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Dyson’s ranks and Maass forms. (English) Zbl 1277.11096

Summary: Motivated by work of Ramanujan, F. J. Dyson [Ramanujan revisited, Proc. Conf., Urbana-Champaign/Illinois 1987, 7–28 (1988; Zbl 0652.10009)] 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
\[ R(w;q):=1+\sum_{n=1}^\infty \sum_{m=-\infty}^\infty N(m,n) w^mq^n=1+\sum_{n=1}^\infty \frac{q^{n^2}}{\prod ^n_{j=1}\left(1-(w+w^{-1})q^j+q^{2j}\right)}\,. \]
We show that if \(\zeta\neq 1\) is a root of unity, then \(R(\zeta,q)\) is essentially the holomorphic part of a weight 1/2 weak Maass form on a subgroup of \(\mathrm{SL}_2(\mathbb Z)\). For integers \(0\leq r<t\), we use this result to determine the modularity of the generating function for \(N(r,t;n)\), the number of partitions of \(n\) whose rank is congruent to \(r\pmod t\). We extend the modularity above to construct an infinite family of vector valued weight 1/2 forms for the full modular group \(\mathrm{SL}_2(\mathbb Z)\), a result which is of independent interest.

MSC:

11P82 Analytic theory of partitions
11F37 Forms of half-integer weight; nonholomorphic modular forms
11P84 Partition identities; identities of Rogers-Ramanujan type

Citations:

Zbl 0652.10009
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References:

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