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

PDFOpen the PDF, 163 pages TeXLaTeX source Source on GitHub Problem page

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.

Start here, by the editorial route

In this paper

Organisation

The argument starts from $2^{N} S = Φ_{N} + R_{N}$ , with $Φ_{N} \in Z$ . A rational $S$ gives a shift $h > 0$ for which $R_{N + h} - R_{N} \in Z$ for every sufficiently large $N$ . Three questions organise the record.

What finite calculation can refute that integrality? The residue test in Definition 1.2 encloses the tail difference in an explicit error interval. Section 5 gives the identities and the independent Farey-window exclusion. Appendix A supplies the complete propagation, Mersenne-residue and rank-one arguments cited in the short paper.

Which information is insufficient to supply such calculations? The short paper determines all finite-level totient relations, with a basis of $k^{e} + 1$ sections in every base $k \geq 2$ . Its integral relation proof recovers the coefficients of every relation. Those are statements about sequences, not their weighted infinite sums. Section 10.8 explains the distinction through carry rank; Section 12.6 constructs a rational $5 / 4$ series retaining the odd totients. Section 2 gives other comparisons, each limited to its stated preserved properties.

What additional estimate would suffice? Section 3 gives the quantified conditions. Section 12.3 separates the weighted exponential-sum estimate from the counting and group-mean bounds, corresponding to the four-estimate problem in the short paper. A second sufficient condition estimates only the average over good indices and needs no group-mean input. Appendix B records the formal declarations; an implication proved there does not establish its remaining hypothesis.

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.

Prefer the manuscript?

PDFOpen the PDF, 163 pages TeXOpen the LaTeX source Problem page

Equations typeset from the exact TeX. Rendered from the LaTeX manuscript source with file fingerprint sha256:2be13b340f9dafc4. Its file fingerprint matched the published source record, so the PDF above, the LaTeX, and this page are one manuscript. Redeploying the site regenerates this page from whatever the public repository holds at that moment.