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#### On Petersson's partition limit formula

##### External Resource

https://doi.org/10.1142/S1793042121500408

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##### Fulltext (public)

2011.14601.pdf

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##### Citation

Castaño-Bernard, C., & Luca, F. (2021). On Petersson's partition limit formula.* International Journal of Number Theory,* *17*(6), 1365-1378.
doi:10.1142/S1793042121500408.

Cite as: https://hdl.handle.net/21.11116/0000-0009-2AF4-5

##### 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.

$\chi=(\frac{\cdot}{p})$. 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.