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The asymptotic behavior of the partition function was first determined by Hardy and Ramanujan in 1918 using their famous circle method. Since then, many new ways of deriving their asymptotic formula have been discovered. I will discuss these and report on my recent work with Karin Ikeda.

As a continuation of our previous lecture, we study the mock theta functions in the context of harmonic Maass forms, beginning with Zwegers' thesis in 2002. We discuss (1) universal mock theta functions, (2) a proof of the mock theta conjectures, and (3) exact formulas for coefficients of mock theta functions, beginning with Dragonette and Andrews.

In the first part of this talk, I will provide a brief introduction to classical summation formulas and their significance in number theory. We will review the foundational contributions of Bochner, Koshliakov, and the seminal work of Chandrasekharan and Narasimhan on summation formulas for a broad class of arithmetical functions.

In the second part, I will present recent developments involving new summation formulas in the theory of harmonic Maass forms. As an application of our summation formula, I will discuss the asymptotic behavior of the Riesz means of the Hurwitz class numbers.\\

This talk is based on recent joint work with Olivia Beckwith, Nikolaos Diamantis, Larry Rolen, and Kalani Thalagoda.

We review the basic properties of hecke operators, Atkin-Lehner involutions and trace operators, and show how these operators act on harmonic Maass forms. We also discuss certain p-adic results that follow from this theory.

We give motivating examples of harmonic Maass forms of half-integral weight, including Poincare series and Zagier's Eisenstein series. We also discuss the sesquiharmonic lift of Zagier's Eisenstein series by Duke, Imamoglu, and Toth.

We give explicit examples of harmonic Maass forms of integral weight, including non-holomorphic Eisenstein series and Weierstraass-mock modular forms. We also discuss Borcherds' obstruction theorem (a consequence of Serre-duality).