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

The sharp rank-one quotient bound

The quotients below are a particular proposed family of rational approximants to a Möbius-weighted value. We determine their minimum and show that they remain a fixed distance above that value. Put

Θ_{2} = \sum_{d \geq 1} \frac{μ (d)}{(2^{d} - 1)^{2}} = S - \frac{1}{2}, Q (e, Y) = \frac{(\sum_{d = 1}^{Y} μ (d) / (2^{d} - 1)^{e + 2})^{2}}{\sum_{d = 1}^{Y} μ (d) / (2^{d} - 1)^{2 e + 2}} .

The divisor-convolution identity $φ = μ * Id$ gives $S = \sum_{d \geq 1} μ (d) 2^{d} / (2^{d} - 1)^{2}$ . Now $2^{d} = (2^{d} - 1) + 1$ and $\sum_{d \geq 1} μ (d) / (2^{d} - 1) = 1 / 2$ , the latter following from $\sum_{d ∣ n} μ (d) = 0$ for $n > 1$ and $1$ for $n = 1$ . These identities give the displayed value of $Θ_{2}$ ; all the sums are absolutely convergent. Amiram Eldar posted the corresponding formula $S = \frac{1}{2} + \sum_{d \geq 1} μ (d) / (2^{d} - 1)^{2}$ , together with its coprimality interpretation, to OEIS A256936 on 15 March 2026 [8], and Steve Fan posted the formula on the problem’s forum thread on 16 May 2026 [9]. The result below gives the exact extremum for this family, refining the earlier lower bound $1 / 480$ in Section 6.6.13.

Theorem A.3 (A lower bound for the rank-one quotients). For $e \geq 1$ and $Y \geq 4$ , the denominator of $Q (e, Y)$ is positive, and the unique minimiser is $(e, Y) = (1, 5)$ . Every admissible quotient and every nonempty finite positive weighted average of such quotients satisfies

Q - Θ_{2} > \frac{21}{320} .

The uniform bound $Q - Θ_{2} > 1 / 15$ is false already at $(1, 5)$ .

Proof. We first bound every denominator away from zero. We then compare $Q (1, 5)$ with the whole range $e \geq 2$ using one uniform estimate, and with $e = 1$ using a seven-term prefix. Put

P_{Y} (r) = \sum_{d = 1}^{Y} \frac{μ (d)}{(2^{d} - 1)^{r}}, Θ_{r} = \sum_{d \geq 1} \frac{μ (d)}{(2^{d} - 1)^{r}}, E_{r} (Y) = \frac{2^{- r Y}}{1 - 2^{- r}} .

Since $| μ (d) | \leq 1$ and $2^{d} - 1 \geq 2^{d - 1}$ ,

| Θ_{r} - P_{Y} (r) | \leq E_{r} (Y), | P_{Y} (r) - P_{Z} (r) | \leq E_{r} (Z) (Y \geq Z) .

For $r \geq 4$ , separating the terms $d = 1, 2$ gives $Θ_{r} = 1 - 3^{- r} + T_{r}$ , where $| T_{r} | \leq [2^{r} (2^{r} - 1)]^{- 1}$ . Thus

α := \frac{6373}{6480} = 1 - \frac{1}{81} - \frac{1}{240} \leq Θ_{r} < 1, E_{r} (Y) \leq ε := \frac{1}{61440} (Y \geq 4) .

For the strict upper bound, the tail bound is smaller than $3^{- r}$ : $2^{r} (2^{r} - 1) > 3^{r}$ for $r \geq 4$ . In particular $P_{Y} (2 e + 2) \geq α - ε > 0$ for every admissible $e, Y$ , proving denominator positivity.

Write $q_{*} = Q (1, 5)$ . Using $(μ (1), \dots, μ (7)) = (1, - 1, - 1, 0, - 1, 1, - 1)$ gives

q_{*} = \frac{35076077250375200}{37573118933633199} .

For $e \geq 2$ , both exponents in $Q (e, Y)$ are at least four, so the uniform bounds just proved apply to the numerator and denominator. For $e = 1$ , the numerator instead contains $P_{Y} (3)$ ; we use the seven-term prefix to bound it. This leaves the two cases $Y = 4, 6$ to compare with $Y = 5$ . The four exact rational comparisons are

\begin{aligned} \frac{(α - ε)^{2}}{1 + ε} & > q_{*} & (e \geq 2), \\ \frac{79049881}{84671559} = Q (1, 4) & > q_{*} & (e = 1, Y = 4), \\ \frac{1043700953468949889}{1117992059749781919} = Q (1, 6) & > q_{*} & (e = 1, Y = 6), \\ \frac{(P_{7} (3) - 1 / 1835008)^{2}}{P_{7} (4) + 1 / 251658240} & > q_{*} & (e = 1, Y \geq 7) . \end{aligned}

Each comparison follows by multiplication by positive denominators. The first left-hand side bounds every quotient in its range from below. So does the last, by the tail estimates with $Z = 7$ ; the numerator lower bound is positive before it is squared. The middle two rows give the quotients themselves. These ranges cover every admissible pair except $(1, 5)$ , proving its unique minimality.

It remains to compare the minimum with $Θ_{2}$ . An upper bound for $Θ_{2}$ proves the stated gap; a lower bound disproves the stronger proposed gap. The same tail estimate at $r = 2$ gives

P_{6} (2) - \frac{1}{3072} \leq Θ_{2} \leq P_{7} (2) + \frac{1}{12288} .

The exact prefixes are

P_{6} (2) = \frac{3309559}{3814209}, P_{7} (2) = \frac{53376062902}{61519376961} .

Comparison of fractions gives

q_{*} - (P_{7} (2) + \frac{1}{12288}) > \frac{21}{320}, q_{*} - (P_{6} (2) - \frac{1}{3072}) < \frac{1}{15} .

The minimum proves the lower gap for every quotient; the second inequality proves failure of the proposed $1 / 15$ bound at $(1, 5)$ . Multiplying the strict lower gap by positive weights, summing over a nonempty finite set and dividing by the positive total weight proves the assertion for weighted averages. ◻

This theorem treats averages after the quotients are formed. Signed coefficients, coupling before the quotient, and higher-rank kernels require separate arguments. For rational approximants $p_{j} / q_{j}$ in lowest terms, with $q_{j} > 0$ , a sufficient arithmetic target is

0 < q_{j} | Θ_{2} - p_{j} / q_{j} | ⟶ 0.

The denominator sets the scale of the required error. Convergence alone does not control their product. For the elementary dyadic prefixes, the available bound is only $q_{j} | S - p_{j} / q_{j} | \leq N + 2$ at depth $N$ , not a quantity tending to zero. A prescribed approximation family therefore needs a sharper error or reduced-denominator bound. When approximants may be chosen freely, continued-fraction convergents show that the criterion is equivalent to irrationality, just as (8) is. The theorem above excludes the displayed rank-one quotients and their positive averages, not other approximation families.

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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