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 residue margin at an LCM jump

Theorem 9.37 (Arbitrarily large prime-power LCM jumps). For every $t_{0} \in N$ there is $t \geq t_{0}$ with $H (t) < H (t + 1)$ . One may take $t = p - 1$ for any prime $p > t_{0}$ . The positions $t = 2^{a} - 1$ are also strict LCM jumps for $a \geq 1$ : $2^{a}$ is the next required power of $2$ . The restriction excludes $a = 0$ , since $H (0) = H (1) = 1$ . The second condition in the next definition asks for the additional residue margin along this explicit sequence. Knowing these jump positions does not establish that margin.

Definition 9.38 (The remaining residue margin). For $t \geq 3$ , put $H = H_{t}$ and define

\begin{aligned} m & = ⌊ \log_{2} H ⌋ + 10, \\ d & = (D (H, H + 1, m) - D (H, H, m)) mod 2^{m}, \\ σ_{t} & = min {d - 2^{m - 5}, 2^{m} - 2^{m - 5} - d} . \end{aligned}

Thus $d$ is the difference of two adjacent window numerators, reduced modulo $2^{m}$ , not the residue of $D (H, H, m)$ alone. The integer $σ_{t}$ is nonnegative exactly when $2^{m - 5} \leq d \leq 2^{m} - 2^{m - 5}$ . The choice of $m$ makes the error bound $3 H + 2 m + 5$ smaller than $2^{m - 5}$ ; nonnegativity is the additional arithmetic requirement, not a consequence of that choice. canonicalAdjacentSuffixCentralSlack canonicalAdjacentSuffixDepth diagonalAdjacentSuffixResidue

The first unproved condition asks for every $t_{0}$ for an index $t \geq max (3, t_{0})$ with $H_{t} < H_{t + 1}$ and $σ_{t + 1} \geq 0$ . The second restricts these endpoints to powers of two: for every $a_{0}$ , there must be $a \geq max (2, a_{0})$ with $σ_{2^{a}} \geq 0$ . They are defined in Lean (CanonicalAdjacentSuffixPostJumpSlackSupply, CanonicalAdjacentSuffixPowerTwoPostJumpSlackSupply). Each condition implies $S \notin Q$ (irrational_totientSeries_of_canonicalAdjacentSuffixPostJumpSlackSupply, irrational_totientSeries_of_canonicalAdjacentSuffixPowerTwoPostJumpSlackSupply). The implications are proved; both conditions are unproved.

Remark 9.39 (Finite numerical evidence). All $40$ strict jumps up to endpoint $113$ pass, closest margin $\approx 0.000221$ of the modulus at $t = 100$ ; the power-of-two endpoints $4, 8, 16, 32$ pass with slack fractions $0.41, 0.036, 0.40, 0.19$ . This is a finite computation; both conditions remain unproved at arbitrarily large endpoints. A strict jump has $t + 1 = p^{k}$ for a prime $p$ and $k \geq 1$ , with $H_{t + 1} = p H_{t}$ (periodLcm_succ_eq_prime_mul_of_strict_jump). Thus $σ_{t + 1}$ uses the new height $p H_{t}$ , not the pre-jump height $H_{t}$ . Estimating its change would require information about the adjacent-window residue, not just the LCM ratio. Moreover $σ_{t} = σ_{u}$ whenever $H_{t} = H_{u}$ (canonicalAdjacentSuffixCentralSlack_eq_of_periodLcm_eq). Thus the value changes only when the LCM height changes. This reduces the number of distinct values to study; it does not determine their signs.

Proposition 9.40 (A sufficient inequality). If, for every $a_{0} \in N$ , there is $a \geq max (2, a_{0})$ with $σ_{2^{a}} \geq 0$ , then $S \notin Q$ . This is the sufficient condition defined above, not a claim that the inequality holds at arbitrarily large indices. Equality at the edge is allowed: the condition is nonnegativity, not strict positivity.

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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