## Zeros of Dirichlet $L$-functions over Function Fields

with S. J. Miller, K. Pratt and M. Trinh

Communications in Number Theory and Physics, Vol. 8, No. 3 (2014), pp. 511-539.

Communications in Number Theory and Physics, Vol. 8, No. 3 (2014), pp. 511-539.

**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 $Q \rightarrow\infty$, 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.