Skip to main content
SpringerLink
Log in

Uniform asymptotic formulas for the Fourier coefficients of the inverse of theta functions

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

In this paper, we use basic asymptotic analysis to establish some uniform asymptotic formulas for the Fourier coefficients of the inverse of Jacobi theta functions. In particular, we answer and improve some problems suggested and investigated by Bringmann, Manschot, and Dousse. As applications, we establish the asymptotic monotonicity properties for the rank and crank of the integer partitions introduced and investigated by Dyson, Andrews, and Garvan.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. In [7], \(a_{m,k}(n)\) is denoted as \(M_k(m,n)\).

  2. For the definition of asymptotic sequence, see [36, Section 2.1(v)].

References

  1. Alvarez-Gaumé, L., Moore, G., Vafa, C.: Theta functions, modular invariance, and strings. Commun. Math. Phys. 106(1), 1–40 (1986)

    Article  MathSciNet  Google Scholar 

  2. Andrews, G.E.: Generalized Frobenius partitions. Mem. Am. Math. Soc. 49(301), 44 (1984)

    MathSciNet  MATH  Google Scholar 

  3. Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook. Part III. Springer, New York (2012)

    Book  Google Scholar 

  4. Andrews, G.E., Garvan, F.G.: Dyson’s crank of a partition. Bull. Am. Math. Soc. 18(2), 167–171 (1988)

    Article  MathSciNet  Google Scholar 

  5. Berkovich, A., Garvan, F.G.: Some observations on Dyson’s new symmetries of partitions. J. Combin. Theory Ser. A 100(1), 61–93 (2002)

    Article  MathSciNet  Google Scholar 

  6. Berndt, B.C., Kim, B.: Asymptotic expansions of certain partial theta functions. Proc. Am. Math. Soc. 139(11), 3779–3788 (2011)

    Article  MathSciNet  Google Scholar 

  7. Bringmann, K., Dousse, J.: On Dyson’s crank conjecture and the uniform asymptotic behavior of certain inverse theta functions. Trans. Am. Math. Soc. 368(5), 3141–3155 (2016)

    Article  MathSciNet  Google Scholar 

  8. Bringmann, K., Manschot, J.: Asymptotic formulas for coefficients of inverse theta functions. Commun. Number Theory Phys. 7(3), 497–513 (2013)

    Article  MathSciNet  Google Scholar 

  9. Bringmann, K., Folsom, A., Milas, A.: Asymptotic behavior of partial and false theta functions arising from Jacobi forms and regularized characters. J. Math. Phys. 58(1), 011702 (2017)

    Article  MathSciNet  Google Scholar 

  10. Bringmann, K., Jennings-Shaffer, C., Mahlburg, K.: On a Tauberian theorem of Ingham and Euler–Maclaurin summation. arXiv:1910.03036

  11. Chan, S.H., Mao, R.: Inequalities for ranks of partitions and the first moment of ranks and cranks of partitions. Adv. Math. 258, 414–437 (2014)

    Article  MathSciNet  Google Scholar 

  12. Ciolan, A.: Ranks of overpartitions: asymptotics and inequalities. J. Math. Anal. Appl. 480(2), 123444 (2019)

    Article  MathSciNet  Google Scholar 

  13. Curtright, T.L., Thorn, C.B.: Symmetry patterns in the mass spectra of dual string models. Nuclear Phys. B 274(3–4), 520–558 (1986)

    Article  MathSciNet  Google Scholar 

  14. Dousse, J., Mertens, M.H.: Asymptotic formulae for partition ranks. Acta Arith. 168(1), 83–100 (2015)

    Article  MathSciNet  Google Scholar 

  15. Dyson, F.J.: Some guesses in the theory of partitions. Eureka 8, 10–15 (1944)

    MathSciNet  Google Scholar 

  16. Dyson, F.J.: A new symmetry of partitions. J. Combin. Theory 7, 56–61 (1969)

    Article  MathSciNet  Google Scholar 

  17. Dyson, F.J.: Mappings and symmetries of partitions. J. Combin. Theory Ser. A 51(2), 169–180 (1989)

    Article  MathSciNet  Google Scholar 

  18. Eichler, M., Zagier, D.: The Theory of Jacobi Forms. Progress in Mathematics, vol. 55. Birkhäuser Boston Inc., Boston (1985)

  19. Fu, S., Tang, D.: On a generalized crank for \(k\)-colored partitions. J. Number Theory 184, 485–497 (2018)

    Article  MathSciNet  Google Scholar 

  20. Garvan, F.G.: New combinatorial interpretations of Ramanujan’s partition congruences mod \(5,7\) and \(11\). Trans. Am. Math. Soc. 305(1), 47–77 (1988)

    MathSciNet  MATH  Google Scholar 

  21. Garvan, F.G.: Generalizations of Dyson’s rank and non-Rogers-Ramanujan partitions. Manuscripta Math. 84(3–4), 343–359 (1994)

    Article  MathSciNet  Google Scholar 

  22. Göttsche, L.: The Betti numbers of the Hilbert scheme of points on a smooth projective surface. Math. Ann. 286(1–3), 193–207 (1990)

