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 doubling identity

Write $S = \sum_{n \geq 1} φ (n) / 2^{n}$ , $R_{N} = \sum_{m \geq 1} φ (N + m) / 2^{m}$ for the local tail (Erdos249257.TotientTailPeriodKiller.totientTail), and $Φ_{N} = \sum_{n \leq N} φ (n) 2^{N - n} \in Z$ for the integer prefix, so that $2^{N} S = Φ_{N} + R_{N}$ . For $h \geq 1$ set

α_{h} := (2^{h} - 1) S .

Lemma 12.1 (The doubling identity). For all $N \geq 0$ and $h \geq 1$ ,

R_{N + 1} = 2 R_{N} - φ (N + 1), R_{N + h} - R_{N} = 2^{N} α_{h} - (Φ_{N + h} - Φ_{N}),

so $R_{N + h} - R_{N} \equiv 2^{N} α_{h} (\mod 1)$ , and consequently the exact first character of the tail difference is the $\times 2$ orbit of one real number:

e (R_{N + h} - R_{N}) = e (2^{N} α_{h}), e (x) := \exp (2 π i x) .

Hence, for fixed $h$ , the phases ${R_{N + h} - R_{N} mod 1}_{N \geq 0}$ are the forward orbit of $α_{h} mod 1$ under $x \mapsto 2 x$ .

The recurrence $R_{N + 1} = 2 R_{N} - φ (N + 1)$ is one line from the definition and is used below only for intuition.

These identities express the phases as a lacunary exponential sum in one real variable. The next proposition transfers an estimate for that orbit to the truncated phases, accounting for the truncation error. The identities themselves are already in the cited sources.

Proposition 12.2 (Transfer to the doubling orbit). Suppose that

\forall h \geq 1 \forall X_{0} \exists X \geq max (X_{0}, 1) : \sum_{N = X}^{2 X - 1} \cos (2 π 2^{N} α_{h}) \leq \frac{89}{100} X .

This hypothesis implies that $S$ is irrational.

Proof. Fix $h \geq 1$ and $X_{0}$ , and take $X$ as supplied. Choose $L \geq h$ with $2^{L} \geq 1024 (2 X + h + L + 2)$ ; such $L$ exists because $2^{L} / L \to \infty$ , and it satisfies the size inequality $16 (2 X + h + L + 2) \leq 2^{L}$ a fortiori. For every $N < 2 X$ the proved truncation bound (Erdos249257.TotientTailPeriodKiller.norm_windowFirstExp_sub_tailOrbitFirstExp_lt) gives $‖ E (h, N, L) - e (2^{N} α_{h}) ‖ < 2 π (N + L + h + 2) / 2^{L} \leq 2 π / 1024 < 1 / 100$ , so, by the corresponding cosine comparison (Erdos249257.TotientTailPeriodKiller.windowFirstCos_le_add_of_tailOrbitGap), $Re E (h, N, L) \leq \cos (2 π 2^{N} α_{h}) + 1 / 100$ . Summing over the $X$ values $N \in [X, 2 X)$ gives $\sum_{N} Re E (h, N, L) \leq \frac{89}{100} X + \frac{1}{100} X = \frac{9}{10} X$ , so Theorem 9.4 produces $N \in [X, 2 X)$ with $C (h, N, L)$ , and $N \geq X \geq X_{0}$ . This is exactly the quantified certificate condition of Theorem 6.47. ◻

The proof above makes the truncation margin explicit. The full implication is also proved in Lean (ErdosProblems.Erdos249.irrational_totient_series_of_tailOrbitBlockGap). It uses a real-part estimate for the exact phases and spends $1 / 100$ on truncation. Compare the sufficient norm condition of Theorem 6.59. This is a separately sufficient condition. It is not asserted to be equivalent to the $21 / 25$ norm bound or to the four separate estimates stated above. The constants account for different quantities and different error margins.

