On Petersson's partition limit formula

TL;DR

This paper explores a new approach to prove Petersson's limit formula for partition ratios involving quadratic fields, addressing Grosswald's conjecture and discussing related monotonicity conjectures.

Contribution

It proposes a novel method to prove Petersson's limit formula using Tauberian theorems and discusses related monotonicity conjectures in partition theory.

Findings

Suggests an approach to prove Petersson's formula via Tauberian methods.

Discusses a natural monotonicity conjecture related to partition ratios.

Connects partition problems with invariants of quadratic fields.

Abstract

For each prime $p\equiv 1\pmod{4}$ consider the Legendre character $\chi=(\frac{\cdot}{p})$. Let $p_\pm(n)$ be the number of partitions of $n$ into parts $\lambda>0$ such that $\chi(\lambda)=\pm 1$. Petersson proved a beautiful limit formula for the ratio of $p_+(n)$ to $p_-(n)$ as $n\to\infty$ expressed in terms of important invariants of the real quadratic field $\mathbb{Q}(\sqrt{p})$. 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-Erd\H{o}s.