On Petersson's partition limit formula

Abstract

For each prime $p\equiv 1(\mathrm{mod}4)$ 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-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.

Online Ready - International Journal of Number Theory