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

PDFOpen the PDF, 163 pages TeXLaTeX source Source on GitHub Problem page

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.

Start here, by the editorial route

In this paper

Equivalent expressions and their denominator bounds

The finite bound in Theorem 5.6 also applies to the two expressions for $S - 1 / 2$ below, with half the denominator bound. Amiram Eldar posted the coprimality interpretation of Proposition 5.9 in revision 28 of OEIS A256936 on 15 March 2026, and the Möbius-square formula of Proposition 5.14 in revision 31 [8]. Steve Fan posted the latter formula on the problem’s forum thread on 16 May 2026 [9]. We derive both identities by Möbius inversion and geometric summation.

Proposition 5.9 (Fair-coin coprimality form). Let $X, Y$ be independent random variables with $Pr (X = n) = Pr (Y = n) = 2^{- n}$ for $n \geq 1$ (independent fair-coin waiting times). Then

S = \frac{1}{2} + Pr (gcd (X, Y) = 1) = \frac{1}{2} + \sum_{\begin{matrix} a, b \geq 1 \\ gcd (a, b) = 1 \end{matrix}} 2^{- (a + b)} .

Equivalently, on the visible lattice: summing $2^{- (a + b)}$ over the half-open coprime pairs ( $a \geq 1$ , $b \geq 0$ , $gcd (a, b) = 1$ ) recovers $\sum_{n} φ (n) / 2^{n}$ exactly, with no boundary correction, because the visible-point count on the half-open antidiagonal at height $n$ equals $φ (n)$ for every $n$ , including $n = 0, 1$ .

Proposition 5.10 (The gcd-layer normalisation). For independent fair-coin waiting times as above, $\sum_{g \geq 1} Pr (gcd (X, Y) = g) = 1$ exactly; and for every $d > 0$ , $Pr (d ∣ X \land d ∣ Y) = 1 / (2^{d} - 1)^{2}$ .

Theorem 5.11 (Denominator exclusion for the coprimality-probability form). Let

Q_{1} := ⌊ \frac{Q_{0}}{2} ⌋ = 39 819 823 323 350 687 661 677 887 437 915 526.

For every $a \in Z$ , $d \in N$ with $0 < d \leq Q_{1}$ : the visible coprime-pair probability $Pr (gcd (X, Y) = 1)$ is not equal to $a / d$ .

Theorem 5.12 (Denominator exclusion for the Möbius-square form). With $Q_{1}$ as in Theorem 5.11: for every $a \in Z$ , $d \in N$ with $0 < d \leq Q_{1}$ , the signed series $T := \sum_{m \geq 1} μ (m) / (2^{m} - 1)^{2} = S - \frac{1}{2}$ (see §5.5) is not equal to $a / d$ .

Remark 5.13 (Why the bound halves, and why this is not new information). $Q_{1} = ⌊ Q_{0} / 2 ⌋$ arithmetically: $Q_{0}$ is odd ( $Q_{0} = 79 639 646 646 701 375 323 355 774 875 831 053$ ), so $Q_{0} / 2 = 39 819 823 323 350 687 661 677 887 437 915 526.5$ and $Q_{1}$ is its floor. The halving is the cost of transporting the known bound through the affine shift $S = \frac{1}{2} + T$ (resp. $S = \frac{1}{2} + Pr (gcd (X, Y) = 1)$ ) on a denominator-exclusion statement: excluding all denominators $\leq Q_{1}$ for $T$ follows from excluding all denominators $\leq Q_{0}$ for $S$ (a denominator- $d$ value of $T$ with $d \leq Q_{1}$ yields a denominator dividing $2 d \leq Q_{0}$ for $S$ ). Theorems 5.11 and 5.12 are therefore the same finite Farey record as Theorem 5.6, transported through Proposition 5.9 and Proposition 5.14 respectively; not independent evidence. The converse finite implication is not asserted: subtracting $1 / 2$ can double a denominator, so the $Q_{1}$ exclusion for $T$ alone need not recover the full $Q_{0}$ exclusion for $S$ .

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.

Prefer the manuscript?

PDFOpen the PDF, 163 pages TeXOpen the LaTeX source Problem page

Equations typeset from the exact TeX. Rendered from the LaTeX manuscript source with file fingerprint sha256:2be13b340f9dafc4. Its file fingerprint matched the published source record, so the PDF above, the LaTeX, and this page are one manuscript. Redeploying the site regenerates this page from whatever the public repository holds at that moment.