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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References

D. Duverney and Y. Tachiya, Refinement of the Chowla–Erdős method and linear independence of certain Lambert series, Forum Math. 31 (2019), no. 6, 1557–1566, DOI. Page numbers refer to the linked author preprint.
H. Kaneko, Y. Suzuki, and Y. Tachiya, Refinements of Erdős’s irrationality criterion for certain sparse infinite series, arXiv:2601.20743v1 (2026).
T. Tao and J. Teräväinen, Quantitative correlations and some problems on prime factors of consecutive integers, arXiv:2512.01739v2 (submitted December 2025, revised April 2026).
V. Kovač and T. Tao, On several irrationality problems for Ahmes series, Acta Math. Hungar. 175 (2025), no. 2, 572–608, DOI. Page numbers refer to arXiv:2406.17593v4.
P. Erdős, On arithmetical properties of Lambert series, J. Indian Math. Soc. 12 (1948), 63–66.
The Formal Conjectures Authors, FormalConjectures.ErdosProblems.257, Lean source at commit f776d2f, 2025, accessed 13 September 2026.
P. Erdős, On the irrationality of certain series, Math. Student 36 (1968), 222–226 (issued 1969); five-page scan.
P. Erdős, Some problems and results on the irrationality of the sum of infinite series, J. Math. Sci. 10 (1975), 1–7.
K. Barreto, J. Kang, S.-H. Kim, V. Kovač, and S. Zhang, Irrationality of rapidly converging series: a problem of Erdős and Graham, arXiv:2601.21442v3 (2026), to appear in Bull. London Math. Soc.
Z. Nitecki, Subsum sets: intervals, Cantor sets, and Cantorvals, arXiv:1106.3779v2 (2013).
F. Luca and Y. Tachiya, Linear independence results for the values of divisor functions series, RIMS Kôkyûroku No. 2014 (2017), 138–150. Theorem A restates their periodic-sequence theorem (DOI).
F. Luca and Y. Tachiya, Irrationality of Lambert series associated with a periodic sequence, International Journal of Number Theory 10 (2014), no. 3, 623–636. doi:10.1142/S1793042113501121.
F. Luca and Y. Tachiya, Linear independence of certain Lambert series, Proceedings of the American Mathematical Society 142 (2014), no. 10, 3411–3419. doi:10.1090/S0002-9939-2014-12102-2.
W. Van Assche, Little $q$ -Legendre polynomials and irrationality of certain Lambert series, The Ramanujan Journal 5 (2001), 295–310. doi:10.1023/A:1012930828917; arXiv:math/0101187v1.
H. Hornich, Über beliebige Teilsummen absolut konvergenter Reihen, Monatshefte für Mathematik und Physik 49 (1941), 316–320. doi:10.1007/BF01707309.
W. van Doorn and V. Kovač, Lacunary sequences whose reciprocal sums represent all rational numbers in an interval, Acta Arith. 223 (2026), 275–295. DOI. Page references use https://arxiv.org/abs/2509.24971v3.

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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[ref-1] D. Duverney and Y. Tachiya, Refinement of the Chowla–Erdős method and linear independence of certain Lambert series, Forum Math. 31 (2019), no. 6, 1557–1566, DOI. Page numbers refer to the linked author preprint.

[ref-2] H. Kaneko, Y. Suzuki, and Y. Tachiya, Refinements of Erdős’s irrationality criterion for certain sparse infinite series, arXiv:2601.20743v1 (2026).

[ref-3] T. Tao and J. Teräväinen, Quantitative correlations and some problems on prime factors of consecutive integers, arXiv:2512.01739v2 (submitted December 2025, revised April 2026).

[ref-4] V. Kovač and T. Tao, On several irrationality problems for Ahmes series, Acta Math. Hungar. 175 (2025), no. 2, 572–608, DOI. Page numbers refer to arXiv:2406.17593v4.

[ref-5] P. Erdős, On arithmetical properties of Lambert series, J. Indian Math. Soc. 12 (1948), 63–66.

[ref-6] The Formal Conjectures Authors, FormalConjectures.ErdosProblems.257, Lean source at commit f776d2f, 2025, accessed 13 September 2026.

[ref-7] P. Erdős, On the irrationality of certain series, Math. Student 36 (1968), 222–226 (issued 1969); five-page scan.

[ref-8] P. Erdős, Some problems and results on the irrationality of the sum of infinite series, J. Math. Sci. 10 (1975), 1–7.

[ref-9] K. Barreto, J. Kang, S.-H. Kim, V. Kovač, and S. Zhang, Irrationality of rapidly converging series: a problem of Erdős and Graham, arXiv:2601.21442v3 (2026), to appear in Bull. London Math. Soc.

[ref-10] Z. Nitecki, Subsum sets: intervals, Cantor sets, and Cantorvals, arXiv:1106.3779v2 (2013).

[ref-11] F. Luca and Y. Tachiya, Linear independence results for the values of divisor functions series, RIMS Kôkyûroku No. 2014 (2017), 138–150. Theorem A restates their periodic-sequence theorem (DOI).

[ref-12] F. Luca and Y. Tachiya, Irrationality of Lambert series associated with a periodic sequence, International Journal of Number Theory 10 (2014), no. 3, 623–636. doi:10.1142/S1793042113501121.

[ref-13] F. Luca and Y. Tachiya, Linear independence of certain Lambert series, Proceedings of the American Mathematical Society 142 (2014), no. 10, 3411–3419. doi:10.1090/S0002-9939-2014-12102-2.

[ref-14] W. Van Assche, Little $q$ -Legendre polynomials and irrationality of certain Lambert series, The Ramanujan Journal 5 (2001), 295–310. doi:10.1023/A:1012930828917; arXiv:math/0101187v1.

[ref-15] H. Hornich, Über beliebige Teilsummen absolut konvergenter Reihen, Monatshefte für Mathematik und Physik 49 (1941), 316–320. doi:10.1007/BF01707309.

[ref-16] W. van Doorn and V. Kovač, Lacunary sequences whose reciprocal sums represent all rational numbers in an interval, Acta Arith. 223 (2026), 275–295. DOI. Page references use https://arxiv.org/abs/2509.24971v3.