Mathematical Research Letters

Volume 25 (2018)

Number 2

A unified approach to partial and mock theta functions

Pages: 659 – 675

DOI: http://dx.doi.org/10.4310/MRL.2018.v25.n2.a16

Author

Robert C. Rhoades (Department of Mathematics, Stanford University, Stanford, California, U.S.A.; and Susquehanna International Group, Bala Cynwyd, Pennsylvania, U.S.A.)

Abstract

The theta functions\[\sum_{n \in \mathbb{Z}} \psi (n) n^{\nu} e^{2 \pi in^2 z} \; \textrm{,}\]with $\psi$ a Dirichlet character and $\nu = 0, 1$, have played an important role in the theory of holomorphic modular forms and modular $L$-functions. A partial theta function is defined by a sum over part of the integer lattice, such as ${\textstyle \sum_{n \gt 0}} \psi (n) n^{\nu} e^{2 \pi in^2 z}$. Such sums typically fail to have modular properties. We give an analytic construction which unifies these partial theta functions with the mock theta functions introduced by Ramanujan.

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The author was supported by an NSF fellowship during the writing of this paper.

Received 17 February 2012

Published 5 July 2018