# Twisted $2k$th moments of primitive Dirichlet $L$-functions: beyond the diagonal

## Location

In this joint work with Caroline Turnage-Butterbaugh, 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 infinity. We approximate the twisted $2k$th moment of this family using Dirichlet polynomials of length between $Q$ and $Q^2$. Assuming the Generalized Lindelof Hypothesis, we prove an asymptotic formula for these approximations. Our result agrees with the prediction of Conrey, Farmer, Keating, Rubinstein, and Snaith, and provides the first rigorous evidence beyond the diagonal terms for their conjectured asymptotic formula for the general $2k$th moment of this family. The main device we use in our proof is the asymptotic large sieve developed by Conrey, Iwaniec, and Soundararajan.