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

Rational approximation with a squared-Mersenne remainder

The first-order Lambert term in the diagonal tail difference can be summed exactly. The approximation below leaves only a squared-Mersenne tail to estimate. Small error is only half of a denominator-clearing argument. If a target $α = a / b$ is rational, then every unequal rational $p / q$ satisfies $q | α - p / q | = | a q - b p | / b \geq 1 / b$ . Thus nonzero errors tending to zero after multiplication by the reduced denominators would prove irrationality. The identity and geometric bound below give explicit errors, but not that comparison with the denominators.

Proposition 6.128 (An exact rational approximation formula). For integers $H, D \geq 0$ , with the integer prefixes $Φ_{N}$ defined above,

\begin{aligned} R_{2 H} - R_{H} & - [Φ_{H} - Φ_{2 H} + 2^{H} (2^{H} - 1) (\frac{1}{2} + \sum_{d = 1}^{D} \frac{μ (d)}{(2^{d} - 1)^{2}})] \\ = 2^{H} (2^{H} - 1) \sum_{d > D} \frac{μ (d)}{(2^{d} - 1)^{2}} . \end{aligned}

This follows by substituting the squared-denominator identity for $S$ into $R_{2 H} - R_{H} = 2^{H} (2^{H} - 1) S + Φ_{H} - Φ_{2 H}$ . There is no required ordering between $D$ and $H$ .

Proposition 6.129 (A geometric tail bound). For every integer $D \geq 0$ ,

| \sum_{d > D} \frac{μ (d)}{(2^{d} - 1)^{2}} | \leq \frac{4}{3 (2^{D + 1} - 1)^{2}} .

Indeed, $| μ (d) | \leq 1$ and $2^{D + 1 + j} - 1 \geq 2^{j} (2^{D + 1} - 1)$ for $j \geq 0$ . The sum of the resulting geometric majorant is $\sum_{j \geq 0} 4^{- j} = 4 / 3$ .

Observation 6.130 (One-sided and symmetric error bounds). Suppose $n > D$ , $μ (d) = 0$ for $D < d < n$ , and $| μ (n) | = 1$ . Removing that first nonzero summand gives the sharper enclosure

| μ (n) \sum_{d > D} \frac{μ (d)}{(2^{d} - 1)^{2}} - \frac{1}{(2^{n} - 1)^{2}} | \leq \frac{4}{3 (2^{n + 1} - 1)^{2}} .

The radius is smaller than the positive centre, so the tail has sign $μ (n)$ . The interval’s width, divided by the width of the preceding symmetric enclosure, is

{(\frac{2^{D + 1} - 1}{2^{n + 1} - 1})}^{2} < \frac{1}{4} .

The cited theorem gives this signed enclosure (Erdos249257.SquaredMersenneDiagonalEnclosure.mobiusSquareTail_directed_of_first_support). Multiplication by $2^{H} (2^{H} - 1)$ and translation by the rational approximation above enclose $R_{2 H} - R_{H}$ . If the resulting interval contains no integer, the tail difference is nonintegral. For the symmetric bound, the linked statement requires the distance from its rational centre to every integer to exceed its error radius (Erdos249257.SquaredMersenneDiagonalEnclosure.scaleFullTarget_miss_of_lambert_projected_separation). The enclosure does not establish that strict separation. In particular, shrinking an interval around a rational approximation does not, without a denominator comparison, show that the interval misses every integer.

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.