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.
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Alvarez-Gaumé, L., Moore, G., Vafa, C.: Theta functions, modular invariance, and strings. Commun. Math. Phys. 106(1), 1–40 (1986)
Andrews, G.E.: Generalized Frobenius partitions. Mem. Am. Math. Soc. 49(301), 44 (1984)
Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook. Part III. Springer, New York (2012)
Andrews, G.E., Garvan, F.G.: Dyson’s crank of a partition. Bull. Am. Math. Soc. 18(2), 167–171 (1988)
Berkovich, A., Garvan, F.G.: Some observations on Dyson’s new symmetries of partitions. J. Combin. Theory Ser. A 100(1), 61–93 (2002)
Berndt, B.C., Kim, B.: Asymptotic expansions of certain partial theta functions. Proc. Am. Math. Soc. 139(11), 3779–3788 (2011)
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)
Bringmann, K., Manschot, J.: Asymptotic formulas for coefficients of inverse theta functions. Commun. Number Theory Phys. 7(3), 497–513 (2013)
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)
Bringmann, K., Jennings-Shaffer, C., Mahlburg, K.: On a Tauberian theorem of Ingham and Euler–Maclaurin summation. arXiv:1910.03036
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)
Ciolan, A.: Ranks of overpartitions: asymptotics and inequalities. J. Math. Anal. Appl. 480(2), 123444 (2019)
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)
Dousse, J., Mertens, M.H.: Asymptotic formulae for partition ranks. Acta Arith. 168(1), 83–100 (2015)
Dyson, F.J.: Some guesses in the theory of partitions. Eureka 8, 10–15 (1944)
Dyson, F.J.: A new symmetry of partitions. J. Combin. Theory 7, 56–61 (1969)
Dyson, F.J.: Mappings and symmetries of partitions. J. Combin. Theory Ser. A 51(2), 169–180 (1989)
Eichler, M., Zagier, D.: The Theory of Jacobi Forms. Progress in Mathematics, vol. 55. Birkhäuser Boston Inc., Boston (1985)
Fu, S., Tang, D.: On a generalized crank for \(k\)-colored partitions. J. Number Theory 184, 485–497 (2018)
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)
Garvan, F.G.: Generalizations of Dyson’s rank and non-Rogers-Ramanujan partitions. Manuscripta Math. 84(3–4), 343–359 (1994)
Göttsche, L.: The Betti numbers of the Hilbert scheme of points on a smooth projective surface. Math. Ann. 286(1–3), 193–207 (1990)
Göttsche, L.: Theta functions and Hodge numbers of moduli spaces of sheaves on rational surfaces. Commun. Math. Phys. 206(1), 105–136 (1999)
Hammond, P., Lewis, R.: Congruences in ordered pairs of partitions. Int. J. Math. Math. Sci. 45–48, 2509–2512 (2004)
Hausel, T., Villegas, F.R.: Cohomology of large semiprojective hyperkähler varieties. Astérisque 370, 113–156 (2015)
Jacobi, C.G.J.: Fundamenta nova theoriae functionum ellipticarum. Cambridge University Press, Cambridge Library Collection - Mathematics (2012)
Ji, K.Q., Zang, W.J.T.: Unimodality of the Andrews–Garvan–Dyson cranks of partitions. arXiv:1811.07321
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)
Kim, B., Kim, E., Seo, J.: Asymptotics for \(q\)-expansions involving partial theta functions. Discrete Math. 338(2), 180–189 (2015)
Korpas, G., Manschot, J.: Donaldson-Witten theory and indefinite theta functions. J. High Energy Phys. 11, 083 (2017)
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)
Males, J.: Asymptotic equidistribution and convexity for partition ranks. Ramanujan J. (2020). https://doi.org/10.1007/s11139-019-00202-8
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)
Mao, R.: Asymptotic inequalities for \(k\)-ranks and their cumulation functions. J. Math. Anal. Appl. 409(2), 729–741 (2014)
Moore, G.: Modular forms and two-loop string physics. Phys. Lett. B 176(3–4), 369–379 (1986)
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
Parry, D., Rhoades, R.C.: On Dyson’s crank distribution conjecture and its generalizations. Proc. Am. Math. Soc. 145(1), 101–108 (2017)
Rademacher, H., Zuckerman, H.S.: On the Fourier coefficients of certain modular forms of positive dimension. Ann. Math. 39(2), 433–462 (1938)
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)
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)
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
Zhou, N.H.: On the distribution of rank and crank statistics for integer partitions. Res. Number Theory 5(2):Art. 18, 8 (2019)
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The authors would like to thank the anonymous referees for their very helpful comments and suggestions.
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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)
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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
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DOI: https://doi.org/10.1007/s11139-021-00409-8