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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Log-concavity of the Möbius-weighted series

Put $Θ_{r} = \sum_{n \geq 0} μ (n + 1) / (2^{n + 1} - 1)^{r}$ , so that $Θ_{2} = S - 1 / 2$ . We prove strict log-concavity for every integer $r \geq 1$ . This orders the values of the family without determining their rationality.

Definition 6.120. For each integer $r \geq 1$ , put

Θ_{r} = \sum_{n \geq 0} \frac{μ (n + 1)}{(2^{n + 1} - 1)^{r}} = 1 - 3^{- r} + \sum_{n \geq 0} \frac{μ (n + 3)}{(2^{n + 3} - 1)^{r}} .

The first two summands isolate the dominant contribution. The remaining absolute sum is at most $1 / (2^{r} (2^{r} - 1))$ , by comparison with $\sum_{n \geq 2} 2^{- r n}$ . In particular, $Θ_{r} > 0$ for $r \geq 1$ : $1 - 3^{- r} \geq 2 / 3$ and this tail bound is at most $1 / 2$ . Erdos249257.SignedQMomentObstruction.mobiusMersenneTheta Erdos249257.SignedQMomentObstruction.mobiusMersenneTwoAtom Erdos249257.SignedQMomentObstruction.mobiusMersenneTailAfterTwo

Theorem 6.121 (The values at exponents one and two). The first two values are $Θ_{1} = 1 / 2$ and $Θ_{2} = S - 1 / 2$ .

The second equality is the Möbius-square identity credited in §5.4; whether $Θ_{2}$ is rational is #249 itself.

Theorem 6.122 (The first two summands give a positive Hankel gap). $(1 - 3^{- (r + 1)})^{2} - (1 - 3^{- r}) (1 - 3^{- (r + 2)}) = 4 / 3^{r + 2}$ , so the rational sequence $1 - 3^{- r}$ is strictly log-concave for every integer $r \geq 1$ .

Theorem 6.123 (Strict log-concavity for all integer $r \geq 1$ ). For every integer $r \geq 1$ ,

Θ_{r} Θ_{r + 2} < Θ_{r + 1}^{2} .

Equivalently, the determinant of $(\begin{matrix} Θ_{r} & Θ_{r + 1} \\ Θ_{r + 1} & Θ_{r + 2} \end{matrix})$ is negative. The two-summand comparison gives an elementary proof, detailed below; the same strict inequality is also proved in Lean.

A quantitative comparison. For this proof put $a_{r} = 1 - 3^{- r}$ and $ε_{r} = 1 / (2^{r} (2^{r} - 1))$ . The preceding bound gives $| Θ_{r} - a_{r} | \leq ε_{r}$ . Since $0 < a_{r} < 1$ , expansion of the two products gives

Θ_{r + 1}^{2} - Θ_{r} Θ_{r + 2} \geq \frac{4}{3^{r + 2}} - B_{r}, B_{r} = 2 ε_{r + 1} + ε_{r + 1}^{2} + ε_{r} + ε_{r + 2} + ε_{r} ε_{r + 2} .

Here $ε_{r + 1} < ε_{r} / 4$ , so $B_{r + 1} < B_{r} / 4$ . The rational comparison

B_{5} = \frac{50106593}{32001951744} < \frac{4}{3^{7}}

therefore proves the inequality for all $r \geq 5$ : the error decreases faster than the positive main term. For $r = 1, 2, 3, 4$ , truncate each $Θ_{j}$ at $n = 8$ . Its omitted absolute sum is at most

\frac{1}{(2^{9} - 1)^{j} (1 - 2^{- j})} .

Using $Θ_{1} = 1 / 2$ and these rational intervals, the four gaps $Θ_{r + 1}^{2} - Θ_{r} Θ_{r + 2}$ are respectively greater than $1 / 4$ , $1 / 16$ , $1 / 64$ and $1 / 200$ . These finite rational comparisons complete the ordinary proof. ◻

Observation 6.124. The inequality excludes a representation $Θ_{r} = \int_{0}^{\infty} x^{r} d ν (x)$ by a positive measure with these finite moments: Cauchy–Schwarz would give $Θ_{r + 1}^{2} \leq Θ_{r} Θ_{r + 2}$ . It does not by itself imply irrationality of $Θ_{2}$ . Indeed, the rational sequence $1 - 3^{- r}$ satisfies the same strict log-concavity inequalities. Thus the determinant sign supplies an obstruction to a positive-moment representation, not a rational-approximation estimate.

The accompanying formal results include a rectangular determinant expansion derived from the Leibniz formula (Erdos249257.SignedQMomentObstruction.det_mul_rectangular), and the following elementary parity fact: in a finite sum of dyadic fractions, if the largest denominator exponent occurs once and its numerator is odd, clearing that denominator gives an odd numerator (Erdos249257.SignedQMomentObstruction.scaled_dyadic_sum_odd).

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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