Communications in Number Theory and Physics

Volume 8 (2014)

Number 3

Zeros of Dirichlet $L$-functions over function fields

Pages: 511 – 539



Julio C. Andrade (Institut des Hautes Études Scientifiques (IHÉS), Bures-sur-Yvette, France; and Depto. Matematica, PUC, Rio De Janeiro, Brazil)

Steven J. Miller (Department of Mathematics and Statistics, Williams College, Williamstown, Massachusetts, U.S.A.)

Kyle Pratt (Department of Mathematics, Brigham Young University, Provo, Utah, U.S.A.)

Minh-Tam Trinh (Department of Mathematics, Princeton University, Princeton, New Jersey, U.S.A.)


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.

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