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

Averaging over the block parameter.

For each fixed $h$ , an average of the nonnegative, normalised block norms can establish the full-block condition. It is enough to prove

\sum_{k = 1}^{K} 2^{- k} | \sum_{N = 2^{k}}^{2^{k + 1} - 1} e (D (h, N, L_{k}) / 2^{L_{k}}) | = o (K),

where every $L_{k}$ satisfies the size inequality for $X = 2^{k}$ . Indeed, the number of $k \leq K$ for which the normalised norm exceeds $21 / 25$ is at most $25 / 21$ times the displayed sum, hence is $o (K)$ . There are therefore arbitrarily large admissible blocks satisfying Definition 3.1. This argument does not establish the four-term condition: that condition requires all four separate budgets with common $X, L, s, η$ .

The displayed average is a Cesàro average of norms on dyadic blocks. It is not the signed logarithmically weighted correlation appearing in the cited Chowla–Elliott theorems. The position of the absolute value matters: cancellation between different blocks would not bound this sum of norms. Invoking entropy decrement would therefore require an additional argument yielding this particular estimate.

The finite phase depends on the $L + h$ totient values at offsets $1, \dots, L + h$ , with the coefficients in (3). Although its absolute value is $1$ , it is not a multiplicative function of $N$ . Both the number of shifts and their dyadic denominators grow with $L$ . The cited fixed-shift correlation theorems do not supply the uniform bounds for this growing family. No such extension is proved here.

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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