# The first moment of cusp form L-functions in weight aspect on average

## TL;DR

This paper investigates the average behavior of the first moment of cusp form L-functions in weight aspect, improving error estimates and extending the mollifier length beyond previous limits.

## Contribution

It introduces a new approach using Legendre polynomials to estimate error terms, enabling a longer mollifier length in the analysis.

## Key findings

- Extended the mollifier length to 2 from the previous limit of 1.
- Provided a new representation formula for error terms involving Legendre polynomials.
- Achieved more precise asymptotic estimates for the first moment of L-functions.

## Abstract

We study the asymptotic behaviour of the twisted first moment of central $L$-values associated to cusp forms in weight aspect on average. Our estimate of the error term allows extending the logarithmic length of mollifier $\Delta$ up to 2. The best previously known result, due to Iwaniec and Sarnak, was $\Delta<1$. The proof is based on a representation formula for the error in terms of Legendre polynomials.