Abstract
We exploit transformations relating generalized q-series, infinite products, sums over integer partitions, and continued fractions, to find partition-theoretic formulas to compute the values of constants such as π, and to connect sums over partitions to the Riemann zeta function, multiple zeta values, and other number-theoretic objects.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alladi, K, Erdős, P: On an additive arithmetic function. Pacific J. Math. 71(2), 275–294 (1977).
Alladi, K: Partition identities involving gaps and weights. Trans. Amer. Math. Soc. 349(12), 5001–5019 (1997). doi:10.1090/S0002-9947-97-01831-X.
Andrews, GE: The Theory of Partitions. Cambridge Mathematical Library, p. 255. Cambridge University Press, Cambridge (1998). Reprint of the 1976 original.
Berndt, BC: Number Theory in the Spirit of Ramanujan. Student Mathematical Library, vol. 34, p. 187. American Mathematical Society, Providence, RI (2006). doi:10.1090/stml/034. http://dx.doi.org/10.1090/stml/034.
Besser, A, Furusho, H: The double shuffle relations for p-adic multiple zeta values. In: Primes and Knots. Contemp. Math., vol. 416, pp. 9–29. Amer. Math. Soc., Providence, RI (2006). doi:10.1090/conm/416/07884. http://dx.doi.org/10.1090/conm/416/07884.
Bloch, S, Okounkov, A: The character of the infinite wedge representation. Adv. Math. 149(1), 1–60 (2000). doi:10.1006/aima.1999.1845.
Chamberland, M, Straub, A: On gamma quotients and infinite products. Adv. Appl. Math. 51(5), 546–562 (2013). doi:10.1016/j.aam.2013.07.003.
Dunham, W: Euler: the Master of Us All. The Dolciani Mathematical Expositions, vol. 22, p. 185. Mathematical Association of America, Washington, DC (1999).
Fine, NJ: Basic Hypergeometric Series and Applications. Mathematical Surveys and Monographs, vol. 27, p. 124. American Mathematical Society, Providence, RI (1988). doi:10.1090/surv/027. http://dx.doi.org/10.1090/surv/027.
Griffin, M, Ono, K, Warnaar, SO: A framework of Rogers-Ramanujan identities and their arithmetic properties. Duke Math. J. Accepted for publication. arXiv:1401.7718.
Hoffman, ME: Multiple harmonic series. Pacific J. Math. 152(2), 275–290 (1992).
MacMahon, PA: Combinatory Analysis. Two volumes (bound as one), pp. 302–340. Chelsea Publishing Co.,New York (1960).
Ramanujan, S: Collected Papers of Srinivasa Ramanujan. In: Hardy, GH, Aiyar PVS, Wilson, BM (eds.)Third printing of the 1927 original, With a new preface and commentary by Bruce C. Berndt. AMS Chelsea Publishing, Providence, RI (2000).
Rolen, L, Schneider, RP: A “strange” vector-valued quantum modular form. Arch. Math. (Basel). 101(1), 43–52 (2013). doi:10.1007/s00013-013-0529-9.
Zagier, D: Partitions, quasimodular forms, and the Bloch–Okounkov theorem. Ramanujan J, 1–24 (2015). doi:10.1007/s11139-015-9730-8.
Acknowledgments
I wish to thank the following colleagues of mine for their help in the writing of this paper: Jackson Morrow, for extensive assistance in typesetting and editing; Larry Rolen, for offering editorial comments and references; and Andrew Sills, for historical background related to MacMahon’s work. I am also grateful to the anonymous referees, whose suggestions strengthened the piece. Moreover, I would like to express gratitude to my Ph.D. advisor, Ken Ono, for his interest and insight during the course of this study.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Schneider, R. Partition zeta functions. Res. number theory 2, 9 (2016). https://doi.org/10.1007/s40993-016-0039-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40993-016-0039-5