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Thursday, March 12, 2015

**Abstract:** Building on ground-breaking work of Hardy and Ramanujan, Rademacher proved an exact formula for the values of the ordinary partition function. More recently, Bruinier and Ono obtained an algebraic formula for these values. We study the smallest parts function. Introduced by Andrews, its generating function is a prototypical example of a mock modular form of weight 3/2. Using automorphic methods, we obtain an exact formula and an algebraic formula for its values. The convergence of the exact formula is not obvious, and requires power savings estimates for sums of Kloosterman sums attached to a multiplier. These are proved with spectral methods following an argument of Goldfeld-Sarnak. (Joint work with Nick Andersen)