For each prime p ≡ 1(mod 4) consider the Legendre character χ = ( ⋅ p). Let p±(n) be the number of partitions of n into parts λ > 0 such that χ(λ) = ±1. Petersson proved a beautiful limit formula for the ratio of p+(n) to p−(n) as n →∞ expressed in terms of important invariants of the real quadratic field K = ℚ(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.

中文翻译：

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关于彼得森的分区极限公式

对于每个素数p ≡ 1(模组 4)考虑勒让德角色χ = ( ⋅ p). 让p±(n)是的分区数n分成几部分λ > 0这样χ(λ) = ±1. 彼得森证明了一个漂亮的极限公式p+(n)到p-(n)作为n →∞用实二次场的重要不变量表示ķ = ℚ(p). 但他的证明并不具有启发性，格罗斯瓦尔德使用斯托尔茨-塞萨罗定理的陶伯逆反函数推测了一个更自然的证明。在本文中，我们提出了一种解决 Grosswald 猜想的方法。我们讨论了一个单调性猜想，它在 Bateman-Erdős 的单调性定理的背景下看起来很自然。