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.

Keywords:

• Ratios Conjecture
• Random Matrix Theory
• Gaussian Primes
• Hecke L-functions

Acknowledgments

This work emerged from a summer project developed and guided by E. Waxman, as part of the 2017 SMALL Undergraduate Research Project at Williams College. We thank Zeev Rudnick for advice, and for suggesting this problem, as well as Bingrong Huang and J. P. Keating for helpful discussions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Notes

1 See also [17].

2 Here, and elsewhere, we allow for a slight abuse of notation: α and β denote coordinates of M_K, as well as coordinates of the point at which the derivative is then evaluated.

3 A function $f:\Omega \subset {\mathbb{C}}^2\mapsto \mathbb{C}$ is said to be homolorphic if it is holomorphic in each variable separately.

Additional information

Funding

The summer research was supported by NSF Grant DMS1659037. Chen was moreover supported by Princeton University, and Miller was supported by NSF Grant DMS1561945. Waxman was supported by the European Research Council under the European Union’s Seventh Framework Program (FP7/2007-2013)/ERC grant agreement no 320755., as well as by the Czech Science Foundation GAČR, grant 17-04703Y.