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.
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 [Citation17].
2 Here, and elsewhere, we allow for a slight abuse of notation: α and β denote coordinates of MK, as well as coordinates of the point at which the derivative is then evaluated.
3 A function is said to be homolorphic if it is holomorphic in each variable separately.