Number theoretic properties of generating functions related to Dyson’s rank for partitions into distinct parts

Abstract

Let Q(n) denote the number of partitions of n into distinct parts. We show that Dyson's rank provides a combinatorial interpretation of the well-known fact that Q(n) is almost always divisible by 4. This interpretation gives rise to a new false theta function identity that reveals surprising analytic properties of one of Ramanujan's mock theta functions, which in turn gives generating functions for values of certain Dirichlet L-functions at nonpositive integers.