Asymptotic formulas for the second moments of $L$-series associated to holomorphic cusp forms on the critical line

TL;DR

This paper derives new uniform asymptotic formulas for the second moments of L-series associated with holomorphic cusp forms on the critical line, advancing understanding of their distribution and behavior.

Contribution

It introduces novel asymptotic formulas for the second moments of L-series of cusp forms, extending previous results to a uniform setting across parameters.

Findings

Established new asymptotic formulas for second moments

Provided uniform estimates across parameters

Enhanced understanding of L-series behavior on the critical line

Abstract

New uniform asymptotic formulas are obtained for the second moment of $L$-series of cusp forms of even weight $2k\ge2$ with respect to the congruence subgroup $\Gamma_0(N).$