Twisted $2k$th moments of primitive Dirichlet $L$functions: beyond the diagonal
Abstract
We study the family of Dirichlet $L$functions of all even primitive characters of conductor at most $Q$, where $Q$ is a parameter tending to $\infty$. For an arbitrary positive integer $k$, we approximate the twisted $2k$th moment of this family by using Dirichlet polynomial approximations of $L^k(s,\chi)$ of length $X$, with $Q<X<Q^2$. Assuming the Generalized Lindelöf Hypothesis, we prove an asymptotic formula for these approximations of the twisted moments. Our result agrees with the prediction of Conrey, Farmer, Keating, Rubinstein, and Snaith for this family of $L$functions, and provides the first rigorous evidence beyond the diagonal terms for their conjectured asymptotic formula for the general $2k$th moment of this family.
 Publication:

arXiv eprints
 Pub Date:
 May 2022
 DOI:
 10.48550/arXiv.2205.00641
 arXiv:
 arXiv:2205.00641
 Bibcode:
 2022arXiv220500641B
 Keywords:

 Mathematics  Number Theory;
 11M06
 EPrint:
 99 pages