A Refined Conjecture for the Variance of Gaussian Primes Across Sectors

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.

arXiv comment47 pages. Minor revisions. Appendix with relevant Mathematica code, included. Accepted for publication in Experimental Mathematics

References in corpus (3)

• Conjectures for the integral moments and ratios of L-functions over function fields
• On the variance of sums of arithmetic functions over primes in short intervals and pair correlation for L-functions in the Selberg class
• Low-lying zeros of quadratic Dirichlet $L$-functions: A transition in the Ratios Conjecture

Cited by in corpus (1)

• Discrepancies in the distribution of Gaussian primes