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 $L\left(\frac{1}{2},{\chi }_D\right)$ in the symplectic family of Dirichlet $L$-functions associated with the hyperelliptic curve of genus $\delta$ over a fixed finite field ${\mathbb{F}}_q$ in the limit as $\delta \to \infty$. Unconditionally, we show that the distribution of $\log \left|L\left(\frac{1}{2},{\chi }_D\right)\right|$ is asymptotically bounded above by the Gaussian distribution of mean $\frac{1}{2}\log °(D)$ 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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