Mathematical Research Letters

Volume 25 (2018)

Number 2

A unified approach to partial and mock theta functions

Pages: 659 – 675

DOI: https://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.

Received 17 February 2012

Accepted 17 February 2012

Published 5 July 2018