Geometric Moments and Rational Lambert Values: Proofs and Further Results

Will Cook

Long paper

Geometric Moments and Rational Lambert Values: Proofs and Further Results

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

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Summary. The record proves the fixed-base Hankel asymptotic for Zudilin's normalised moments and the rational-base irrationality criterion log b/log a<0.4056830213840605…. The determinant proof separates geometric node configurations from positive weights; the irrationality proof instead specialises Zudilin's 2004 polynomial forms after cancellation. Coefficient moments, evaluated lattices and congruence conditions distinguish integer divisibility from nonzero remainder decay. A broader ratio-limit argument and a rational-value model remain unformalised. None of these arguments settles $F (3 / 2)$ .

This paper covers the sharp fixed-base Hankel asymptotic and rational-base irrationality proofs, coefficient moments and arithmetic obstructions, with ordinary ratio-limit and rational-value models distinguished from checked results.

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 arithmetical properties of Lambert series, J. Indian Math. Soc. (N.S.) 12 (1948), 63–66.
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. B. Borwein, On the irrationality of $\sum 1 / (q^{n} + r)$ , J. Number Theory 37 (1991), no. 3, 253–259, doi:10.1016/S0022-314X(05)80041-1.
P. B. Borwein, On the irrationality of certain series, Math. Proc. Cambridge Philos. Soc. 112 (1992), no. 1, 141–146, doi:10.1017/S030500410007081X. Van Assche cites Lemma 2 for the neighbouring little- $q$ -Legendre evaluation used above.
T. Amdeberhan and D. Zeilberger, $q$ -Apéry irrationality proofs by $q$ -WZ pairs, Adv. Appl. Math. 20 (1998), no. 2, 275–283, doi:10.1006/aama.1997.0565; arXiv:math/9804122v1. Page references are to arXiv:math/9804122v1.
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.
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.
W. Zudilin, A determinantal approach to irrationality, arXiv:1507.05697v2. Propositions 1–2 are cited from v2; the preceding Heine-expansion locators retain their separately cited v1 edition.
P. Bundschuh and W. Zudilin, Rational approximations to a $q$ -analogue of $π$ and some other $q$ -series, in H. P. Schlickewei, K. Schmidt and R. F. Tichy (eds.), Diophantine Approximation, Dev. Math. 16, Springer, 2008, pp. 123–139, doi:10.1007/978-3-211-74280-8_6.
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.
D. Duverney, À propos de la série $\sum_{n \geq 1} x^{n} / (q^{n} - 1)$ , J. Théor. Nombres Bordeaux 8 (1996), no. 1, 173–181. Théorème 2 on p. 174 gives the rational-base region $\log | s | / \log | r | < \frac{1}{3} (1 - 3 / π^{2}) = 0.2320 \dots$ for this series; Théorème 1, for a general numerator, is weaker.
T. Matala-aho, K. Väänänen and W. Zudilin, New irrationality measures for $q$ -logarithms, Math. Comp. 75 (2006), no. 254, 879–889, doi:10.1090/S0025-5718-05-01812-0. The hypothesis $p = 1 / q \in Z \ {0, \pm 1}$ is carried in the abstract on p. 879 and in both theorem statements on p. 880, where the authors also record that their methods do not sharpen the $q$ -harmonic case of [8].
B. Adamczewski and J. P. Bell, A problem about Mahler functions, Ann. Sc. Norm. Super. Pisa Cl. Sci. 17 (2017), no. 4, 1301–1355; arXiv:1303.2019v1, 2013. Theorem 1.1 on p. 6 of arXiv:1303.2019v1: over a field of characteristic zero, a power series is both $k$ - and $ℓ$ -Mahler for multiplicatively independent $k, ℓ$ if and only if it is a rational function. Bell and Smertnig cite it as Theorem 1.3 of the journal version [27].
G. Rhin and C. Viola, On a permutation group related to $ζ (2)$ , Acta Arith. 77 (1996), no. 1, 23–56, doi:10.4064/aa-77-1-23-56.
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.
J. Coussement and C. Smet, Irrationality proof of certain Lambert series using little $q$ -Jacobi polynomials, J. Comput. Appl. Math. 233 (2009), no. 3, 680–690, doi:10.1016/j.cam.2009.02.036; arXiv:math/0701345v1. Page references are to arXiv:math/0701345v1.
J. Koizumi and A. Yokoi, Apéry-type approximations and irrationality measures for certain $q$ -series, arXiv:2608.26918v1, 27 August 2026.
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).
F. Luca and Y. Tachiya, Linear independence results for the values of divisor functions series, RIMS Kôkyûroku No. 2014 (2017), 138–150. Theorem A on p. 139 restates the periodic-coefficient irrationality theorem; Example 1 on p. 140 gives the divisor-function specialization.
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.
I. Rivin, Zero Coefficients of Rational Power Series and Rational Lambert Series, arXiv:2604.25151v1, 28 April 2026. Theorem 1.1 is on p. 2 and proved on pp. 6–7; the periodic-coefficient Corollary 6.4 is on p. 9.
V. Kovač and T. Tao, On several irrationality problems for Ahmes series, Acta Math. Hungar. 175 (2025), no. 2, 572–608, doi:10.1007/s10474-025-01528-0; arXiv:2406.17593v4. Page references are to arXiv:2406.17593v4.
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 $k \geq 2$ .
F. W. J. Olver et al. (eds.), NIST Digital Library of Mathematical Functions, dlmf.nist.gov/17.2.E37, Eq. 17.2.37 in §17.2(iii), accessed 15 September 2026.
The Formal Conjectures Authors, FormalConjectures.ErdosProblems.1049, Lean source at commit f776d2f, 2026, accessed 28 July 2026. The irrationality declarations are unproved; the Lambert-series identity is proved.
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).
R. Kozhan and M. Vaktnäs, Christoffel transform and multiple orthogonal polynomials, arXiv:2407.13946v1 (2024). The moment-transform comparison uses the introduction, (3)–(4), of v1.
C. Krattenthaler, A determinant identity for moments of orthogonal polynomials that implies Uvarov’s formula for the orthogonal polynomials of rationally related densities, arXiv:2103.03969v1 (2021). Page references are to this version.
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.
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.
D. Duverney, Arithmetical functions and irrationality of Lambert series, AIP Conf. Proc. 1385 (2011), 5–16, doi:10.1063/1.3630035. Theorem and page references are to the 12-page author manuscript.
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.
H. Liang, J. Remmel and S. Zheng, Stieltjes moment sequences of polynomials, arXiv:1710.05795v1, 2017.
M. Pétréolle, A. D. Sokal and B.-X. Zhu, Lattice paths and branched continued fractions: An infinite sequence of generalizations of the Stieltjes–Rogers and Thron–Rogers polynomials, with coefficientwise Hankel-total positivity, Mem. Amer. Math. Soc. 291 (2023), no. 1450, doi:10.1090/memo/1450; arXiv:1807.03271v2. Theorem and page references use that preprint version.
J. Middeke, D. J. Jeffrey and C. Koutschan, Common Factors in Fraction-Free Matrix Decompositions, Math. Comput. Sci. 15 (2021), no. 4, 589–608, doi:10.1007/s11786-020-00495-9; arXiv:2005.12380v1. Section and theorem references use the preprint.
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 arithmetical properties of Lambert series, J. Indian Math. Soc. (N.S.) 12 (1948), 63–66.

