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Thursday, September 20, 2018

**Abstract:** Let f(q) be the well-known third order mock theta function of Ramanujan. In 1964, George Andrews proved an asymptotic formula for the Fourier coefficients of f(q), and he made two conjectures about his asymptotic series (these coefficients have an important combinatorial interpretation). The first of these conjectures was proved in 2009 by Bringmann and Ono. Here we prove the second conjecture, and we obtain a power savings bound in Andrews’ original asymptotic formula. The proofs rely on uniform bounds for sums of Kloosterman sums which follow from the spectral theory of Maass forms of half integral weight and in particular from a new estimate which we derive for the Fourier coefficients of such forms. This is joint work with Alexander Dunn.