Zeros of Dirichlet L- Function Over Function Fields

Abstract

In this paper we consider the Zeros of Dirichlet L- Function Over Function Fields. We compute 1- and 2-level statistics of the analogous family of Dirichlet L-functions over Fq(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. Our statements are more general and also include error terms. In concluding, we discuss an Fq(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-Rudnickresults.