Hankel Determinants of Geometric Moments and Rational Lambert Values

Will Cook

Short paper

Hankel Determinants of Geometric Moments and Rational Lambert Values

Erdős #1049 By Will Cook September 2026 Authorship, AI use and citation

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Summary. Positive geometric moments and a uniform partition-sum bound give the fixed-base asymptotic of Zudilin’s normalised Hankel determinants, including its constant and power correction. A separate calculation of degrees after cancellation in his 2004 forms extends the rational-base irrationality region. It includes every positive integral power of $31 / 4$ and excludes $3 / 2$ .

This paper covers the fixed-base Hankel asymptotic and rational-base irrationality theorem; the long paper develops exact orders, content and endpoint obstructions, and the remaining problem at $3 / 2$ .

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #1049, which remains open.

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References

P. Erdős, On the irrationality of certain series: problems and results, in A. Baker (ed.), New Advances in Transcendence Theory, Cambridge UP, 1988, pp. 102–109, doi:10.1017/CBO9780511897184.009.
P. Bundschuh and K. Väänänen, Arithmetical investigations of a certain infinite product, Compositio Math. 91 (1994), no. 2, 175–199.
W. Zudilin, Remarks on irrationality of $q$ -harmonic series, Manuscripta Math. 107 (2002), no. 4, 463–477, doi:10.1007/s002290200249.
W. Zudilin, Heine’s basic transform and a permutation group for $q$ -harmonic series, Acta Arith. 111 (2004), no. 2, 153–164, doi:10.4064/aa111-2-4. Page references are to the printed journal pages.
W. Zudilin, On the irrationality of generalized $q$ -logarithm, arXiv:1601.02688; Res. Number Theory 2 (2016), Art. 15, doi:10.1007/s40993-016-0042-x. Page references are to arXiv:1601.02688v2. The remark that the results extend to non-integer $p = r / s$ , $| p | > 1$ , under an assumption $\log | r | > c \log | s |$ for a computable $c > 0$ , is in Section 2, p. 4, in the paragraph beginning “Finally, we remark”; no value of $c$ is computed there, and the remark is made for the generalized $q$ -logarithm of that paper.
R. P. Stanley, Smith normal form in combinatorics, J. Combin. Theory Ser. A 144 (2016), 476–495, doi:10.1016/j.jcta.2016.06.013; arXiv:1602.00166v1. Page references are to arXiv:1602.00166v1.
W. Zudilin, A determinantal approach to irrationality, Constr. Approx. 45 (2017), no. 2, 301–310, doi:10.1007/s00365-016-9333-7; arXiv:1507.05697v1. Page and equation references are to arXiv:1507.05697v1.
T. F. Bloom, Erdős Problem #1049, erdosproblems.com/1049. Accessed 28 July 2026, when the page displayed “last edited 28 September 2025”.
J. Bell and D. Smertnig, Mahler series with multiplicative coefficient sequences, arXiv:2603.23456v1, 24 March 2026. Theorem 1.3 is on pp. 2–3; its stated consequences on p. 3 include that the divisor and totient generating series are not $k$ -Mahler for any $k \geq 2$ .
J. Vandehey, On an incomplete argument of Erdős on the irrationality of Lambert series, Integers 13 (2013), Paper A58. Page references are to arXiv:1206.0340v1 (2012).
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:10.1515/forum-2018-0299. Page references are to the authors’ version.
W. Van Assche, Little $q$ -Legendre polynomials and irrationality of certain Lambert series, Ramanujan J. 5 (2001), no. 3, 295–310, doi:10.1023/A:1012930828917. Page references are to arXiv:math/0101187v1.
K. Postelmans and W. Van Assche, Irrationality of $ζ_{q} (1)$ and $ζ_{q} (2)$ , J. Number Theory 126 (2007), no. 1, 119–154, doi:10.1016/j.jnt.2006.11.011. Page references are to arXiv:math/0604312v1 (2006).
C. Krattenthaler, I. Rochev, K. Väänänen and W. Zudilin, On the non-quadraticity of values of the $q$ -exponential function and related $q$ -series, Acta Arith. 136 (2009), no. 3, 243–269, doi:10.4064/aa136-3-4; arXiv:0812.2921v1. Page references are to arXiv:0812.2921v1.
Y. Wang and B.-X. Zhu, Log-convex and Stieltjes moment sequences, Adv. Appl. Math. 81 (2016), 115–127, doi:10.1016/j.aam.2016.06.008. Page references are to arXiv:1612.04114v1.
C. Berg, On powers of Stieltjes moment sequences, II, J. Comput. Appl. Math. 199 (2007), 23–38; arXiv:math/0412340v1. Theorem references use the preprint; Theorem 5.1 treats factorial powers.
A. D. Sokal and J. Walrad, Continued-fraction characterization of Stieltjes moment sequences with support in $[ξ, \infty)$ , arXiv:2404.12131v1, 2024. The classical Stieltjes criterion is recalled on pp. 1–2.
G. H. Golub and J. H. Welsch, Calculation of Gauss Quadrature Rules, Math. Comp. 23 (1969), no. 106, 221–230, doi:10.1090/S0025-5718-69-99647-1.
C. R. Vinroot, Multivariate Rogers–Szegő polynomials and flags in finite vector spaces, arXiv:1011.0984v1, 3 November 2010, abstract accessed 20 September 2026. Cited for the identification of the sum of all $q$ -multinomial coefficients of fixed degree and length with a flag count, and for its recursion generalising the Galois numbers. The factorisation of the moment weights is proved here.

