The f(q) mock theta function conjecture and partition ranks
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- Bringmann, K. & Ono, K. Invent. math. (2006) 165: 243. doi:10.1007/s00222-005-0493-5
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In 1944, Freeman Dyson initiated the study of ranks of integer partitions. Here we solve the classical problem of obtaining formulas for Ne(n) (resp. No(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-Poincaré series whose “holomorphic part” is q-1f(q24), we prove the Andrews-Dragonette conjecture, and as a consequence obtain the desired formulas for Ne(n) and No(n).