Non-vanishing and One Level Density for Dirichlet L-functions Along Short Averages

2024/09/19 by Basak, Debmalya
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2409.12474

Abstract

Assuming the Generalized Riemann Hypothesis, it is known that at least half of the central values L((1)/(2),χ) are non-vanishing as χ ranges over primitive characters modulo q. Unconditionally, this is known on average over both χ modulo q and Q/2 ≤ q ≤ 2Q. We prove that for any δ>0, there exist η₁,η₂>0 depending on δ such that the non-vanishing proportion for L((1)/(2),χ) as χ ranges modulo q with q varying in short intervals of size Q^1-η₁ around Q and in arithmetic progressions with moduli up to Q^η₂ is larger than (1)/(2)-δ. Furthermore, by studying the one-level density of low-lying zeros of L(s, χ), we show that under the Generalized Riemann Hypothesis, non-vanishing proportions exceeding (1)/(2) can be obtained while still averaging over short ranges of q.

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