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

TL;DR

This paper investigates the average behavior of Zagier L-series associated with real quadratic fields, deriving asymptotic formulas and applying these results to improve error estimates in the prime geodesic theorem.

Contribution

It provides new asymptotic expansions and omega results for the averages of Zagier L-series, linking these to prime geodesic theorem error terms.

Findings

Derived asymptotic expansions for the average size of Zagier L-series

Established omega results indicating oscillatory behavior of the averages

Connected error terms in L-series averages to improvements in prime geodesic theorem estimates

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.