j} \leq d \leq A_{j}$ , overlap and cover $[- A_{j} w_{j} - T_{j}, A_{j} w_{j} + T_{j}]$ . Successively choose a digit whose interval contains the remaining target. The residual tends to zero with $T_{j}$ , so the digits represent every value in $[- T, T]$ . This is the finite-choice interval principle of Crmarić and Kovač [36], here verified for signed finite differences. It also shows $T$ can be made arbitrarily small, while positive, by increasing the initial cutoff.

Congruences and moments. For every polynomial $p$ of degree less than $k$ ,

\begin{matrix} (2) & \sum_{l = 0}^{k} (- 1)^{l} (\binom{k}{l}) p (n + l t) = 0. \end{matrix}

This follows by iterating the forward-difference operator, which lowers polynomial degree. Eventually $q ∣ M_{j}$ , so every later coefficient and every partial block sum vanish modulo $q$ . Complete blocks have mass zero, giving the cumulative assertion. Eventually also $q ∣ t_{j}$ and $k_{j} > d$ . Each such block lies in one residue class modulo $q$ and its moment of degree $d$ is zero by (2). The restriction on the cutoff is needed here: outside $H$ it cannot split a block, so the moment sum contains only complete blocks. All sufficiently late complete blocks cancel, leaving just the integer contribution of the finitely many early blocks. If $q \leq P$ and $d < P$ , there are no exceptional early blocks. These cutoffs depend on the block parameters, not on the later choice of digits.

Sparsity and blocks. Within the $j$ th block the gaps in $S$ are $t_{j} \to \infty$ ; between blocks they are at least $s_{j + 1} - \sqrt{s_{j + 1}} \to \infty$ . Uniform spacing after a finite prefix implies upper Banach density zero. To estimate the hulls without assuming regular variation of $f$ , take any hull meeting $[X, 2 X + m)$ , $m \leq X$ . Its center lies in $[X / 2, 3 X]$ eventually and its preceding center is at least $X / 4$ , since its width and spacing are $O (\log X)$ . Thus $s_{j} \geq ℓ (X)$ for every such block. There are $O (X / ℓ (X) + 1)$ of them, and

W_{j} + 1 \leq 2 \sqrt{s_{j}} \leq \frac{2 s_{j + 1}}{\sqrt{ℓ (X)}} .

The following gaps telescope to $O (X)$ over these consecutive centers. Consequently the total hull lengths are $O (X / \sqrt{ℓ (X)})$ . A hull can meet length- $m$ blocks at at most $m + W_{j} + 1$ starts. Coupling the two empirical laws at the same start proves the stated total-variation bound; the extra $m / X$ is absorbed by $m / ℓ (X)$ . The hull estimate at $m = 1$ , followed by dyadic summation, gives ordinary density zero for $H$ . But $W_{j} \to \infty$ , so $H$ contains arbitrarily long solid intervals and its upper Banach density is one.

Radial smoothness. The number of differences increases with the block index, so any fixed derivative order is eventually controlled. Fix a root $ζ$ of order $q$ , and integers $u, A \geq 0$ . Choose $B \geq u + A + 2$ . For all sufficiently late blocks, $q ∣ t_{j}$ and $k_{j} \geq B$ . Factor

P_{j} (x ζ) = (1 - x)^{B} R_{j} (x), R_{j} (x) = d_{j} M_{j} ζ^{n_{j}} x^{n_{j}} (1 + x + \dots + x^{t_{j} - 1})^{B} (1 - x^{t_{j}})^{k_{j} - B} .

Its coefficient norm is at most $\sqrt{g (n_{j - 1})} t_{j}^{B} = O_{B} (n_{j}^{1 / 2})$ , because $g (n) \leq (n / 2)^{1 / 4}$ and $t_{j} = O (\sqrt{\log n_{j}})$ . All exponents lie between $n_{j}$ and $2 n_{j}$ eventually. For $x \in [1 / 2, 1)$ ,

\sum_{j} | R_{j}^{(v)} (x) | \leq C_{B, v} \sum_{n \geq 1} n^{v + 1 / 2} x^{n} \leq C_{B, v}^{'} (1 - x)^{- v - 3 / 2} .

The series and derivatives converge normally on compact subintervals. After differentiating $(1 - x)^{B} \sum_{j} R_{j} (x)$ $u$ times, every term is $O ((1 - x)^{B - u - 3 / 2}) = O ((1 - x)^{A + 1 / 2})$ . Thus, after subtracting a finite initial polynomial, all prescribed derivative orders vanish to any prescribed order at the boundary. Integration of the continuous derivative limits gives a single smooth extension. For an integer polynomial $\sum c_{n} z^{n}$ , its radial Taylor coefficient of order $u$ at $ζ$ is $\sum c_{n} (\binom{n}{u}) ζ^{n} \in Z [ζ]$ . Every such coefficient stabilizes after finitely many blocks. The first $P$ vanish at roots of order at most $P$ .

Entire Dirichlet continuation. Here we apply the same finite differences to $y^{- s}$ rather than to a polynomial. Fix $(q, c)$ and a compact set of complex $s$ , with $Re s \geq - R$ . Choose an integer $B > R + 3 / 2$ . All sufficiently late blocks align modulo $q$ and have $k_{j} \geq B$ . The derivative of $y^{- s}$ of order $B$ is $(- 1)^{B} (s) (s + 1) \dots (s + B - 1) y^{- s - B}$ . Repeated integration expresses its $B$ th finite difference as an integral over $[0, t_{j}]^{B}$ . The remaining differences contribute at most $2^{k_{j} - B}$ , so the absolute Dirichlet contribution of one polynomial is at most

C | d_{j} | M_{j} 2^{k_{j} - B} t_{j}^{B} n_{j}^{R - B} \leq C^{'} n_{j}^{R - B + 1 / 2} .

