Geodesic
The $n$-level densities of low-lying zeros of quadratic Dirichlet $L$-functions
Jake Levinson1 ; Steven J. Miller2
1 Department of Mathematics University of Michigan Ann Arbor, MI 48109, U.S.A.
2 Department of Mathematics and Statistics Williams College Williamstown, MA 01267, U.S.A.
Acta Arithmetica, Tome 161 (2013) no. 2, pp. 145-182.

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Previous work by Rubinstein and Gao computed the $n$-level densities for families of quadratic Dirichlet $L$-functions for test functions $\widehat{f}_1, \dots, \widehat{f}_n$ supported in $\sum_{i=1}^n |u_i| 2$, and showed agreement with random matrix theory predictions in this range for $n \le 3$ but only in a restricted range for larger $n$. We extend these results and show agreement for $n \le 7$, and reduce higher $n$ to a Fourier transform identity. The proof involves adopting a new combinatorial perspective to convert all terms to a canonical form, which facilitates the comparison of the two sides.
DOI : 10.4064/aa161-2-3
Keywords: previous work rubinstein gao computed n level densities families quadratic dirichlet l functions test functions widehat dots widehat supported sum showed agreement random matrix theory predictions range only restricted range larger extend these results agreement reduce higher fourier transform identity proof involves adopting combinatorial perspective convert terms canonical form which facilitates comparison sides

Jake Levinson 1 ; Steven J. Miller 2

1 Department of Mathematics University of Michigan Ann Arbor, MI 48109, U.S.A.
2 Department of Mathematics and Statistics Williams College Williamstown, MA 01267, U.S.A. 
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Jake Levinson; Steven J. Miller. The $n$-level densities of low-lying zeros of
 quadratic Dirichlet $L$-functions. Acta Arithmetica, Tome 161 (2013) no. 2, pp. 145-182. doi : 10.4064/aa161-2-3. http://geodesic.mathdoc.fr/articles/10.4064/aa161-2-3/

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