[ref-2] 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-3] P. B. Borwein, On the irrationality of $\sum 1 / (q^{n} + r)$ , J. Number Theory 37 (1991), no. 3, 253–259, doi:10.1016/S0022-314X(05)80041-1.

[ref-4] P. B. Borwein, On the irrationality of certain series, Math. Proc. Cambridge Philos. Soc. 112 (1992), no. 1, 141–146, doi:10.1017/S030500410007081X. Van Assche cites Lemma 2 for the neighbouring little- $q$ -Legendre evaluation used above.

[ref-5] T. Amdeberhan and D. Zeilberger, $q$ -Apéry irrationality proofs by $q$ -WZ pairs, Adv. Appl. Math. 20 (1998), no. 2, 275–283, doi:10.1006/aama.1997.0565; arXiv:math/9804122v1. Page references are to arXiv:math/9804122v1.

[ref-6] P. Bundschuh and K. Väänänen, Arithmetical investigations of a certain infinite product, Compositio Math. 91 (1994), no. 2, 175–199.

[ref-7] W. Zudilin, Remarks on irrationality of $q$ -harmonic series, Manuscripta Math. 107 (2002), no. 4, 463–477, doi:10.1007/s002290200249.

[ref-8] 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-9] 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-10] 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-11] W. Zudilin, A determinantal approach to irrationality, arXiv:1507.05697v2. Propositions 1–2 are cited from v2; the preceding Heine-expansion locators retain their separately cited v1 edition.

[ref-12] P. Bundschuh and W. Zudilin, Rational approximations to a $q$ -analogue of $π$ and some other $q$ -series, in H. P. Schlickewei, K. Schmidt and R. F. Tichy (eds.), Diophantine Approximation, Dev. Math. 16, Springer, 2008, pp. 123–139, doi:10.1007/978-3-211-74280-8_6.

[ref-13] 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-14] D. Duverney, À propos de la série $\sum_{n \geq 1} x^{n} / (q^{n} - 1)$ , J. Théor. Nombres Bordeaux 8 (1996), no. 1, 173–181. Théorème 2 on p. 174 gives the rational-base region $\log | s | / \log | r | < \frac{1}{3} (1 - 3 / π^{2}) = 0.2320 \dots$ for this series; Théorème 1, for a general numerator, is weaker.

