Irrationality criteria for Lambert subseries

Will Cook

Short paper

Irrationality criteria for Lambert subseries

Erdős #257 By Will Cook 30 September 2026 Authorship, AI use and citation

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Summary. The series $\sum_{a \in A} (b^{a} - 1)^{- 1}$ is irrational for every integer $b \geq 2$ and infinite A contained in E∪V when E satisfies a finite-prime weighted summability condition and V has a summable positive divisor cover. The two classes are incomparable, and their union reaches supports in neither class. A common average over dyadic lengths produces one index with both displacements small. Arbitrary infinite supports at base two remain unresolved.

This paper covers the weighted, divisor-cover and mixed support criteria with complete proofs and method limits; the long paper treats finite denominators, the achievement set and rational membership.

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #257, which remains open.

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In this paper

Reciprocal-summable supports

Erdős stated the following extension of his pairwise-coprime theorem [7]. It follows from Theorem 1.2, since $h / (2^{h} - 1) \leq 1$ , but the direct argument explains why reciprocal summability makes the averaging simpler.

Theorem A.1 (reciprocal-summable supports). Let $A \subseteq N_{> 0}$ be infinite. If

\sum_{a \in A} \frac{1}{a} < \infty,

then $X_{A} (b)$ is irrational for every integer $b \geq 2$ .

Erdős also discussed weaker conditions [7]. We do not identify our weighted hypothesis with the conditions he suggested.

Reciprocal summability permits two successive limits: first the observation length grows with the modulus fixed, then the modulus grows until each fixed exponent divides it. The two interchanges use different summable bounds; neither requires a period for the infinite support.

Fix the integer base $b \geq 2$ and put

w_{b, d} (N) = \frac{b^{N mod d}}{b^{d} - 1}, T_{N}^{(b)} = \sum_{d \in A} w_{b, d} (N) .

Then $T_{N}^{(b)} - T_{0}^{(b)} = Δ_{b, A} (N)$ . For a fixed positive integer $Q$ , the orbit of $Q$ modulo $d$ consists of $d / g$ residues, where $g = (Q, d)$ . Summing the geometric progression gives

\begin{matrix} (8) & M_{Q, b} (d) := lim_{X \to \infty} \frac{1}{X} \sum_{m = 1}^{X} w_{b, d} (Q m) = \frac{g}{d (b^{g} - 1)} \leq \frac{1}{d} . \end{matrix}

The example $b = 2$ , $Q = 4$ , $d = 6$ in Section 2 has mean $1 / 9$ for $w_{2, 6}$ , but $2 / 21$ for its displacement. When $d ∣ Q$ , $w_{b, d} (Q m) = w_{b, d} (0)$ and the displacement vanishes.

To interchange the sum over exponents and this limit, we use a bound summable in $d$ for the fixed modulus $Q$ . Counting positive multiples of $d$ gives

\begin{aligned} \frac{1}{X} \sum_{m = 1}^{X} w_{b, d} (Q m) & = \sum_{r \geq 1} b^{- r} \frac{# {1 \leq m \leq X : d ∣ Q m + r}}{X} \\ \leq \sum_{r \geq 1} b^{- r} \frac{Q + r}{d} = \frac{Q / (b - 1) + b / (b - 1)^{2}}{d} \leq \frac{Q + 2}{d} . \end{aligned}

Indeed, the counted multiples are distinct and at most $Q X + r$ . For this fixed $Q$ , the majorant $(Q + 2) / d$ is summable over $A$ . Dominated convergence therefore takes the observation-length limit inside the sum over $d$ , giving the Cesàro mean $\sum_{d \in A} M_{Q, b} (d)$ . This majorant is not uniform in growing $Q$ .

Next let $Q_{t} = lcm (1, \dots, t)$ . For each fixed $d$ , eventually $d ∣ Q_{t}$ and $M_{Q_{t}, b} (d) = (b^{d} - 1)^{- 1}$ . A second dominated-convergence argument, using $M_{Q_{t}, b} (d) \leq 1 / d$ , yields

\begin{matrix} (9) & \sum_{d \in A} M_{Q_{t}, b} (d) ⟶ X_{A} (b) . \end{matrix}

The limiting displacement means therefore tend to zero. Choose $t$ , then a sufficiently long finite average, then a term no larger than that average. The resulting arbitrarily small positive displacements contradict the rationality bound $1 / q$ in (D). This proves Theorem A.1 at every integer base.

Powerful integers (all prime exponents at least two) have the form $u^{2} v^{3}$ , so their reciprocal sum is at most $ζ (2) ζ (3)$ . The theorem therefore applies to every infinite subset. For full perfect-power supports, compare Duverney–Tachiya [1] and Luca–Tachiya’s earlier linear-independence results [13].

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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