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On the One-Level Density Conjecture for Quadratic Dirichlet L-Functions

  Published:2006-08-01
 Printed: Aug 2006
  • A. E. Õzlük
  • C. Snyder
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Abstract

In a previous article, we studied the distribution of ``low-lying" zeros of the family of quad\-ratic Dirichlet $L$-functions assuming the Generalized Riemann Hypothesis for all Dirichlet $L$-functions. Even with this very strong assumption, we were limited to using weight functions whose Fourier transforms are supported in the interval $(-2,2)$. However, it is widely believed that this restriction may be removed, and this leads to what has become known as the One-Level Density Conjecture for the zeros of this family of quadratic $L$-functions. In this note, we make use of Weil's explicit formula as modified by Besenfelder to prove an analogue of this conjecture.
MSC Classifications: 11M26 show english descriptions Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses 11M26 - Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses
 

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