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

Sources, verification and reproducibility

For Erdős’s reciprocal-summable extension and fractional-part argument, see pp. 222 and 226 of [7]. The periodic theorem [12] is also stated in Luca and Tachiya’s account [11], and Hornich’s strict-tail theorem is proved in Nitecki’s exposition [10]. These expositions are the sources used here for those two results; the original articles were not independently retrieved. The Formal Conjectures file [6] is statement-level prior art, not a proof dependency.

Formal bindings concern statements, not every printed argument. The mixed implication has a formal proof; a separate formal construction of a separating support $V$ uses finite sets of squarefree divisors. The $A_{⋆}$ calculation, printed fresh-prime construction, Corollary 5.2, signed finite-denominator extension and Theorem D.1 have ordinary arguments here, without independent human review. The strictness corollary has no formal binding in the record. The companion paper locates the separate comparisons for its three logarithmic-sampling results. No historical priority is asserted for the support comparison.

Reproducing the weighted theorem.

The two declarations for Theorem 1.2 are in the public source at commit 91ca3405. That snapshot pins Lean 4.29.1 in lean-toolchain and its Mathlib revision in lake-manifest.json. From a complete checkout, with elan installed, build the two modules with:

lake exe cache get
python3 scripts/lean_fast_build.py --jobs 2 \
  ErdosProblems.Erdos257.PaperCompleteR8.WeightedReturn \
  ErdosProblems.Erdos257.PaperCompleteR8.WeightedHereditaryClaim

The cache download is optional. The modules contain divisibilityWeightedClaim and finitePrimeWeighted_fixedBase_hereditary, respectively; their precise statements and associated checks are linked above.

The supplementary declarations are collected in the companion paper’s final source section, grouped by support criteria, finite denominators, integer recurrences and achievement sets.

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.

Cite and contact. Cite this paper by its title, author and date above, with its PDF; cite the earlier sources it uses for a mathematical result. For software, use the release citation and give the commit used. Contact Will with questions or corrections.

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