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

The lower bound for rank-one quotients

The finite-sum quotients below stay a positive distance above $Θ_{2} = S - 1 / 2$ , uniformly in their exponent and truncation length. Consequently neither parameter can make the associated rational linear form tend to zero. Appendix A.4 gives the sharp minimum and its ordinary proof, strengthening the bound recorded here.

Theorem 6.156 (A uniform positive gap for rank-one quotients). For integers $r \geq 2$ and $Y \geq 1$ , write

Θ_{r} = \sum_{d \geq 1} \frac{μ (d)}{(2^{d} - 1)^{r}}, t (Y, r) = \sum_{d = 1}^{Y} \frac{μ (d)}{(2^{d} - 1)^{r}} .

For every $e \geq 1$ and $Y \geq 4$ ,

\frac{t (Y, e + 2)^{2}}{t (Y, 2 e + 2)} - Θ_{2} > \frac{1}{480} .

Here $Θ_{2} = S - 1 / 2$ . For $r \geq 3$ , the proof uses $1429 / 1512 \leq Θ_{r} < 1$ and $| t (Y, r) - Θ_{r} | \leq 1 / 3584$ when $Y \geq 4$ . In particular, these inequalities make the denominator positive.

Proposition 6.157 (The bound is preserved by positive normalised averaging). For a nonempty finite family of admissible quotients with positive weights summing to $1$ , the weighted average still exceeds $Θ_{2}$ by more than $1 / 480$ . If an admissible quotient is $p / q$ in lowest terms, with $q > 0$ , then $| q Θ_{2} - p | > q / 480$ .

Observation 6.158. Writing an admissible quotient in lowest terms as $p / q$ , with $q > 0$ , gives $| q Θ_{2} - p | > q / 480 \geq 1 / 480$ . Such forms cannot tend to zero. The conclusion concerns exactly the displayed quotients and their positive normalised averages. It makes no assertion about other rational approximants or about signed averages.

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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