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

Remaining arithmetic questions

The remaining questions ask for witnesses with unbounded indices. We collect them in residue and exponential-sum form, retaining the quantifiers needed to imply irrationality of $S = \sum_{n \geq 0} φ (n) 2^{- n}$ . Erdős Problem #249 remains open.

This section collects quantified conditions on the scaled tail $R_{N} = \sum_{j \geq 0} φ (N + 1 + j) 2^{- (j + 1)}$ that suffice for irrationality. The implication theorems and the arithmetic assumptions have different status: the former are proved in the indicated sources, while the latter are not supplied for the actual totient sequence. Erdos249257.TotientTailPeriodKiller.totientTail

Each condition is stated with its quantifiers and linked to its formal source. The comparison distinguishes proved equivalences from one-way sufficient conditions. It is not an exhaustive list of possible methods. A finite list of certificates does not stand in for a condition beyond every threshold, and an implication does not establish its antecedent.

Recall that $H_{t} = H (t) = lcm (1, \dots, t)$ (Erdos249257.TotientTailPeriodKiller.periodLcm). For nonnegative integers $h, N, L$ , put

D (h, N, L) = \sum_{j = 0}^{L - 1} (φ (N + h + 1 + j) - φ (N + 1 + j)) 2^{L - 1 - j} .

Erdos249257.TotientTailPeriodKiller.windowDiscrepancy The certificate $C (h, N, L)$ is the pair of strict inequalities

N + h + L + 2 < D (h, N, L) mod 2^{L} < 2^{L} - (N + h + L + 2) .

Erdos249257.TotientTailPeriodKiller.certifiedKill This condition is decidable. Thus the least nonnegative residue must lie more than $N + h + L + 2$ from either endpoint. For fixed $h, N$ , this excluded radius grows linearly with $L$ , while its proportion of the modulus tends to zero.

Certificate completeness

The original quantified residue condition

Multiples, diagonals, and LCM grids

The block exponential-sum condition

The actual tail difference on the LCM diagonal

An equivalent test using two leading residue bits

Farey growth and the failed generic rank argument

Summary of implications

Remaining estimates

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.

Cite and contact. Cite this paper by its title, author and date above, with its PDF; cite the earlier sources it uses for a mathematical result. For software, use the release citation and give the commit used. Contact Will with questions or corrections.

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