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

A general squared-Lambert identity

The same geometric summation applies to any weight with the stated linear bound.

Proposition 5.16 (The squared-Lambert identity). For $w : N \to R$ with $| w (d) | \leq d$ for all $d > 0$ , and $0 \leq r < 1$ :

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

The identity follows by expanding $r^{d} / (1 - r^{d})$ as a geometric series and interchanging absolutely convergent sums. It is the squared analogue of the classical divisor-sum Lambert identity $\sum_{n} a_{n} q^{n} / (1 - q^{n}) = \sum_{m} (\sum_{d ∣ m} a_{d}) q^{m}$ [16].

At $r = 1 / 2$ , the choices $w = 1$ and $w = φ$ give the following two identities. They help distinguish the role of the weight from that of the squared denominator.

Proposition 5.17 (The divisor-sum identity).

\sum_{d \geq 1} \frac{1}{(2^{d} - 1)^{2}} = \sum_{n \geq 1} \frac{σ (n) - τ (n)}{2^{n}} = ζ_{q} (2) - ζ_{q} (1) at q = \frac{1}{2},

where $σ$ is the sum-of-divisors function and $τ$ the number-of-divisors function. The identity is machine-checked. With $ζ_{q} (s) = \sum_{n \geq 1} n^{s - 1} q^{n} / (1 - q^{n})$ , Postelmans and Van Assche prove that $1, ζ_{q} (1), ζ_{q} (2)$ are linearly independent over $Q$ for $q = 1 / p$ with an integer $p \geq 2$ [12]; at $q = \frac{1}{2}$ this gives the irrationality of the value $ζ_{q} (2) - ζ_{q} (1)$ . Two irrational numbers can have a rational difference, so the separate irrationality of the two values would not suffice.

Proposition 5.18 (The gcd-moment identity).

\sum_{d \geq 1} \frac{φ (d)}{(2^{d} - 1)^{2}} = \sum_{n \geq 1} (P (n) - n) \cdot 2^{- n} = E [gcd (X, Y)],

where $P (n) := \sum_{e ∣ n} φ (e) \cdot (n / e) = (φ * Id) (n)$ is Pillai’s gcd-sum function $P (n) = \sum_{k \leq n} gcd (k, n)$ [15] and $X, Y$ are the independent fair-coin waiting times of Proposition 5.9. The identity itself is machine-checked; irrationality of $E [gcd (X, Y)]$ is not proved here. This is a different series from $S$ .

Remark 5.19 (Related series). Writing $L (f) := \sum_{d} f (d) / (2^{d} - 1)$ and $L_{2} (f) := \sum_{d} f (d) / (2^{d} - 1)^{2}$ , the preceding identities give $L (μ) = \frac{1}{2}$ (rational), $L (1) = E$ (Erdős–Borwein, irrational), $L (φ) = 2$ (rational); $L_{2} (μ) = S - \frac{1}{2}$ (irrationality open; this is #249), $L_{2} (1) = ζ_{q} (2) - ζ_{q} (1)$ (irrational by the cited theorem, Proposition 5.17), $L_{2} (φ) = E [gcd (X, Y)]$ (irrationality open, Proposition 5.18). These examples show why the denominator shape alone does not determine irrationality: changing the weight can produce either rational or irrational values.

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.

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