Remarks on the distribution of Dirichlet $L$-functions along cosets

Abstract

In a previous work with B. Garcia, the author considered the asymptotic for the second moment of Dirichlet $L$-functions along cosets, and exhibited a surprising secondary main term that is not predicted by the recipe of Conrey, Farmer, Keating, Rubinstein, and Snaith. In this paper, we re-examine this problem and propose a modified recipe that correctly predicts this secondary main term. The original recipe gives the incorrect answer for this family because the root number is not always independent of the Dirichlet series coefficients along certain cosets, and our proposed fix simply takes this feature into account. In addition, we consider a handful of other problems related to Dirichlet $L$-functions along cosets. One goal is to reformulate Heath-Brown's $q$-analog of van der Corput's shifting method in terms of cosets, which leads to an upper bound on a hybrid second moment. We also revisit the classical van der Corput bound and view it (in more modern terms) as an amplified second moment of a trigonometric polynomial.