Schur's partition theorem and mixed mock modular forms

Abstract

We study families of partitions with gap conditions that were introduced by Schur and Andrews, and describe their fundamental connections to combinatorial q-series and automorphic forms. In particular, we show that the generating functions for these families naturally lead to deep identities for theta functions and Hickerson's universal mock theta function, which provides a very general answer to Andrews' Conjecture on the modularity of the Schur-type generating function. Furthermore, we also complete the second part of Andrews' speculation by determining the asymptotic behavior of these functions. In particular, we use Wright's Circle Method in order to prove families of asymptotic inequalities in the spirit of the Alder-Andrews Conjecture. As a final application, we prove the striking result that the universal mock theta function can be expressed as a conditional probability in a certain natural probability space with an infinite sequence of independent events.