论文标题
关于彼得森的分区限制公式
On Petersson's partition limit formula
论文作者
论文摘要
对于每个prime $ p \ equiv 1 \ pmod {4} $考虑legendre targue $χ=（\ frac {\ cdot} {p}）$。令$ p_ \ pm（n）$为$ n $ of parts $λ> 0 $的分区数，这样$χ（λ）= \ pm 1 $。彼得森证明了$ p _+（n）$与$ p _-（n）$的比率为$ n \ to \ infty $的美丽限制公式，该$ to \ n \ to \ infty $根据真正的Quadratic Field $ \ mathbb {q}（\ sqrt {p}）$表示。但是他的证明并没有启发性，格罗斯瓦尔德（Grosswald）猜想使用tauberian converse stolz-cesàro定理来证明更自然的证据。在本文中，我们建议一种解决格罗斯瓦尔德猜想的方法。我们讨论了一个单调性的猜想，它在贝特曼·埃德斯（Bateman-Erds）的单调定理的背景下看起来很自然。
For each prime $p\equiv 1\pmod{4}$ consider the Legendre character $χ=(\frac{\cdot}{p})$. Let $p_\pm(n)$ be the number of partitions of $n$ into parts $λ>0$ such that $χ(λ)=\pm 1$. 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.