    Article  MathSciNet  Google Scholar 

  23. Göttsche, L.: Theta functions and Hodge numbers of moduli spaces of sheaves on rational surfaces. Commun. Math. Phys. 206(1), 105–136 (1999)

    Article  MathSciNet  Google Scholar 

  24. Hammond, P., Lewis, R.: Congruences in ordered pairs of partitions. Int. J. Math. Math. Sci. 45–48, 2509–2512 (2004)

    Article  MathSciNet  Google Scholar 

  25. Hausel, T., Villegas, F.R.: Cohomology of large semiprojective hyperkähler varieties. Astérisque 370, 113–156 (2015)

    MATH  Google Scholar 

  26. Jacobi, C.G.J.: Fundamenta nova theoriae functionum ellipticarum. Cambridge University Press, Cambridge Library Collection - Mathematics (2012)

    Book  Google Scholar 

  27. Ji, K.Q., Zang, W.J.T.: Unimodality of the Andrews–Garvan–Dyson cranks of partitions. arXiv:1811.07321

  28. Jin, S., Jo, S.: The asymptotic formulas for coefficients and algebraicity of Jacobi forms expressed by infinite product. J. Math. Anal. Appl. 471(1–2), 623–646 (2019)

    Article  MathSciNet  Google Scholar 

  29. Kim, B., Kim, E., Seo, J.: Asymptotics for \(q\)-expansions involving partial theta functions. Discrete Math. 338(2), 180–189 (2015)

    Article  MathSciNet  Google Scholar 

  30. Korpas, G., Manschot, J.: Donaldson-Witten theory and indefinite theta functions. J. High Energy Phys. 11, 083 (2017)

    Article  MathSciNet  Google Scholar 

  31. Males, J.:. The asymptotic profile of an eta-theta quotient related to entanglement entropy in string theory. Res. Number Theory, 6(1):Paper No. 15, 14 (2020)

  32. Males, J.: Asymptotic equidistribution and convexity for partition ranks. Ramanujan J. (2020). https://doi.org/10.1007/s11139-019-00202-8

    Article  MathSciNet  MATH  Google Scholar 

  33. Manschot, J., Rolon, J.M.Z.: The asymptotic profile of \(\chi _y\)-genera of Hilbert schemes of points on \({K}3\) surfaces. Commun. Number Theory Phys. 9(2), 413–436 (2015)

    Article  MathSciNet  Google Scholar 

  34. Mao, R.: Asymptotic inequalities for \(k\)-ranks and their cumulation functions. J. Math. Anal. Appl. 409(2), 729–741 (2014)

    Article  MathSciNet  Google Scholar 

  35. Moore, G.: Modular forms and two-loop string physics. Phys. Lett. B 176(3–4), 369–379 (1986)

    Article  MathSciNet  Google Scholar 

  36. NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.25 of 2019-12-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds

  37. Parry, D., Rhoades, R.C.: On Dyson’s crank distribution conjecture and its generalizations. Proc. Am. Math. Soc. 145(1), 101–108 (2017)

    Article  MathSciNet  Google Scholar 

  38. Rademacher, H., Zuckerman, H.S.: On the Fourier coefficients of certain modular forms of positive dimension. Ann. Math. 39(2), 433–462 (1938)

    Article  MathSciNet  Google Scholar 

  39. Ramanujan, S.: Some properties of \(p(n)\), the number of partitions of \(n\) [Proc. Cambridge Philos. Soc. 19 (1919), 207–210]. In Collected papers of Srinivasa Ramanujan, pp. 210–213. AMS Chelsea Publ., Providence (2000)

  40. Yoshioka, K.: The Betti numbers of the moduli space of stable sheaves of rank \(2\) on \({\mathbf{P}}^2\). J. Reine Angew. Math. 453, 193–220 (1994)

    MathSciNet  MATH  Google Scholar 

  41. Zagier, D.: The Mellin transform and other useful analytic techniques, Appendix to E. Zeidler, Quantum field theory. I. Basics in mathematics and physics. Springer, Berlin, 2006. A bridge between mathematicians and physicists

  42. Zhou, N.H.: On the distribution of rank and crank statistics for integer partitions. Res. Number Theory 5(2):Art. 18, 8 (2019)

Download references

Acknowledgements

The authors would like to thank the anonymous referees for their very helpful comments and suggestions.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences and Shanghai Key Laboratory of PMMP, East China Normal University, 500 Dongchuan Road, Shanghai, 200241, People’s Republic of China

    Zhi-Guo Liu & Nian Hong Zhou

Authors
  1. Zhi-Guo Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nian Hong Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nian Hong Zhou.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by the National Science Foundation of China (Grant No. 11971173) and Science and Technology Commission of Shanghai Municipality (Grant No. 13dz2260400)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, ZG., Zhou, N.H. Uniform asymptotic formulas for the Fourier coefficients of the inverse of theta functions. Ramanujan J 57, 1085–1123 (2022). https://doi.org/10.1007/s11139-021-00409-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-021-00409-8

Keywords

Mathematics Subject Classification

Search

Navigation