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

Cancellation on a full block

If every window residue lies close to an endpoint, its complex phase lies close to $1$ . A saving in the norm of a block sum must therefore produce a certificate.

Definition 3.1 (The full-block norm condition). Write $e (x) := \exp (2 π i x)$ and $E (h, N, L) := e ((D (h, N, L) mod 2^{L}) / 2^{L})$ . The block norm condition asserts that

\forall h \geq 1 \forall X_{0} \exists X, L : max (X_{0}, 1) \leq X, 16 (2 X + h + L + 2) \leq 2^{L}, ‖ \sum_{N \in [X, 2 X)} E (h, N, L) ‖ \leq \frac{21}{25} X .

It implies $S \notin Q$ . Erdos249257.TotientTailPeriodKiller.DTWFirstHarmonicNormGap Erdos249257.TotientTailPeriodKiller.irrational_totient_series_of_first_harmonic_norm_gap

A bound on the real part also suffices: it is enough that $\sum_{N \in [X, 2 X)} Re E (h, N, L) \leq \frac{9}{10} X$ (Erdos249257.TotientTailPeriodKiller.exists_certifiedKill_of_first_harmonic_gap; for the implication from $21 / 25$ to $9 / 10$ , Erdos249257.TotientTailPeriodKiller.exists_certifiedKill_of_first_harmonic_norm_gap). Unpacked, the required object is a constant-saving cancellation estimate for the dyadically weighted totient-difference exponential sum

\sum_{X \leq N < 2 X} e (\frac{\sum_{j < L} (φ (N + h + 1 + j) - φ (N + 1 + j)) 2^{L - 1 - j}}{2^{L}}), L subject to the displayed size inequality .

No theorem in this record establishes this estimate with the required quantifiers. The displayed size inequality imposes a lower bound on $L$ ; it does not require $L = \log_{2} X + O (1)$ . A depth of that order would be a possible regime for the estimate, not an additional proved conclusion. A finite instance would not suffice. The norm hypothesis requires cancellation: if all phases are $1$ , the norm is $X$ and the bound fails; a balanced set of opposite phases has sum zero. If instead every phase is $- 1$ , the real part is $- X$ and satisfies the real-part bound, while the norm is $X$ and fails the norm bound. Thus a real-part estimate is strictly weaker for general phase families; these examples do not distinguish the two quantified conditions for the actual totient sequence. The same depth $L$ must work across the whole block.

Comparison with the preceding counterexamples.

The earlier counterexamples do not establish or refute this block estimate. The following distinctions explain why those particular arguments do not settle it.

The parameters grow. The size condition forces the truncation depth to increase with $X$ , so this is not a test of a fixed prefix.
The sample is prescribed: all $N \in [X, 2 X)$ are summed at one depth. Choosing two favourable indices does not establish this block estimate.
The sum uses the accumulated totient differences. It does not prescribe individual values through a congruence construction.
The estimate concerns the actual phases. Neither a bound on carry rank nor agreement only in parity gives it.
The proved implication uses the real-part threshold $9 / 10$ and the constant in the size bound $16$ . The stronger norm bound $21 / 25$ implies that real-part bound. These constants do not assert any distribution theorem for the totient phases.

Formal sources for these implications: Erdos249257.TotientTailPeriodKiller.windowFirstCos_gt_of_not_certifiedKill Erdos249257.TotientTailPeriodKiller.exists_certifiedKill_of_first_harmonic_gap_subset

Remark 3.2 (The finite-prefix example does not choose a method). The finite-prefix example does not select an analytic method. It shows only that a proof valid for every sequence $0 \leq c (n) \leq n$ cannot deduce the required assertion from one fixed prefix. An argument using additional identities of $φ$ , a theorem propagating a finite base case, a Farey estimate, or a suitable exponential sum could provide information not contained in that prefix. The earlier finite audit cannot rank all such possibilities.

Remark 3.3 (Finite certificate observations). In the historical table of $28$ diagonal certificates through $t = 64$ , the reported depths exceed the necessary depth bound by a small additive constant. A separate enumeration at the indices where the least common multiple increases records a closest margin from the central interval’s boundary of approximately $0.000221$ of the modulus, at $t = 100$ . Both are finite observations. Neither establishes nor refutes equidistribution or the estimate at arbitrarily large scales.

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.

Cite and contact. Cite this paper by its title, author and date above, with its PDF; cite the earlier sources it uses for a mathematical result. For software, use the release citation and give the commit used. Contact Will with questions or corrections.

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