The f(q) mock theta function conjecture and partition ranks
成果类型：
Article
署名作者：
Bringmann, Kathrin; Ono, Ken
署名单位：
University of Wisconsin System; University of Wisconsin Madison
刊物名称：
INVENTIONES MATHEMATICAE
ISSN/ISSBN：
0020-9910
DOI：
10.1007/s00222-005-0493-5
发表日期：
2006
页码：
243-266
关键词：
摘要：
In 1944, Freeman Dyson initiated the study of ranks of integer partitions. Here we solve the classical problem of obtaining formulas for N-e(n) (resp. N-o(n)), the number of partitions of n with even (resp. odd) rank. Thanks to Rademacher's celebrated formula for the partition function, this problem is equivalent to that of obtaining a formula for the coefficients of the mock theta function f(q), a problem with its own long history dating to Ramanujan's last letter to Hardy. Little was known about this problem until Dragonette in 1952 obtained asymptotic results. In 1966, G.E. Andrews refined Dragonette's results, and conjectured an exact formula for the coefficients of f(q). By constructing a weak Maass-Poincare series whose holomorphic part is q(-1) f(q(24)), we prove the Andrews-Dragonette conjecture, and as a consequence obtain the desired formulas for N-e(n) and N-o(n).
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