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# Communications in Number Theory and Physics

## Volume 8 (2014)

### Number 3

### Zeros of Dirichlet $L$-functions over function fields

Pages: 511 – 539

DOI: http://dx.doi.org/10.4310/CNTP.2014.v8.n3.a3

#### Authors

#### Abstract

Random matrix theory has successfully modeled many systems in physics and mathematics, and often analysis in one area guides development in the other. Hughes and Rudnick computed one-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 the random matrix theory. We compute one- and two-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. Our statements are more general and also include error terms. In concluding, we discuss an $\mathbb{F}_q(T)$-analog 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.