Selberg's Central limit theorem for quadratic Dirichlet L-functions over function fields

2021/05/23 by Darbar, Pranendu, Lumley, Allysa · 1 citation
#11G20 #11K65 #11T35 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2105.10863

Abstract

In this article, we study the logarithm of the central value L((1)/(2), χ_D) in the symplectic family of Dirichlet L-functions associated with the hyperelliptic curve of genus δ over a fixed finite field \mathbbF_q in the limit as δ→ ∞. Unconditionally, we show that the distribution of log |L((1)/(2), χ_D)| is asymptotically bounded above by the Gaussian distribution of mean (1)/(2)log °(D) and variance log °(D). Assuming a mild condition on the distribution of the low-lying zeros in this family, we obtain the full Gaussian distribution.

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• Linear Combinations of Logarithms of L-functions over Function Fields at Microscopic Shifts and Beyond

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