Prime geodesics and averages of the Zagier $L$-series

Abstract

The Zagier L-series encode data of real quadratic fields. We study the average size of these L-series, and prove asymptotic expansions and omega results for the expansion. We then show how the error term in the asymptotic expansion can be used to obtain error terms in the prime geodesic theorem.

Spectral decomposition formula and moments of symmetric square L-functions

“…was studied in [3] in connection with the prime geodesic theorem. See also [4], [5], [7], [19], [20] for related results. Furthermore, sums of the form (1.2) appear in the explicit formulas for the first moments of symmetric square L-functions associated to holomorphic (see [21], [2]) or Maass (see [1]) cusp forms.…”