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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The simultaneous inequalities.

Use the four finite sums defined in Section 3.2; in particular, each group mean averages the selected prime phase, not the residual weight. For every $h \geq 1$ there must exist $s \geq 1$ and $η \in (0, 1)$ such that for every $X_{0}$ there are $X \geq max (X_{0}, 1)$ and $L$ with $h \leq L - s$ , $16 (2 X + h + L + 2) \leq 2^{L}$ , and all four of

\begin{aligned} Re \sum_{N \in G} w_{N} (z_{N} - {\bar{z}}_{m_{N}}) & \leq \frac{14}{25} X, \\ ‖ \sum_{N \in G} w_{N} {\bar{z}}_{m_{N}} ‖ & \leq \frac{1}{100} X, \\ ‖ \sum_{N \in A ∖ G} E (h, N, L) ‖ & \leq \frac{1}{100} X, \\ ‖ \sum_{\begin{matrix} X \leq N < 2 X \\ N \notin A \end{matrix}} E (h, N, L) ‖ & \leq \frac{8}{25} X . \end{aligned}

Their simultaneous validity remains unproved; together they imply irrationality (Erdos249257.TotientTailPeriodKiller.DTWPivotResidualDecorrelation, Erdos249257.irrational_totient_series_of_pivotResidualDecorrelation). The decomposition is an exact identity (Erdos249257.TotientTailPeriodKiller.windowFirstExp_sum_eq_pivot_decomposition). The last two sums are bounded by counting indices, since every phase has modulus one. The mean contribution is bounded by the triangle inequality once a uniform group-mean estimate is available. The remaining sum retains a varying weight after centring, and requires a separate real-part estimate. The parameters $s, η$ must be fixed before $X_{0}$ , with the estimates valid for the same $X, L$ ; separate successful blocks do not suffice. For $X \geq 4$ and $s = L - h$ , the single cofactor group $m_{N} = 1$ , not the full assigned set, consists of the indices for which $N + h + 1$ is prime (Erdos249257.TotientTailPeriodKiller.mem_pivotFiber_one_overlap_iff).

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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