For 2 < k ∈ ½ℤ, we define lifts of cuspidal Poincaré series in Sk(Γ0(N)) to weight 2 − k harmonic weak Maass forms. This construction answers a question of Dyson by providing the general framework “explaining” Ramanujan's mock theta functions. As an application, we show that the number of partitions of a positive integer n is the “trace” of singular moduli of a Maass form arising from the lift of a weight 4 cusp form corresponding to a Calabi–Yau threefold.
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