A Refined Conjecture for the Variance of Gaussian Primes across Sectors. Issue 1 (2nd January 2023) - Chen, Ryan C.; Kim, Yujin H.; Lichtman, Jared D.; Miller, Steven J.; Shubina, Alina; Sweitzer, Shannon; Waxman, Ezra; Winsor, Eric; Yang, Jianing
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.