[ref-15] T. Matala-aho, K. Väänänen and W. Zudilin, New irrationality measures for $q$ -logarithms, Math. Comp. 75 (2006), no. 254, 879–889, doi:10.1090/S0025-5718-05-01812-0. The hypothesis $p = 1 / q \in Z \ {0, \pm 1}$ is carried in the abstract on p. 879 and in both theorem statements on p. 880, where the authors also record that their methods do not sharpen the $q$ -harmonic case of [8].

[ref-16] B. Adamczewski and J. P. Bell, A problem about Mahler functions, Ann. Sc. Norm. Super. Pisa Cl. Sci. 17 (2017), no. 4, 1301–1355; arXiv:1303.2019v1, 2013. Theorem 1.1 on p. 6 of arXiv:1303.2019v1: over a field of characteristic zero, a power series is both $k$ - and $ℓ$ -Mahler for multiplicatively independent $k, ℓ$ if and only if it is a rational function. Bell and Smertnig cite it as Theorem 1.3 of the journal version [27].

[ref-17] G. Rhin and C. Viola, On a permutation group related to $ζ (2)$ , Acta Arith. 77 (1996), no. 1, 23–56, doi:10.4064/aa-77-1-23-56.

[ref-18] 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-19] J. Coussement and C. Smet, Irrationality proof of certain Lambert series using little $q$ -Jacobi polynomials, J. Comput. Appl. Math. 233 (2009), no. 3, 680–690, doi:10.1016/j.cam.2009.02.036; arXiv:math/0701345v1. Page references are to arXiv:math/0701345v1.

[ref-20] J. Koizumi and A. Yokoi, Apéry-type approximations and irrationality measures for certain $q$ -series, arXiv:2608.26918v1, 27 August 2026.

[ref-21] 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-22] F. Luca and Y. Tachiya, Linear independence results for the values of divisor functions series, RIMS Kôkyûroku No. 2014 (2017), 138–150. Theorem A on p. 139 restates the periodic-coefficient irrationality theorem; Example 1 on p. 140 gives the divisor-function specialization.

[ref-23] 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-24] I. Rivin, Zero Coefficients of Rational Power Series and Rational Lambert Series, arXiv:2604.25151v1, 28 April 2026. Theorem 1.1 is on p. 2 and proved on pp. 6–7; the periodic-coefficient Corollary 6.4 is on p. 9.

[ref-25] V. Kovač and T. Tao, On several irrationality problems for Ahmes series, Acta Math. Hungar. 175 (2025), no. 2, 572–608, doi:10.1007/s10474-025-01528-0; arXiv:2406.17593v4. Page references are to arXiv:2406.17593v4.

[ref-26] T. F. Bloom, Erdős Problem #1049, erdosproblems.com/1049. Accessed 28 July 2026, when the page displayed “last edited 28 September 2025”.

[ref-27] 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-28] F. W. J. Olver et al. (eds.), NIST Digital Library of Mathematical Functions, dlmf.nist.gov/17.2.E37, Eq. 17.2.37 in §17.2(iii), accessed 15 September 2026.

[ref-29] The Formal Conjectures Authors, FormalConjectures.ErdosProblems.1049, Lean source at commit f776d2f, 2026, accessed 28 July 2026. The irrationality declarations are unproved; the Lambert-series identity is proved.

[ref-30] 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-31] R. Kozhan and M. Vaktnäs, Christoffel transform and multiple orthogonal polynomials, arXiv:2407.13946v1 (2024). The moment-transform comparison uses the introduction, (3)–(4), of v1.

[ref-32] C. Krattenthaler, A determinant identity for moments of orthogonal polynomials that implies Uvarov’s formula for the orthogonal polynomials of rationally related densities, arXiv:2103.03969v1 (2021). Page references are to this version.

[ref-33] 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-34] 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-35] D. Duverney, Arithmetical functions and irrationality of Lambert series, AIP Conf. Proc. 1385 (2011), 5–16, doi:10.1063/1.3630035. Theorem and page references are to the 12-page author manuscript.

[ref-36] 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-37] 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-38] H. Liang, J. Remmel and S. Zheng, Stieltjes moment sequences of polynomials, arXiv:1710.05795v1, 2017.

[ref-39] M. Pétréolle, A. D. Sokal and B.-X. Zhu, Lattice paths and branched continued fractions: An infinite sequence of generalizations of the Stieltjes–Rogers and Thron–Rogers polynomials, with coefficientwise Hankel-total positivity, Mem. Amer. Math. Soc. 291 (2023), no. 1450, doi:10.1090/memo/1450; arXiv:1807.03271v2. Theorem and page references use that preprint version.

[ref-40] J. Middeke, D. J. Jeffrey and C. Koutschan, Common Factors in Fraction-Free Matrix Decompositions, Math. Comput. Sci. 15 (2021), no. 4, 589–608, doi:10.1007/s11786-020-00495-9; arXiv:2005.12380v1. Section and theorem references use the preprint.

[ref-41] 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-42] 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.