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

The Möbius-weighted identity

Subtracting $1 / 2$ converts the totient series to a Möbius-weighted squared-Lambert series without changing rationality.

Proposition 5.14 (Möbius-square reduction).

S = \sum_{n \geq 1} \frac{φ (n)}{2^{n}} = \frac{1}{2} + \sum_{d \geq 1} \frac{μ (d)}{(2^{d} - 1)^{2}},

where $μ$ is the Möbius function, so $μ (d) \in {- 1, 0, 1}$ for every $d$ . Consequently Erdős #249 $⟺$ $T := \sum_{d \geq 1} μ (d) / (2^{d} - 1)^{2} \notin Q$ , since adding a rational number does not change irrationality.

Remark 5.15 (Why the bounded coefficients matter; the Erdős-1948 regime). The identity of Proposition 5.14 rewrites $S$ (equivalently $T$ ) as a Lambert-type series with squared denominators and bounded Möbius weights. To see the constant $1 / 2$ directly, expand

\sum_{d \geq 1} \frac{μ (d)}{2^{d} - 1} = \sum_{n \geq 1} \frac{\sum_{d ∣ n} μ (d)}{2^{n}} = \frac{1}{2} .

Combining this identity with $φ = μ * Id$ and $2^{d} = (2^{d} - 1) + 1$ proves the formula for $T$ ; absolute convergence justifies the rearrangements. Thus the corresponding first-power sum $L (μ) := \sum_{d} μ (d) / (2^{d} - 1)$ is rational, whereas $L (1) = \sum_{d \geq 1} 1 / (2^{d} - 1) = E$ is the Erdős–Borwein constant. Erdős proved its irrationality in 1948 [11], and it is also proved in Lean (irrational_erdosBorwein_series).

In contrast, the coefficients of the original power series satisfy $0 \leq φ (n) \leq n$ and are unbounded. Bounded Möbius weights do not make the new denominators powers of a fixed base, so they do not turn this representation into a bounded-digit expansion. The function $φ$ has average order $6 n / π^{2}$ , equivalently $\sum_{k \leq x} φ (k) \sim 3 x^{2} / π^{2}$ , but there is no pointwise estimate $φ (n) = Θ (n)$ . The near-integer irrationality criterion (formalised generically as irrational_of_int_mul_near_int and its base-power specialisation irrational_of_pow_mul_near_int) applies to either representation if its nonzero approximation errors can be established. Boundedness of the coefficients does not itself establish those errors. Erdős’s 1948 argument is itself a proof for the specific Lambert series: it chooses arguments by a system of congruences so that the divisor counts along a block are divisible by increasing powers of the base, and a tail bound then forces a long run of zero digits [11]. Boundedness of a signed weight supplies none of that divisibility input.

The rational identity $L (μ) = 1 / 2$ is already a warning: bounded signed coefficients alone do not imply irrationality. The later denominator calculations retain the Möbius terms whose indices do not divide the chosen LCM; those terms cannot be discarded when transferring an estimate.

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.