On Petersson’s partition limit formula

Abstract

For each prime [Formula: see text] consider the Legendre character [Formula: see text]. Let [Formula: see text] be the number of partitions of [Formula: see text] into parts [Formula: see text] such that [Formula: see text]. Petersson proved a beautiful limit formula for the ratio of [Formula: see text] to [Formula: see text] as [Formula: see text] expressed in terms of important invariants of the real quadratic field [Formula: see text]. But his proof is not illuminating and Grosswald conjectured a more natural proof using a Tauberian converse of the Stolz–Cesàro 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–Erdős.