Averages of twisted L-functions

TL;DR

This paper employs a relative trace formula on GL(2) to evaluate sums of twisted modular L-functions across the critical strip, providing bounds that match Lindelof hypothesis predictions in certain regimes.

Contribution

It introduces a novel application of the relative trace formula to compute averages of twisted L-functions with explicit bounds in the large weight or level limits.

Findings

Sum of twisted L-functions is nonzero for large weight or level.

Derived bounds at the central point match Lindelof hypothesis predictions.

Method provides new insights into the distribution of twisted L-values.

Abstract

We use a relative trace formula on GL(2) to compute a sum of twisted modular L-functions anywhere in the critical strip, weighted by a Fourier coefficient and a Hecke eigenvalue. When the weight k or level N is sufficiently large, the sum is nonzero. Specializing to the central point, we show in some cases that the resulting bound for the average is as good as that predicted by the Lindelof hypothesis in the k and N aspects.