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 estimates

Two explicit estimates for $φ$ remain: weighted cancellation in the exponential sums and a residue inequality involving the final totient difference. Neither is derived here from the cited methods. Other approaches to the irrationality question are not excluded.

Exponential sums.

For each fixed $h > 0$ and every threshold $X_{0}$ , the block norm condition asks for some $X \geq max (X_{0}, 1)$ and $L$ satisfying both the modulus bound and

| \sum_{N \in [X, 2 X)} e (D (h, N, L) / 2^{L}) | \leq \frac{21}{25} X .

The saving $21 / 25$ is fixed, but the chosen block and depth may depend on $h$ and $X_{0}$ ; uniformity of these choices in $h$ is not required. The preceding four-term decomposition and the proved counting bounds identify contributions to this sum. By Theorem 3.5(b) the two remaining weighted estimates may be replaced by the single bound $Re \sum_{N \in G} E (h, N, L (X)) \leq 603 X / 1000$ at $s = 26$ , $η = 1 / 1000$ and minimal depth, for arbitrarily large $X$ ; that bound is open. The record therefore gives an exact reduction and some of its estimates; the displayed cancellation bound remains unproved.

The last totient difference.

The endpoint criteria compare

δ_{2^{a}} (2 q + 2) = φ (2 H + 2 q + 2) - φ (H + 2 q + 2), H = H_{2^{a}},

with twice the centred residue $u_{a, q}$ . An appropriate inequality, proved for arbitrarily large $a$ with the stated bounds on $q$ , would contradict the carry interval required by integrality. The exact identity explains why such an inequality would suffice; it does not supply the inequality. The finite examples through $t = 82$ , and the short-window examples at $a = 4, 6$ , do not establish an asymptotic pattern.

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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