Abstract

We derive a refined conjecture for the variance of Gaussian primes across sectors, with a power saving error term, by applying the L-functions Ratios Conjecture. We observe a bifurcation point in the main term, consistent with the Random Matrix Theory (RMT) heuristic previously proposed by Rudnick and Waxman. Our model also identifies a second bifurcation point, undetected by the RMT model, that emerges upon taking into account lower order terms. For sufficiently small sectors, we moreover prove an unconditional result that is consistent with our conjecture down to lower order terms.

中文翻译：

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跨部门高斯素数方差的改进猜想

摘要

我们通过应用L函数比值猜想，推导出跨扇区高斯素数方差的精细猜想，其中包含节能误差项。我们在主项中观察到一个分叉点，这与 Rudnick 和 Waxman 先前提出的随机矩阵理论 (RMT) 启发式一致。我们的模型还确定了 RMT 模型未检测到的第二个分叉点，该分叉点是在考虑低阶项时出现的。对于足够小的扇区，我们还证明了一个无条件的结果，该结果与我们的猜想一致，直到低阶项。