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

Approximating the nondivisor terms

The terms with $d ∣ H$ form a finite rational contribution to $R_{2 H} - R_{H}$ . To use it, we must also approximate the terms with $d ∤ H$ . The approximation and its error are explicit; for a cutoff $D \geq 2 H$ , only separation of the resulting rational centre from the integers remains to be checked.

For $d \geq 1$ and $N \geq 0$ , let $a_{d} (N) = d - (N mod d)$ be the least positive shift from $N$ to a multiple of $d$ . Summing the multiples of $d$ in the totient tail gives the rational term

κ_{d} (N) = μ (d) 2^{d - a_{d} (N)} (\frac{N + a_{d} (N)}{d (2^{d} - 1)} + \frac{1}{(2^{d} - 1)^{2}}), R_{N} = \sum_{d \geq 1} κ_{d} (N) .

Indeed, substitute $φ (n) = \sum_{d ∣ n} μ (d) n / d$ into $R_{N}$ . For a fixed $d$ , the admissible shifts are $a_{d} (N) + k d$ , $k \geq 0$ ; their contribution is $μ (d) \sum_{k \geq 0} ((N + a_{d} (N)) / d + k) 2^{- a_{d} (N) - k d}$ , which is the displayed expression. The interchange of sums is justified by

\sum_{j \geq 1} 2^{- j} \sum_{d ∣ N + j} | μ (d) | \frac{N + j}{d} \leq \sum_{j \geq 1} (N + j)^{2} 2^{- j} < \infty .

Thus subtraction and finite splitting of these series are valid. The terms are signed: for example, $κ_{2} (0) = - 4 / 9$ . This identity is proved in Lean (Erdos249257.TotientShiftedMobiusForeignBridge.totientTail_eq_tsum_positiveForeignResidueKernel).

Proposition 6.142 (The finite divisor sum). For an integer $H > 0$ , the contribution of the divisor indices to $R_{2 H} - R_{H}$ is

A_{H} = H \sum_{d ∣ H} \frac{μ (d)}{d (2^{d} - 1)} .

For $d ∣ H$ , both $a_{d} (H)$ and $a_{d} (2 H)$ equal $d$ . Hence $κ_{d} (2 H) - κ_{d} (H) = H μ (d) / (d (2^{d} - 1))$ . Summing over the divisors gives $A_{H}$ , not the whole tail difference: the nondivisor terms must still be included.

For $H > 0$ and an integer cutoff $D \geq 0$ , set

\begin{aligned} P_{H, D} & = \sum_{\begin{matrix} 1 \leq d \leq D \\ d ∤ H \end{matrix}} (κ_{d} (2 H) - κ_{d} (H)), \\ ε_{H, D} & = 2^{H} (2^{H} - 1) (\frac{2}{2^{D}} + \frac{4}{3 \cdot 4^{D}}) . \end{aligned}

Here $A_{H}$ includes all divisor indices, even when $D < H$ . The following implication therefore keeps the error estimate as an explicit hypothesis; it will be proved below when $D \geq 2 H$ .

Proposition 6.143 (Separation larger than the error implies exclusion). Suppose

| (R_{2 H} - R_{H}) - (A_{H} + P_{H, D}) | \leq ε_{H, D}, | A_{H} + P_{H, D} - z | > ε_{H, D} for every z \in Z .

Then $R_{2 H} - R_{H} \notin Z$ .

Proof. If $R_{2 H} - R_{H} = z \in Z$ , the two hypotheses give opposite inequalities for $| A_{H} + P_{H, D} - z |$ . ◻

Why the error hypothesis holds for $D \geq 2 H$ .

Every divisor of $H$ then lies below the cutoff. For $d > D$ , both $H$ and $2 H$ are smaller than $d$ , so the omitted increment simplifies to

κ_{d} (2 H) - κ_{d} (H) = μ (d) 2^{H} (2^{H} - 1) (\frac{1}{2^{d} - 1} + \frac{1}{(2^{d} - 1)^{2}}) .

Since $| μ (d) | \leq 1$ and $2^{d} - 1 \geq 2^{d - 1}$ ,

\begin{aligned} | (R_{2 H} - R_{H}) - (A_{H} + P_{H, D}) | & \leq 2^{H} (2^{H} - 1) \sum_{d > D} (\frac{2}{2^{d}} + \frac{4}{4^{d}}) \\ = ε_{H, D} . \end{aligned}

Thus for $H > 0$ and $D \geq 2 H$ the error bound needs no additional convergence hypothesis. The same estimate is proved in Lean (Erdos249257.TotientShiftedMobiusForeignBridge.controlledForeignProjection_of_twice_le); together with the displayed separation inequality, it gives the cited nonintegrality consequence (Erdos249257.TotientShiftedMobiusForeignBridge.scaleFullTarget_miss_of_projected_separation_of_twice_le_of_forall_int).

For example, at $H = 1$ and $D = 5$ the rational centre and radius are

A_{1} + P_{1, 5} = \frac{304282}{423801}, ε_{1, 5} = \frac{49}{384}, \frac{1}{2} < A_{1} + P_{1, 5} - ε_{1, 5} < A_{1} + P_{1, 5} + ε_{1, 5} < 1.

Thus this finite calculation excludes an integral value of $R_{2} - R_{1}$ . The cutoff restriction alone does not ensure success: at $H = 1, D = 4$ , the centre $346 / 441$ is within the radius $25 / 96$ of $1$ . For fixed $H$ the radius tends to zero as $D \to \infty$ , but an irrationality argument still needs the required family of separated centres. Neither a single example nor convergence of the error supplies it.

When $D \geq 2 H$ , the first-order part of the omitted sum can also be included exactly: $\sum_{d > D} μ (d) / (2^{d} - 1) = 1 / 2 - \sum_{d \leq D} μ (d) / (2^{d} - 1)$ . Adding $2^{H} (2^{H} - 1)$ times this rational correction to $A_{H} + P_{H, D}$ gives the squared-Mersenne approximation above (Erdos249257.TotientShiftedMobiusForeignBridge.renormalizedResidueAgreement_of_twice_le). This explains the relation between the two approximations and the improved squared-denominator remainder; it does not provide their integer separation.

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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Approximating the nondivisor terms

Why the error hypothesis holds for 𝐷 ≥2𝐻D≥2H.

About this paper

Why the error hypothesis holds for $D \geq 2 H$ .