On Petersson's partition limit formula

Abstract

For each prime p 14 consider the Legendre character =(·p). Let p(n) be the number of partitions of n into parts λ>0 such that (λ)= 1. Petersson proved a beautiful limit formula for the ratio of p+(n) to p-(n) as n∞ expressed in terms of important invariants of the real quadratic field Q(p). But his proof is not illuminating and Grosswald conjectured a more natural proof using a Tauberian converse of the Stolz-Ces\`aro theorem. In this paper we suggest an approach to address Grosswald's conjecture. We discuss a monotonicity conjecture which looks quite natural in the context of the monotonicity theorems of Bateman-Erdos.