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We estimate the $1$-level density of low-lying zeros of $L(s,\chi)$ with $\chi$ ranging over primitive Dirichlet characters of conductor $\in [Q/2,Q]$ and for test functions whose Fourier transform is supported in $[- 2 - 50/1093, 2 + 50/1093]$.
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C. P. Hughes and Z. Rudnick, Linear statistics of low-lying zeros of L L -functions , Q. J. Math. 54
2003
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by same author, The average analytic rank of elliptic curves , Duke Math. J. 122
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H. Iwaniec and E. Kowalski, Analytic number theory , American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004
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V. Chandee, Y. Lee, S.-C. Liu, and M. Radziwiłł, Simple zeros of primitive Dirichlet L L -functions and the asymptotic large sieve , Q. J. Math. 65
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P. Sarnak, S. W. Shin, and N. Templier, Families of L L -functions and their symmetry , Families of automorphic forms and the trace formula, Simons Symp., Springer, [Cham], 2016, pp. 531–578
2016
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by same author, Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method , Proc. London Math. Soc. (3) 114
2017
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É. Fouvry and M. Radziwiłł, Another application of Linnik’s dispersion method , Chebyshevskiĭ Sb. 19
2018
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A. Granville and K. Soundararajan, Large character sums: Burgess’s theorem and zeros of L L -functions , J. Eur. Math. Soc. (JEMS) 20
2018
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K. Pratt, Average non-vanishing of Dirichlet L L -functions at the central point , Algebra Number Theory 13
2019
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M. Watkins, Comments on Deuring’s zero-spacing phenomenon , J. Number Theory 218
2021
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