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

Denominators of Möbius sums

Theorem 9.44. For every integer $t \geq 5$ , with $P_{t} = {p prime : t / 2 < p \leq t}$ as above,

2^{⌊ t / 2 ⌋} \leq \prod_{p \in P_{t}} (2^{p} - 1) \leq den (H_{t} β_{H_{t}}) .

Moreover, some $p \in P_{t}$ satisfies

2^{⌊ t / 2 ⌋} \leq 2^{p} - 1 < 2^{t}, 2^{p} - 1 ∣ den (H_{t} β_{H_{t}}) .

Indeed, Bertrand’s postulate makes $P_{t}$ nonempty. Its members satisfy $p - 1 \geq ⌊ t / 2 ⌋$ and $p \leq t$ , giving the size bounds. The divisibility follows from Proposition 6.6; divisibility of a positive reduced denominator also gives the product inequality.

Remark 9.45 (Growth of these denominators does not give approximation of $S$ ). The number $H_{t} β_{H_{t}}$ is the finite rational contribution to a totient-tail decomposition. Its complementary term can still cancel its fractional part. The stated denominator bound supplies neither an error estimate for approximation of $S$ nor nonintegrality of $R_{2 H_{t}} - R_{H_{t}}$ .

To apply the diagonal criterion in Theorem 6.49, one must establish this tail nonintegrality at arbitrarily large LCM scales. Denominator growth alone does not supply that condition.

Proposition 9.46 (The additional approximation hypothesis). A sequence of rationals $u_{t}$ proves irrationality of $S$ if

u_{t} \neq S for all sufficiently large t, den (u_{t}) | S - u_{t} | ⟶ 0.

This is the rational-separation criterion of Proposition 6.22: if $S = a / b$ were reduced, every unequal $u_{t}$ would instead satisfy $den (u_{t}) | S - u_{t} | \geq 1 / b$ . An upper bound tending to zero for this product is needed. A lower bound for the denominator cannot establish it; improving a lower bound does not change the actual approximation error.

A different argument could deduce a residue certificate from the surviving divisor and further information about the complementary term. The denominator theorem alone contains no such separation statement. Neither sufficient argument is established by its lower bound, and these are not claimed to exhaust possible approaches.

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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