n -level density of the low-lying zeros of quadratic Dirichlet L -functions.

By Peng Gao. 2008

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The Density Conjecture of Katz and Sarnak associates a classical compact group to each reasonable family of $L$-functions. Under the assumption of the Generalized Riemann Hypothesis, Rubinstein computed the $n$-level density of low-lying zeros for the family of quadratic Dirichlet $L$-functions in the case that the Fourier transform $\hat{f}(u)$ of any test function $f$ is supported in the region $\sum^n_{j=1}u_j < 1$ and showed that the result agrees with the Density Conjecture. In this paper, we improve Rubinstein's result on computing the $n$-level density for the Fourier transform $\hat{f}(u)$ being supported in the region $\sum^n_{j=1}u_j < 2$.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6839eb3b34bd5fdd99be58675ba2caf96571d9ed