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

Speaker: Allysa Lumley

Date: Mon, Jul 25, 2022

Location: PIMS, University of Northern British Columbia

Conference: Moments of L-functions Workshop

Subject: Mathematics, Number Theory

Class: Scientific

CRG: L-Functions in Analytic Number Theory


In this talk, we will discuss 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 $g$ over a fixed finite field $\mathbb{F}_q$ in the limit as $g\to \infty$. Unconditionally, we show that the distribution of $\log \big|L\left(\frac{1}{2}, \chi_D\right)\big|$ is asymptotically bounded above by the full Gaussian distribution of mean $\frac{1}{2}\log \deg(D)$ and variance $\log \deg(D)$, and also $\log \big|L\left(\frac{1}{2}, \chi_D\right)\big|$ is at least $94.27 \%$ Gaussian distributed. 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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