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 four-term bound for the real part

We isolate the phase of one term $φ (m p)$ with $p > m$ prime, then group indices with the same cofactor $m$ . The resulting identity separates three tasks: count the excluded indices, bound the group means, and estimate the remaining weighted, mean-subtracted sum. Only the last of these remains open after the stated group-mean input. The identity itself supplies none of the estimates.

Fix $h, s \geq 1$ , $L \geq s + h$ , $X \geq 1$ and $0 < η < 1$ , and put $t = L - s + 1$ . For $X \leq N < 2 X$ , let $p_{N}$ be the largest prime factor of $N + t$ and put $m_{N} = (N + t) / p_{N}$ . Assign the index $N$ to this factorisation when

m_{N} \leq ⌊ \frac{\sqrt{X}}{2} ⌋, p_{N} > 2 ⌊ \sqrt{X} ⌋ .

Write $A$ for the assigned indices and $G = {N \in A : φ (m_{N}) \geq η m_{N}}$ . The two inequalities ensure $p_{N} > m_{N} \geq 1$ , so $φ (N + t) = φ (m_{N}) (p_{N} - 1)$ .

Since $h < t \leq L$ , the selected term occurs in both truncated tails: it has coefficient $2^{- (t - h)}$ in the shifted tail and $- 2^{- t}$ in the unshifted one. Their sum is $(2^{h} - 1) 2^{- t}$ . On the assigned indices, set

z_{N} = e (\frac{(2^{h} - 1) φ (m_{N}) (p_{N} - 1)}{2^{t}}), w_{N} = \frac{E (h, N, L)}{z_{N}} .

Both factors have absolute value $1$ . Let ${\bar{z}}_{m}$ be the average of $z_{N}$ over the assigned indices with cofactor $m$ , taking the average to be zero when the group is empty. The four contributions are then defined by the identity

\begin{aligned} \sum_{N = X}^{2 X - 1} E (h, N, L) & = \underset{mean-subtracted contribution}{\underset{⏟}{\sum_{N \in G} w_{N} (z_{N} - {\bar{z}}_{m_{N}})}} + \underset{mean contribution}{\underset{⏟}{\sum_{N \in G} w_{N} {\bar{z}}_{m_{N}}}} \\ + \underset{excluded-cofactor contribution}{\underset{⏟}{\sum_{N \in A ∖ G} E (h, N, L)}} + \underset{unassigned contribution}{\underset{⏟}{\sum_{\begin{matrix} X \leq N < 2 X \\ N \notin A \end{matrix}} E (h, N, L)}} . \end{aligned}

This follows by partitioning the indices and using $E (h, N, L) = w_{N} z_{N}$ . Subtracting the group mean is exact algebra; it does not make the two factors independent. The chosen prime need not be confined to the selected argument: with $X = 16$ , $h = s = 1$ , $L = 20$ and $N = 18$ , the argument is $N + t = 38 = 2 \cdot 19$ , which satisfies both assignment inequalities, but the same prime also divides $N + 1 = 19$ . Thus the factorisation at $N + t$ alone gives no separation from the other totient values in the phase.

Definition 3.4 (Four simultaneous inequalities). For every $h > 0$ there are $s > 0$ and $η \in (0, 1)$ such that for every $X_{0}$ there are $X, L$ with $max (X_{0}, 1) \leq X$ , $h \leq L - s$ , $16 (2 X + h + L + 2) \leq 2^{L}$ , and all four of

\begin{aligned} Re \sum_{N \in G} w_{N} (z_{N} - {\bar{z}}_{m_{N}}) & \leq \frac{14}{25} X, \\ ‖ \sum_{N \in G} w_{N} {\bar{z}}_{m_{N}} ‖ & \leq \frac{1}{100} X, \\ ‖ \sum_{N \in A ∖ G} E (h, N, L) ‖ & \leq \frac{1}{100} X, \\ ‖ \sum_{\begin{matrix} X \leq N < 2 X \\ N \notin A \end{matrix}} E (h, N, L) ‖ & \leq \frac{8}{25} X . \end{aligned}

It implies $S \notin Q$ . Erdos249257.TotientTailPeriodKiller.DTWPivotResidualDecorrelation Erdos249257.TotientTailPeriodKiller.irrational_totient_series_of_pivotResidualDecorrelation

First fix $h$ , then choose $s, η$ ; these must remain fixed as the threshold $X_{0}$ increases. At each threshold all four bounds must hold for one common $X, L$ , rather than along unrelated sequences of blocks. Their constants add to $14 / 25 + 1 / 100 + 1 / 100 + 8 / 25 = 9 / 10$ , yielding the real-part criterion. This is sufficient for a certificate and does not assert the stronger $21 / 25$ norm criterion. In particular, no independence of the two phase factors follows from the grouping.

The decomposition is an exact identity, proved in Lean (Erdos249257.TotientTailPeriodKiller.windowFirstExp_sum_eq_pivot_decomposition), as is the implication from the four bounds to the real-part bound (Erdos249257.TotientTailPeriodKiller.first_harmonic_gap_of_pivotBudgetAt). The reason a new estimate is needed is visible within one group: $\sum_{N} (z_{N} - {\bar{z}}_{m}) = 0$ , whereas $\sum_{N} w_{N} (z_{N} - {\bar{z}}_{m})$ need not vanish. The residual weight varies with the same prime that determines $z_{N}$ . By contrast, a uniform bound for $| {\bar{z}}_{m} |$ immediately controls the mean contribution because $| w_{N} | = 1$ . Thus small means and weighted cancellation are different statements. A two-point group makes this explicit:

(z_{1}, z_{2}) = (1, - 1), (w_{1}, w_{2}) = (1, - 1), \bar{z} = 0, \sum_{j = 1}^{2} w_{j} (z_{j} - \bar{z}) = 2.

The unweighted mean vanishes while the weighted terms agree exactly. This example explains why an implication is missing; it is not a counterexample to the estimate for the actual totient windows. For the latter, only the real-part bound $14 X / 25$ is required, not the earlier norm bound $X / 2$ . For $X \geq 4$ and $s = L - h$ , the assigned indices with cofactor $m_{N} = 1$ are exactly those for which $N + h + 1$ is prime. This describes one cofactor group, not all assigned indices. The source proves this membership equality (Erdos249257.TotientTailPeriodKiller.mem_pivotFiber_one_overlap_iff), using $φ (m p) = φ (m) (p - 1)$ and $p > m$ . This special choice has $s = L - h$ varying with $L$ ; it illustrates the arithmetic of the groups but does not establish the condition with $s$ fixed before $X_{0}$ .

The four inequalities have not been established together. Propositions 12.6 and 12.7 give the unassigned and excluded-cofactor bounds at minimal admissible depth. At $s = 26$ , $η = 1 / 1000$ , the prime-number-theorem argument discussed in Section 3.3 also bounds the mean contribution for all large $X$ . With that input, the missing estimate is the real-part bound for the mean-subtracted contribution, at the same parameters and depth, on arbitrarily large blocks. Theorem 3.5 instead asks for one bound on $G$ , without a group-mean input.

A nonzero determinant alone does not prove the needed correlation estimate. The comparison in the linked source (Erdos249257.ResidualGaugeObstruction.locked_reconstruction_preserves_nonzero_minor) preserves such a determinant while fixing an entire row of weighted phases to $1$ . A successful application must distinguish that comparison, for example by additional arithmetic information, rather than only by column rescaling. The exact four-term identity supplies no prime-distribution estimate.

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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