Zeros of Dirichlet L-functions over Function Fields

Abstract

Random matrix theory has successfully modeled many systems in physics and mathematics, and often the analysis and results in one area guide development in the other. Hughes and Rudnick computed $1$-level density statistics for low-lying zeros of the family of primitive Dirichlet $L$-functions of fixed prime conductor $Q$, as $Q \to \infty$, and verified the unitary symmetry predicted by random matrix theory. We compute $1$- and $2$-level statistics of the analogous family of Dirichlet $L$-functions over $\mathbb{F}_q(T)$. Whereas the Hughes-Rudnick results were restricted by the support of the Fourier transform of their test function, our test function is periodic and our results are only restricted by a decay condition on its Fourier coefficients. We show the main terms agree with unitary symmetry, and also isolate error terms. In concluding, we discuss an $\mathbb{F}_q(T)$-analogue of Montgomery's Hypothesis on the distribution of primes in arithmetic progressions, which Fiorilli and Miller show would remove the restriction on the Hughes-Rudnick results.