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

Limits of the small-displacement criterion

Small displacement is a sufficient condition for irrationality, not a necessary one. Full support already shows the distinction:

Δ_{2, N_{> 0}} (N) > (2^{N} - 1) \sum_{a > N} 2^{- a} = 1 - 2^{- N} \geq \frac{1}{2} (N \geq 1) .

The full-support value is nevertheless irrational [5]. Thus a proof for arbitrary support cannot rely on making these displacements tend to zero in every case. Other uses of the displacement, integer linear forms and averaging arguments are not excluded.

Full prime support gives a second obstruction. Although every prime singleton is admitted by both support criteria, for the set of all primes $P$ one has $Δ_{2, P} (N) > 1 / 3$ for every $N \geq 1$ : indeed $\sum_{r \geq 1} 2^{- r} ω (N + r) \geq 1$ , whereas $X_{P} (2) \leq \sum_{a \geq 2} (2^{a} - 1)^{- 1} < 2 / 3$ . Here $ω (n)$ is the number of distinct prime divisors of $n$ .

Yet Tao–Teräväinen prove the full-prime case at base $2$ [3]. The paragraph following it states extensions to prime support at every integer base and to full prime-power support at base $2$ , leaving the modifications to the reader. It does not treat arbitrary infinite thinnings. The proposed thinning extension is not a premise of any theorem here.

The logarithmic lower bound is not a converse: replacing a fractional majorant by a logarithmic one need not control averages along the multiples of a prescribed modulus. The counterexample and the bound that remains valid on ordinary initial intervals are proved in Section 13 of the companion paper; those proofs were first worked out in an AI-assisted note of 17 September 2026. These are statements about finite averages; they do not rule out an irrationality criterion based on other uses of logarithmic divisor counts.

For squarefree support, Duverney and Tachiya’s theorem already gives irrationality. It holds at every base $2^{j}$ , $j \geq 1$ , by Corollary 1.2 and Example 1.1 of Duverney and Tachiya [1]. Their corollary covers the $s$ -free products of any pairwise coprime sequence of polynomial growth, and they present it as support for the conjecture of Erdős and Graham stated here as Problem 1.1. For the squarefree support with $1$ removed, the divisor count is $2^{ω (n)} - 1$ , which is odd for $n \geq 2$ . This excludes a first-block-divisibility condition at even bases. Adjoining $1$ removes that parity obstruction while changing the sum by a rational number. The argument concerns that normalisation of the certificate, and gives no obstruction to irrationality of the sum. Details and the other known support families appear in the comparison with known support theorems in the companion paper.

A growth-only criterion gives a different class: if $A = {c_{1} < c_{2} < \dots}$ and $\underset{n}{lim sup} c_{n} / 2^{n} = \infty$ , then $X_{A} (b)$ is irrational for every integer $b \geq 2$ [8]. The same section of the companion paper proves the translation from denominator growth and gives examples showing that this class and reciprocal summability are incomparable. The interval-filling constructions in [9] and the freely chosen lacunary denominators of [16] are not constructions of subseries of these fixed Mersenne weights.

Prime-incidence arguments require additional care as well: multiplying all prime exponents by $2$ creates positive correlations between their divisibility indicators. The identity and covariance calculation appear at the end of the companion paper’s discussion of mixed supports. Size conditions alone do not supply that independence.

Algebraic bases on separated supports

The companion Reading eight Erdős problems together, subsection “Divisibility cuts at every algebraic base”, states a transcendence extension based on the number-field Subspace Theorem. Its hypothesis is an infinite set $H$ with indices $L_{j} < M_{j}$ , where $L_{j} \to \infty$ and $M_{j} - L_{j} \to \infty$ , such that every $n \in H$ with $n \leq L_{j}$ divides $L_{j}$ , and every $n \in H$ with $n > L_{j}$ is a multiple of $M_{j}$ . The claimed conclusion is transcendence of $\sum_{n \in B} w_{n} / (t^{n} - 1)$ for every infinite $B \subseteq H$ , bounded positive integer weights $w_{n}$ and real algebraic $t > 1$ .

Every divisibility chain has such cuts, as does $H_{*} = ⋃_{j} N_{j} {1, \dots, 2^{N_{j}}}$ with $N_{0} = 1$ and $N_{j + 1} = 2 N_{j} lcm (1, \dots, 2^{N_{j}})$ . Each block has a reciprocal sum of at least $1 / 2$ and contains antichains of unbounded size. The weighted sum for one prime is at most $2 t^{2} / (t - 1)^{3}$ at every real $t > 1$ . This connects the example with the weighted criterion, although the separation condition is not asserted for every support satisfying that criterion.

The companion proof is outside this paper and is not a premise of the weighted, cover or mixed theorem. It has two AI proof reviews, no Lean proof of transcendence and no independent human review. No historical priority is asserted for the extension. The case of arbitrary infinite support at base two remains unresolved.

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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