A Refined Conjecture for the Variance of Gaussian Primes across Sectors

Chen, Ryan C.; Kim, Yujin H.; Lichtman, Jared Duker; Miller, Steven J.; Shubina, Alina; Sweitzer, Shannon; Waxman, Ezra; Winsor, Eric; Yang, Jianing

doi:10.1080/10586458.2020.1753598

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2021

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Cited by 2 publications

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“…This family of characters arises in several applications connected to the angular distribution of Gaussian primes [19,20], and, in particular, the zero distribution near the central point across F(K) affects the variance of Gaussian primes across sectors [4,24].…”

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confidence: 99%

Lower Order Terms for the One-Level Density of a Symplectic Family of Hecke L-Functions

Waxman

2019

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In this paper we apply the L-function Ratios Conjecture to compute the one-level density for a symplectic family of L-functions attached to Hecke characters of infinite order. When the support of the Fourier transform of the corresponding test function f reaches 1, we observe a transition in the main term, as well as in the lower order term. Assuming GRH, we then directly calculate main and lower order terms for test functions f such that supp( f) ⊂ (−1, 1), and observe that the result is in agreement with the prediction provided by the Ratios Conjecture. As a corollary, we deduce that, under GRH, at least 75% of these L-functions do not vanish at the central point.

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confidence: 99%