We establish an analogue of a conjecture of Balasubramanian, Conrey, and Heath-Brown for the family of all Dirichlet characters with conductor up to Q. This forms another application of our work in developing an asymptotic large sieve.
"The mean square of the product of a Dirichlet $L$-function and a Dirichlet polynomial." Funct. Approx. Comment. Math. 61 (2) 147 - 177, December 2019. https://doi.org/10.7169/facm/1767