Breaking the 1/2-barrier for the twisted second moment of Dirichlet L-functions

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 1/2-barrier in this problem, and obtain an asymptotic formula provided that the length of the Dirichlet polynomial is less than q(51/101) = q(1/2+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. (C) 2020 Elsevier Inc. All rights reserved.