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arXiv · 2105.10863

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

Pranendu Darbar·Allysa Lumley
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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 $\log \big|L\left(\frac{1}{2}, \chi_D\right)\big|$ is asymptotically bounded above by the Gaussian distribution of mean $\frac{1}{2}\log \deg(D)$ 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.

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Pranendu Darbar, Allysa Lumley. 2021-05-23. Selberg's Central limit theorem for quadratic Dirichlet L-functions over function fields. https://arxiv.org/abs/2105.10863

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