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

Scope of the index

The index compares particular quantified hypotheses and the results that use them. An unsuccessful implication between two entries therefore says nothing by itself about every argument using the same method.

Each entry states its quantifiers in full. A cofinal assertion, with a quantifier of the form $\forall a_{0} \exists a \geq a_{0}$ , may itself be proved or may remain an unproved hypothesis; the text of each entry says which. A cofinal assertion about a different quantity, or with a different order of quantifiers, need not supply the irrationality condition.

The quantified residue condition for #249, stated in Definition 1.2 and proved sufficient (irrational_totient_series_of_certificate_supply), is the following statement:

\forall h \geq 1, \forall N_{0}, \exists N \geq N_{0}, \exists L, C h N L .

The following catalogue compares fourteen proposed approaches to this condition. It starts with the exponential-sum estimate, then gives thirteen further approaches with their individual hypotheses and source references.

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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