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

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 F 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 deg(D) and variance log deg (D). Assuming a mild condition on the distribution of the low-lying zeros in this family, we obtain the full Gaussian distribution.

Weighted central limit theorems for central values of $L$-functions

“…Hough [22] adapted Selberg's method to study the distribution of central values of families of L-functions (see also [11]). For the family of quadratic Dirichlet characters, he established a one-sided central limit theorem; that is, he bounded the proportion of fundamental discriminants, in magnitude < X, for which the logarithm of the normalized central L-value is larger than a given V > 0 by…”