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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.