This is summable over the distinct positive centers. The Dirichlet polynomials of the blocks therefore converge normally on every compact set and define an entire function. In $Re s > 3 / 2$ they agree with the absolutely convergent ungrouped series. At $s = - d$ , all late blocks vanish by their moment identity, leaving exactly $C_{q, c, d, r}$ . These are analytic-continuation values; the original ungrouped series need not converge at negative integers. ◻

Remark 10.9 (Observations that do not determine irrationality). The following data, taken together, do not determine whether $F_{b} (β^{- 1})$ is rational: the fixed prefix and odd coefficients; every fixed eventual coefficient and cumulative congruence; the block laws on scales $o (ℓ (X))$ ; fixed progression moments modulo eventual integer constants outside $H$ ; progression Dirichlet functions modulo entire functions; and radial germs at all roots of unity modulo smooth functions with Taylor coefficients in $Z [ζ]$ . This remains true with exact equality of the first $P$ moment degrees and first $P$ radial Taylor coefficients at moduli and root orders at most $P$ .

Ordinary proof; not formalised in Lean.

Proof. Every constructed sequence has the specified data of $φ$ , including the common exceptional sets, while the nondegenerate interval of attained values contains both rational and irrational numbers. ◻

Taking finite linear combinations of the Dirichlet series on arithmetic progressions shows that every fixed periodic twist changes by an entire function. An entire summand has no negative Laurent coefficients: it therefore preserves every pole and its principal part wherever the original series has meromorphic continuation. The regular part and zeros need not be preserved. In the untwisted case the classical identity gives

\sum_{n \geq 1} b (n) n^{- s} = \frac{ζ (s - 1)}{ζ (s)} + D_{1, 0, r} (s), Re s > 2.

The version used by Coons [37] supplies this function identity, not an irrationality theorem for one value of $F_{φ}$ . Likewise, Habiro’s injectivity of the exact Taylor map for a cyclotomic completion [38] concerns a different, stronger datum. The full sequence of Taylor coefficients is not fixed here: the second Taylor coefficient of $M z^{n} (1 - z^{t})^{2}$ at one is $M t^{2}$ .

The change of sign is indispensable for this construction. An infinitely supported nonnegative integer correction has unbounded Abel limit at one. The Dirichlet series of the correction also cannot extend to an entire function if its defining series converges in a right half-plane. Indeed, choose a positive real $σ$ inside that half-plane. Termwise differentiation gives $(- 1)^{k} D^{(k)} (σ) = \sum_{n} e (n) n^{- σ} (\log n)^{k} \geq 0$ . The entire Taylor series at zero and Tonelli’s theorem then give

D (0) = \sum_{k \geq 0} \frac{σ^{k}}{k!} \sum_{n} e (n) n^{- σ} (\log n)^{k} = \sum_{n} e (n) < \infty .

Integer nonnegativity forces finite support, and a finite correction at an integer reciprocal base cannot change rationality. Signed differences are what allow the stronger observation class above.

Remark 10.10 (The first-harmonic test on the rational members). Let $b (n)$ be integers with $0 \leq b (n) \leq n$ and $F_{b} (1 / 2) = u / (2^{c} v)$ , where $c \geq 0$ and $v$ is odd and positive. Choose $h \geq 1$ with $v ∣ 2^{h} - 1$ . If $X \geq max (c, 1)$ and $16 (2 X + h + L + 2) \leq 2^{L}$ , then for every integer $N \in [X, 2 X)$ ,

\cos (2 π 2^{- L} \sum_{j = 0}^{L - 1} [b (N + h + 1 + j) - b (N + 1 + j)] 2^{L - 1 - j}) \geq \cos (π / 8) > \frac{9}{10} .

Ordinary proof; not formalised in Lean.

Proof. The actual tail $R_{N} = \sum_{j \geq 1} b (N + j) 2^{- j}$ lies in $[0, N + 2]$ and satisfies $R_{N} \equiv 2^{N} F_{b} (1 / 2) (\mod) Z$ . Thus $R_{N + h} - R_{N}$ is an integer for $N \geq c$ . If $D$ denotes the integer window sum in the statement, direct truncation gives $2^{L} (R_{N + h} - R_{N}) - D = R_{N + h + L} - R_{N + L}$ . The right side has absolute value at most $N + h + L + 2$ , so $D / 2^{L}$ is within $1 / 16$ of an integer. The cosine bound follows; its strict comparison with $9 / 10$ follows from $\cos^{2} (π / 8) = (2 + \sqrt{2}) / 4 > 17 / 20 > 81 / 100$ . ◻

This applies to every nonempty subset of the block, including the actual totient-selected set $G$ of Theorem 3.5(b), at $s = 26$ , $η = 1 / 1000$ and its minimal admissible depth $L (X)$ . This set is held fixed rather than recomputed from $b$ . Wherever it has more than $67 X / 100$ elements, the real first-harmonic sum exceeds $603 X / 1000$ . The interpolation interval has an explicit dyadic rational member: for sufficiently large $J$ , the finite prefix $\sum_{n \leq J} φ (n) 2^{- n}$ lies inside it, since the omitted tail is at most $(J + 2) 2^{- J}$ . For this member one may take $h = 1$ . Consequently the preserved observations cannot imply the good-base saving for all admissible sequences. No counterexample to the saving for the actual totient is asserted; additional structure, such as its exact multiplicative relations at the relevant growing scales, is still required.

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Authorship and AI use. Will Cook built and directed the research infrastructure and maintains the public release. He reviewed claims when he could. AI agents did most of the research and drafting. Cook did not independently verify every claim.

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