Averages of long Dirichlet polynomials with modular coefficients

TL;DR

This paper derives an asymptotic formula for the moments of L-functions associated with primitive cusp forms, focusing on long Dirichlet polynomials with modular coefficients, under the Generalized Lindelöf Hypothesis.

Contribution

It provides a new asymptotic formula for twisted moments of long Dirichlet polynomials with modular coefficients, aligning with conjectural predictions.

Findings

Asymptotic formula derived under the G-Lindelöf Hypothesis.

Results agree with the conjectural recipe by Conrey et al.

Focus on moments in the weight aspect for L-functions.

Abstract

We study the moments of $L$-functions associated with primitive cusp forms, in the weight aspect. In particular, we obtain an asymptotic formula for the twisted moments of a \textit{long} Dirichlet polynomial with modular coefficients. This result, which is conditional on the Generalized Lindel\"of Hypothesis, agrees with the prediction of the recipe by Conrey, Farmer, Keating, Rubinstein and Snaith.