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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Coprime-pair sums

Coprime-pair counting gives a further expression for $S$ . The classical Lambert identity alongside it uses different weights and sums to an elementary rational function. That evaluation does not determine the rationality of the totient series.

Theorem 6.144 (Coprime-pair counting and the totient). For every $n \in N$ , $# {(a, b) \in N^{2} : a + b = n, a > 0, gcd (a, b) = 1} = φ (n)$ . Here $b = 0$ is allowed: the boundary pair $(1, 0)$ accounts for the value $φ (1) = 1$ .

Proposition 6.145 (Two lattice sums). For $0 \leq r < 1$ , the two choices of boundary give

\begin{aligned} \sum_{\begin{matrix} a \geq 1, b \geq 0 \\ gcd (a, b) = 1 \end{matrix}} r^{a + b} & = \sum_{n \geq 1} φ (n) r^{n}, \\ \sum_{\begin{matrix} a, b \geq 1 \\ gcd (a, b) = 1 \end{matrix}} r^{a + b} & = \sum_{n \geq 1} φ (n) r^{n} - r . \end{aligned}

The removed pair is $(1, 0)$ . Partitioning all strictly positive pairs by their greatest common divisor gives

\sum_{g \geq 1} \sum_{\begin{matrix} a, b \geq 1 \\ gcd (a, b) = 1 \end{matrix}} r^{g (a + b)} = {(\frac{r}{1 - r})}^{2} .

This total is $1$ exactly when $r = 1 / 2$ .

Theorem 6.146 (The classical coprime-pair Lambert identity). For every $0 \leq r < 1$ , $\sum_{(a, b) coprime, a, b \geq 1} \frac{r^{a + b}}{1 - r^{a + b}} = (\frac{r}{1 - r})^{2}$ , an elementary rational function of $r$ , hence rational at every rational $r$ including $r = 1 / 2$ . This is the classical visible-point identity, and its Lean proof is a formalisation of it rather than a new result: writing each pair $(A, B)$ of positive integers uniquely as $g \cdot (a, b)$ with $gcd (a, b) = 1$ converts the quadrant sum $\sum_{A, B \geq 1} r^{A + B} = (r / (1 - r))^{2}$ into the displayed sum over visible points. With the plain weight $r^{a + b}$ , the same strictly positive index set instead sums to $\sum_{n \geq 1} φ (n) r^{n} - r$ . Thus at $r = 1 / 2$ the Lambert-weighted sum is $1$ , whereas the plain-weight sum is $S - 1 / 2$ , not $S$ . Adding the boundary pair $(1, 0)$ recovers $S$ in the plain-weight sum.

Observation 6.147. The Lambert-weighted sum is rational at every rational $r \in [0, 1)$ . Consequently, coprimality of the indices, the decomposition by greatest common divisor and geometric decay alone do not imply its irrationality. This comparison does not decide the arithmetic nature of the differently weighted sum $S - 1 / 2$ , or of $S$ . A proof using the plain-weight sum must exploit information not contained in those three properties.

About this paper

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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