Zagier's weight $3/2$ mock modular form

TL;DR

This paper provides a detailed proof that Zagier's generating function for Hurwitz class numbers is a mock modular form of weight 3/2, elucidating its transformation properties and shadow in the context of Ramanujan's mock theta functions.

Contribution

It offers a comprehensive proof of Zagier's 1975 result, clarifying the mock modular nature of the Hurwitz class number generating function.

Findings

The generating function of Hurwitz class numbers is a mock modular form of weight 3/2.

The theta function acts as the shadow of this mock modular form.

The paper clarifies the transformation properties of the generating function.

Abstract

Mock modular forms have their origins in Ramanujan's pioneering work on mock theta functions. In a 1975 paper, Zagier proved certain transformation properties of the generating function of the Hurwitz class numbers $H(n)$ for the discriminant $(-n)$. In the modern framework, these results show that the generating function of $H(n)$ is a mock modular form of weight 3/2 with the theta function being the shadow. In this expository paper, we provide a detailed proof of Zagier's result.