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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[ref-1] P. Erdős, On the irrationality of certain series: problems and results, in A. Baker (ed.), New Advances in Transcendence Theory, Cambridge UP, 1988, pp. 102–109, doi:10.1017/CBO9780511897184.009.

[ref-2] P. Bundschuh and K. Väänänen, Arithmetical investigations of a certain infinite product, Compositio Math. 91 (1994), no. 2, 175–199.

[ref-3] W. Zudilin, Remarks on irrationality of $q$ -harmonic series, Manuscripta Math. 107 (2002), no. 4, 463–477, doi:10.1007/s002290200249.

[ref-4] W. Zudilin, Heine’s basic transform and a permutation group for $q$ -harmonic series, Acta Arith. 111 (2004), no. 2, 153–164, doi:10.4064/aa111-2-4. Page references are to the printed journal pages.

[ref-5] W. Zudilin, On the irrationality of generalized $q$ -logarithm, arXiv:1601.02688; Res. Number Theory 2 (2016), Art. 15, doi:10.1007/s40993-016-0042-x. Page references are to arXiv:1601.02688v2. The remark that the results extend to non-integer $p = r / s$ , $| p | > 1$ , under an assumption $\log | r | > c \log | s |$ for a computable $c > 0$ , is in Section 2, p. 4, in the paragraph beginning “Finally, we remark”; no value of $c$ is computed there, and the remark is made for the generalized $q$ -logarithm of that paper.

[ref-6] R. P. Stanley, Smith normal form in combinatorics, J. Combin. Theory Ser. A 144 (2016), 476–495, doi:10.1016/j.jcta.2016.06.013; arXiv:1602.00166v1. Page references are to arXiv:1602.00166v1.

[ref-7] W. Zudilin, A determinantal approach to irrationality, Constr. Approx. 45 (2017), no. 2, 301–310, doi:10.1007/s00365-016-9333-7; arXiv:1507.05697v1. Page and equation references are to arXiv:1507.05697v1.

[ref-8] T. F. Bloom, Erdős Problem #1049, erdosproblems.com/1049. Accessed 28 July 2026, when the page displayed “last edited 28 September 2025”.

[ref-9] J. Bell and D. Smertnig, Mahler series with multiplicative coefficient sequences, arXiv:2603.23456v1, 24 March 2026. Theorem 1.3 is on pp. 2–3; its stated consequences on p. 3 include that the divisor and totient generating series are not $k$ -Mahler for any $k \geq 2$ .

[ref-10] J. Vandehey, On an incomplete argument of Erdős on the irrationality of Lambert series, Integers 13 (2013), Paper A58. Page references are to arXiv:1206.0340v1 (2012).

[ref-11] 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:10.1515/forum-2018-0299. Page references are to the authors’ version.

[ref-12] W. Van Assche, Little $q$ -Legendre polynomials and irrationality of certain Lambert series, Ramanujan J. 5 (2001), no. 3, 295–310, doi:10.1023/A:1012930828917. Page references are to arXiv:math/0101187v1.

[ref-13] K. Postelmans and W. Van Assche, Irrationality of $ζ_{q} (1)$ and $ζ_{q} (2)$ , J. Number Theory 126 (2007), no. 1, 119–154, doi:10.1016/j.jnt.2006.11.011. Page references are to arXiv:math/0604312v1 (2006).

[ref-14] C. Krattenthaler, I. Rochev, K. Väänänen and W. Zudilin, On the non-quadraticity of values of the $q$ -exponential function and related $q$ -series, Acta Arith. 136 (2009), no. 3, 243–269, doi:10.4064/aa136-3-4; arXiv:0812.2921v1. Page references are to arXiv:0812.2921v1.

[ref-15] Y. Wang and B.-X. Zhu, Log-convex and Stieltjes moment sequences, Adv. Appl. Math. 81 (2016), 115–127, doi:10.1016/j.aam.2016.06.008. Page references are to arXiv:1612.04114v1.

[ref-16] C. Berg, On powers of Stieltjes moment sequences, II, J. Comput. Appl. Math. 199 (2007), 23–38; arXiv:math/0412340v1. Theorem references use the preprint; Theorem 5.1 treats factorial powers.

[ref-17] A. D. Sokal and J. Walrad, Continued-fraction characterization of Stieltjes moment sequences with support in $[ξ, \infty)$ , arXiv:2404.12131v1, 2024. The classical Stieltjes criterion is recalled on pp. 1–2.

[ref-18] G. H. Golub and J. H. Welsch, Calculation of Gauss Quadrature Rules, Math. Comp. 23 (1969), no. 106, 221–230, doi:10.1090/S0025-5718-69-99647-1.

[ref-19] C. R. Vinroot, Multivariate Rogers–Szegő polynomials and flags in finite vector spaces, arXiv:1011.0984v1, 3 November 2010, abstract accessed 20 September 2026. Cited for the identification of the sum of all $q$ -multinomial coefficients of fixed degree and length with a flag count, and for its recursion generalising the Galois numbers. The factorisation of the moment weights is proved here.