On Dyson’s crank conjecture and the uniform asymptotic behavior of certain inverse theta functions

Bringmann, Kathrin; Dousse, Jehanne

doi:10.1090/tran/6409

On Dyson’s crank conjecture and the uniform asymptotic behavior of certain inverse theta functions
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by Kathrin Bringmann and Jehanne Dousse PDF

Trans. Amer. Math. Soc. 368 (2016), 3141-3155 Request permission

Abstract:

In this paper we prove a longstanding conjecture by Freeman Dyson concerning the limiting shape of the crank generating function. We fit this function in a more general family of inverse theta functions which play a key role in physics.

References

George E. Andrews and F. G. Garvan, Dyson’s crank of a partition, Bull. Amer. Math. Soc. (N.S.) 18 (1988), no. 2, 167–171. MR 929094, DOI 10.1090/S0273-0979-1988-15637-6
A. O. L. Atkin and P. Swinnerton-Dyer, Some properties of partitions, Proc. London Math. Soc. (3) 4 (1954), 84–106. MR 60535, DOI 10.1112/plms/s3-4.1.84
George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
G. Arken, Modified Bessel functions, Mathematical Methods for Physicists, 3rd ed., Orlando, FL: Academic Press (1985), 610–616.
Kathrin Bringmann, Karl Mahlburg, and Robert C. Rhoades, Asymptotics for rank and crank moments, Bull. Lond. Math. Soc. 43 (2011), no. 4, 661–672. MR 2820152, DOI 10.1112/blms/bdq129
K. Bringmann and J. Manschot, Asymptotic formulas for coefficients of inverse theta functions, submitted for publication.
Kathrin Bringmann and Ken Ono, Dyson’s ranks and Maass forms, Ann. of Math. (2) 171 (2010), no. 1, 419–449. MR 2630043, DOI 10.4007/annals.2010.171.419
F. J. Dyson, Some guesses in the theory of partitions, Eureka 8 (1944), 10–15. MR 3077150
Freeman J. Dyson, Mappings and symmetries of partitions, J. Combin. Theory Ser. A 51 (1989), no. 2, 169–180. MR 1001259, DOI 10.1016/0097-3165(89)90043-5
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher transcendental functions, Vol. I, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla. (1981)
Amanda Folsom and Riad Masri, Equidistribution of Heegner points and the partition function, Math. Ann. 348 (2010), no. 2, 289–317. MR 2672303, DOI 10.1007/s00208-010-0478-6
F. G. Garvan, New combinatorial interpretations of Ramanujan’s partition congruences mod $5,7$ and $11$, Trans. Amer. Math. Soc. 305 (1988), no. 1, 47–77. MR 920146, DOI 10.1090/S0002-9947-1988-0920146-8
Lothar Göttsche, The Betti numbers of the Hilbert scheme of points on a smooth projective surface, Math. Ann. 286 (1990), no. 1-3, 193–207. MR 1032930, DOI 10.1007/BF01453572
Paul Hammond and Richard Lewis, Congruences in ordered pairs of partitions, Int. J. Math. Math. Sci. 45-48 (2004), 2509–2512. MR 2102870, DOI 10.1155/S0161171204311439
G. Hardy and S. Ramanujan, Asymptotic formulae for the distribution of integers of various types, Proc. Lond. Math. Soc. 16 (1918), 112–132.
T. Hausel and Rodriguez-Villegas, Cohomology of large semiprojective hyperkaehler varieties, preprint.
B. Kim, E. Kim, and J. Seo, Asymptotics for $q$-expansions involving partial theta functions, in preparation.
D. H. Lehmer, On the remainders and convergence of the series for the partition function, Trans. Amer. Math. Soc. 46 (1939), 362–373. MR 410, DOI 10.1090/S0002-9947-1939-0000410-9
Karl Mahlburg, Partition congruences and the Andrews-Garvan-Dyson crank, Proc. Natl. Acad. Sci. USA 102 (2005), no. 43, 15373–15376. MR 2188922, DOI 10.1073/pnas.0506702102
Hans Rademacher, Topics in analytic number theory, Die Grundlehren der mathematischen Wissenschaften, Band 169, Springer-Verlag, New York-Heidelberg, 1973. Edited by E. Grosswald, J. Lehner and M. Newman. MR 0364103
Hans Rademacher and Herbert S. Zuckerman, On the Fourier coefficients of certain modular forms of positive dimension, Ann. of Math. (2) 39 (1938), no. 2, 433–462. MR 1503417, DOI 10.2307/1968796
S. Ramanujan, Congruence properties of partitions, Math. Z. 9 (1921), no. 1-2, 147–153. MR 1544457, DOI 10.1007/BF01378341
E. Maitland Wright, Asymptotic Partition Formulae: (II) Weighted Partitions, Proc. London Math. Soc. (2) 36 (1934), 117–141. MR 1575956, DOI 10.1112/plms/s2-36.1.117
E. M. Wright, Stacks. II, Quart. J. Math. Oxford Ser. (2) 22 (1971), 107–116. MR 282940, DOI 10.1093/qmath/22.1.107