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 remaining prime-weighted correlation.

Propositions 12.6 and 12.7 give the unassigned and excluded-cofactor bounds at minimal admissible depth. They must be combined with mean and correlation estimates at the same $X, L, s, η$ ; bounds obtained at unrelated depths cannot be substituted.

With the notation of Section 3.2, the correlation estimate still required is

Re \sum_{N \in G} w_{N} (e (\frac{(2^{h} - 1) φ (m_{N}) (p_{N} - 1)}{2^{t}}) - {\bar{z}}_{m_{N}}) \leq \frac{14}{25} X .

The sum is over the assigned indices with good cofactors, not all primes and cofactors in the interval. By (3), $w_{N}$ is the phase with the selected offset $t$ removed.

Regrouping by $m_{N} = m$ gives weighted sums over the primes $p_{N} = p$ . A bilinear argument would need to separate the $m$ - and $p$ -dependence or estimate the resulting two-variable weight directly. For $N = m p - t$ , that weight depends on $φ (m p + j - t)$ at the other offsets $1 \leq j \leq L + h$ , $j \neq t$ . The identity $φ (m p) = φ (m) (p - 1)$ controls the selected value, not these neighbouring totients. No bound for the resulting correlation is established here; this is the missing step, not a limitation of every bilinear method.

A fixed-offset example illustrates the correlation at issue. At $h = 1$ and offset $t = 2$ , a prime $p \geq 5$ satisfies $φ (p - 1) \equiv 0 (\mod 2)$ and $φ (p) = p - 1$ . The phase of $R_{p - 2}$ is therefore $χ_{- 4} (p)$ times the phase of the following weighted totients, since $e ((p - 1) / 4) = i^{p - 1} = χ_{- 4} (p)$ . One possible fixed-saving estimate, with a fixed $δ > 0$ , would be

Re \sum_{X < p \leq 2 X} χ_{- 4} (p) e (\frac{φ (p + 1)}{8} + \frac{φ (p + 2)}{16} + \dots) \leq (\frac{9}{10} - δ) (π (2 X) - π (X)) .

This fixed-offset example is not Definition 3.4: there $s$ is fixed while $t = L - s + 1$ grows. The example isolates a correlation between a quadratic character at $p$ and the binary phases of totients at $p + 1, p + 2, \dots$ . It is analogous in this respect to shifted correlations studied in the Chowla and Elliott problems, but the analogy does not supply an applicable theorem. The cited theorems have different hypotheses and types of average: Matomäki and Radziwiłł compare short and long averages of real multiplicative functions bounded by $1$ , outside an exceptional set of short intervals [24]; Tao proves logarithmically averaged two-point Chowla and a corrected Elliott statement, the latter under an explicit non-pretentiousness hypothesis on one of the bounded multiplicative functions [25]; and Tao and Teräväinen prove the odd-order logarithmically averaged Chowla conjecture [26]. The phase here depends on $φ$ at a growing window of shifts, so none of these applies directly.

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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