Moments and distribution of values of $L$-functions over function fields inside the critical strip
Volume 201 / 2021
Abstract
Let $q\equiv 1 \pmod 4$ and $\mathbb {F}_q$ be the finite field with $q$ elements. Take $\sigma \in (1/2,1)$ fixed. This article discusses the distribution of large values for $\log L(\sigma ,\chi _D)$ with $\chi _D$ the Kronecker symbol and $D$ varying over the monic square-free polynomials with degree $n\to \infty $. The description for the tail of the distribution of values has a familiar shape when compared to the number field case. However, there is a surprising difference occurring which is unique to the function field setting. These results are achieved by showing the $L$-functions are modelled very well by an appropriate random Euler product. Additionally, we find some $\Omega $-results which are predicted to be best possible by the probabilistic model we use.
