Approximations of $SL(3,\mathbb{Z})$ Hecke-Maass $L$-Functions by short Dirichlet polynomials

TL;DR

This paper demonstrates that the reciprocals of $SL(3,Z)$ Hecke-Maass $L$-functions can be effectively approximated by very short Dirichlet polynomials on average, using a Kuznetsov trace formula variant.

Contribution

It introduces a novel method to approximate reciprocals of high-rank $L$-functions with short Dirichlet polynomials on average.

Findings

Reciprocals of $SL(3,Z)$ Hecke-Maass $L$-functions are approximable by short Dirichlet polynomials.

Employs a Kuznetsov trace formula variant to establish these approximations.

Provides average-case results over spectral parameters and forms.

Abstract

We study averages of $L$-functions associated with Hecke-Maass cusp forms for $SL(3,\mathbb{Z})$, multiplied by Dirichlet polynomials built from the Fourier coefficients of the cusp forms. To prove this, we employ a variant of the Kuznetsov trace formula. In particular, we show that the reciprocals of these $L$-functions can be approximated by very short Dirichlet polynomials, on average over $t$ and over the forms.