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.

Start here, by the editorial route

In this paper

The exponential-sum criterion

We express the truncated tail difference as a phase and ask for cancellation when that phase is summed over a block. The following implications explain how a saving gives a residue certificate and, with the required quantifiers, Definition 1.2.

Definition 9.1 (The truncated difference). For $h, N, L \in N$ ,

D (h, N, L) = \sum_{j = 0}^{L - 1} (φ (N + h + 1 + j) - φ (N + 1 + j)) \cdot 2^{L - 1 - j} \in Z .

This is the depth- $L$ truncation of $2^{L} \cdot (R_{N + h} - R_{N})$ , where $R_{N} = \sum_{j \geq 1} φ (N + j) / 2^{j}$ is the local totient tail (totientTail). windowDiscrepancy

Definition 9.2 (The central-residue condition).

C h N L :\Leftrightarrow (N + h + L + 2) < D (h, N, L) mod 2^{L} < 2^{L} - (N + h + L + 2) .

That is: the residue of $D (h, N, L)$ modulo $2^{L}$ avoids the radius- $(N + h + L + 2)$ neighbourhood of $0$ . It forces the size condition $2 (N + h + L + 2) < 2^{L}$ (certifiedKill_depth_floor). certifiedKill

Definition 9.3 (The complex phase and its real part). Put

Re E (h, N, L) = \cos (2 π \frac{D (h, N, L) mod 2^{L}}{2^{L}}), E (h, N, L) = \exp (i 2 π \frac{D (h, N, L) mod 2^{L}}{2^{L}}) .

The real part of $E$ is the displayed cosine (windowFirstExp_re), and $| E (h, N, L) | = 1$ (norm_windowFirstExp). Thus $E$ is the first additive character of the endpoint discrepancy modulo $2^{L}$ , a standard exponential-sum object. windowFirstCos windowFirstExp

Theorem 9.4 (A real-part bound gives a certificate). For all $h, X, L$ with $0 < X$ and the size condition $16 (2 X + h + L + 2) \leq 2^{L}$ , if

\sum_{N = X}^{2 X - 1} Re E (h, N, L) \leq \frac{9}{10} X,

then $\exists N \in [X, 2 X)$ with $C h N L$ .

Theorem 9.5 (The same implication for a nonempty subset). For any nonempty finite $T \subseteq N$ with $T \subset [0, 2 X)$ and the same size condition, if

\sum_{N \in T} Re E (h, N, L) \leq \frac{9}{10} | T |,

then $\exists N \in T$ with $C h N L$ . This generalises Theorem 9.4: $T$ can be any explicitly chosen nonempty finite subset of the dyadic block, not the whole block, and the proof (an averaging pigeonhole) never uses that $T$ has positive density or comes from a partition.

Theorem 9.6 (A norm bound gives the real-part criterion). The complex norm bound implies the real-part bound ( $| z | \geq Re (z)$ , and $21 / 25 < 9 / 10$ absorbs the slack), so it composes through Theorem 9.4 to the same certificate. Define

block norm condition :\Leftrightarrow \forall h > 0 \forall X_{0} \exists X, L, max (X_{0}, 1) \leq X \land 16 (2 X + h + L + 2) \leq 2^{L} \land

‖ \sum_{N = X}^{2 X - 1} E (h, N, L) ‖ \leq \frac{21}{25} X .

This sufficient condition is unproved. Then

block norm condition ⟹ Irrational (\sum_{n \geq 0} \frac{φ (n)}{2^{n}}),

proved in full, with no further gap, by chaining Theorem 9.6 $\to$ Theorem 9.4 $\to$ Theorem 6.47.

Observation 9.7 (What remains unproved). The implications above do not establish the cofinal block estimate. Three further Lean statements retain a separation or gap hypothesis: one uses the two-element set ${N, M}$ (AdjacentPhaseSeparation.lean), and the other two use a supplied finite set $T$ and the singleton ${N}$ (PivotAntiReconstruction.lean, PivotAntiReconstruction.lean). They prove what follows from the relevant estimate; they do not provide that estimate at arbitrarily large scales. The finite numerical observations in Table 1 are a different kind of evidence and do not discharge this quantified hypothesis.

Remark 9.8 (What the reduction does and does not prove). The required savings are explicit: a real-part bound of $9 X / 10$ , or the stronger norm bound of $21 X / 25$ , on blocks at arbitrarily large scales for every positive shift $h$ . An unspecified nonzero saving is not the stated hypothesis. A bound $o (X)$ would suffice, but it is more than the implication needs.

The inequality $16 (2 X + h + L + 2) \leq 2^{L}$ can be met by increasing $L$ for fixed $X, h$ . Doing so changes the exponential sum, and hence does not preserve an estimate proved at another depth. The arithmetic estimate must hold at a depth satisfying this inequality.

The subset theorem permits any nonempty finite sample inside $[0, 2 X)$ ; no density assumption is required. A prime-defined sample is one possible choice. The four-term decomposition of §3.2 isolates one prime factor and the remaining totient-dependent phase, but does not prove the required cancellation between them.

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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