# Dyson’s ranks and Maass forms | Annals of Mathematics

Abstract Motivated by work of Ramanujan, Freeman Dyson defined the rank of an integer partition to be its largest part minus its number of parts. If N(m,n) denotes the number of partitions of n with rank m, then it turns out that R(w;q):=1+∑n=1∞∑m=−∞∞N(m,n)wmqn=1+∑n=1∞qn2∏nj=1(1−(w+w−1)qj+q2j). We show that if ζ≠1 is a root of unity, then R(ζ;q) is essentially the holomorphic part of a weight 1/2 weak Maass form on a subgroup of SL2(ℤ). For integers 0≤r

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### 数学年刊（Annals of Mathematics）

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