The Katz-Sarnak one-level density prediction for Maass form L-functions of prime level now holds for Fourier support up to 15/8, and up to 2 under the Grand Density Conjecture.
Extending the unconditional support in an Iwaniec-Luo-Sarnak family
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abstract
We study the harmonically weighted one-level density of low-lying zeros of $L$-functions in the family of holomorpic newforms of fixed even weight $k$ and prime level $N$ tending to infinity. For this family, Iwaniec, Luo and Sarnak proved that the Katz--Sarnak prediction for the one-level density holds unconditionally when the support of the Fourier transform of the implied test function is contained in $(-\tfrac32,\tfrac32)$. In this paper, we extend this admissible support to $(-\Theta_k,\Theta_k)$, where $\Theta_2 = 1.866\dots$ and $\Theta_k$ tends monotonically to $2$ as $k$ tends to infinity. This is asymptotically as good as the best known GRH result. The main novelty in our analysis is the use of zero-density estimates for Dirichlet $L$-functions.
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Low-lying zeros in families of Maass form L-functions: an extended density theorem
The Katz-Sarnak one-level density prediction for Maass form L-functions of prime level now holds for Fourier support up to 15/8, and up to 2 under the Grand Density Conjecture.