Corollary 12.3 (A digit version of the analytic condition). Let $ρ_{h} (X)$ denote the proportion of $N \in [X, 2 X)$ with $‖ 2^{N} α_{h} ‖_{R / Z} \geq 1 / 4$ . If for every $h \geq 1$ there are cofinally many $X$ with $ρ_{h} (X) \geq 11 / 100$ , then $S$ is irrational. For nondyadic $α_{h}$ , the condition counted by $ρ_{h} (X)$ is exactly a change between binary digits $N + 1$ and $N + 2$ . Thus the same sufficient hypothesis asks for at least $11 X / 100$ such changes, counted with $X \leq N < 2 X$ , on arbitrarily large blocks for every $h$ .

Proof. The inequality $‖ x ‖_{R / Z} \geq 1 / 4$ gives $\cos (2 π x) \leq 0$ . Every remaining term is at most $1$ , so

\sum_{N = X}^{2 X - 1} \cos (2 π 2^{N} α_{h}) \leq (1 - ρ_{h} (X)) X \leq \frac{89}{100} X .

Proposition 12.2 now applies. For the digit reading, for a nondyadic $α_{h}$ the fractional part of $2^{N} α_{h}$ is in $[1 / 4, 3 / 4]$ exactly when those two binary digits differ. The nondyadic qualification excludes the endpoint ambiguity at $3 / 4$ . ◻

The exact hypothesis is the digit-change inequality in the corollary. It is not a claim about an average run length: the two boundary runs of a finite block need not be complete, and their lengths are not specified by the number of interior digit changes. The finite observations below satisfy the displayed bound at their sampled parameters. They do not establish it on arbitrarily large blocks.

Finite block statistics for $α_{h} = (2^{h} - 1) S$ . The exact digit-change counts in the six rows are $122$ , $1039$ , $119$ , $1028$ , $1047$ and $1034$ ; the proportions and cosine averages are rounded to four decimal places. The interval calculation described below certifies these counts and displayed roundings.
$h$	$X$	$ρ_{h} (X)$	$X^{- 1} \sum_{N \in [X, 2 X)} \cos (2 π 2^{N} α_{h})$	required
$1$	$256$	$0.4766$	$0.0037$	$\leq 0.89$
$1$	$2048$	$0.5073$	$- 0.0043$	$\leq 0.89$
$3$	$256$	$0.4648$	$0.0296$	$\leq 0.89$
$3$	$2048$	$0.5020$	$0.0027$	$\leq 0.89$
$8$	$2048$	$0.5112$	$- 0.0088$	$\leq 0.89$
$13$	$2048$	$0.5049$	$- 0.0195$	$\leq 0.89$

For the numerical enclosure, compute $P = \sum_{n = 1}^{6080} φ (n) 2^{6080 - n}$ . The elementary tail bound gives

P \leq 2^{6080} S \leq P + 6082.

The endpoints have the same quotient on division by $2^{80}$ , which certifies $⌊ 2^{6000} S ⌋$ ; an error bound smaller than one would not alone certify a floor near an integer boundary. Multiplying this interval by $(2^{h} - 1) 2^{N - 6080}$ encloses each phase. None of the resulting intervals crosses a digit-test boundary, and interval evaluation of the cosine certifies the rounded averages. Among the first $5800$ fractional digits, the numbers of runs for $h = 1, 3, 8$ are $2864$ , $2964$ , $2927$ , and the longest runs have lengths $13$ , $12$ , $14$ . These count the two boundary runs as observed, whether or not they continue outside the window. No randomness assumption or asymptotic conclusion is part of these finite calculations.

Observation 12.4. Lemma 12.1 expresses each tail phase in terms of the doubling orbit of $α_{h} = (2^{h} - 1) S$ . The block conditions average those phases, the prime condition samples prescribed prime-indexed positions, and the full-depth condition requires a finite residue certificate for each $(d, N)$ at some multiple of $d$ . Corollary 12.3 gives one sufficient digit-change test for the real-part estimate; it is not an equivalence with the norm bound or with the four separate budgets. Likewise, a bound on a binary run length is not the exact prime-gap condition. Section 12.7 compares the original inequalities.

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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