TY - JOUR

T1 - Breaking the [Formula presented]-barrier for the twisted second moment of Dirichlet L-functions

AU - Bui, Hung M.

AU - Pratt, Kyle

AU - Robles, Nicolas

AU - Zaharescu, Alexandru

N1 - Funding Information:
The second author was supported by NSF grant DMS-1501982 . The authors would like to thank Sandro Bettin and Maksym Radziwiłł for various helpful comments. The authors would like to thank the referees for pointing out very useful clarifications and insights that have increased the quality of the manuscript.
Publisher Copyright:
© 2020 Elsevier Inc.

PY - 2020/8/26

Y1 - 2020/8/26

N2 - We study the second moment of Dirichlet L-functions to a large prime modulus q twisted by the square of an arbitrary Dirichlet polynomial. We break the [Formula presented]-barrier in this problem, and obtain an asymptotic formula provided that the length of the Dirichlet polynomial is less than q51/101=q1/2+1/202. As an application, we obtain an upper bound of the correct order of magnitude for the third moment of Dirichlet L-functions. We give further results when the coefficients of the Dirichlet polynomial are more specialized.

AB - We study the second moment of Dirichlet L-functions to a large prime modulus q twisted by the square of an arbitrary Dirichlet polynomial. We break the [Formula presented]-barrier in this problem, and obtain an asymptotic formula provided that the length of the Dirichlet polynomial is less than q51/101=q1/2+1/202. As an application, we obtain an upper bound of the correct order of magnitude for the third moment of Dirichlet L-functions. We give further results when the coefficients of the Dirichlet polynomial are more specialized.

KW - Dirichlet L-functions

KW - Kloosterman fractions

KW - Large sieve inequality

KW - Moments

KW - Twisted second moment

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U2 - 10.1016/j.aim.2020.107175

DO - 10.1016/j.aim.2020.107175

M3 - Article

AN - SCOPUS:85084524665

VL - 370

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

M1 - 107175

ER -