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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A fixed separation at prime positions

Instead of averaging over a block, one may ask for a fixed gap at prime positions. The lower bound on the prime ensures that the corresponding tail indices lie beyond the required threshold.

Definition 3.6 (A fixed gap at prime positions). For every $h \geq 1$ and every $N_{0}$ there is a prime $p$ with $max (N_{0} + h + 1, h + 5) \leq p$ and $Re e (R_{p - 1} - R_{p - h - 1}) < \frac{9}{10}$ , where $e (x) = \exp (2 π i x)$ . Equivalently: cofinally many primes $p$ at which the totient tail difference across the shift $h$ stays a fixed distance $> \arccos (9 / 10) / (2 π) \approx 0.0718$ from every integer. It implies $S \notin Q$ . DTWNaturalPrimeTailOrbitStrictGap irrational_totient_series_of_naturalPrimeTailOrbitStrictGap

This asks for more than nonintegrality. A nonintegral real number can be arbitrarily close to an integer, whereas the displayed condition requires a fixed positive distance, along prime positions and for each shift $h$ . For example, phases at a half-integer satisfy the numerical gap and phases converging to an integer eventually fail it. These are illustrations of the hypothesis, not assertions about the totient tails. No estimate for these tail differences at prime positions with the stated quantifiers is proved here. Certificate completeness alone does not supply it.

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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