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

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.