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

TL;DR

This work addresses approximating the reciprocals of $L$-functions $L_{F_j}(s)$ attached to $SL(3,\mathbb{Z})$ Hecke--Maass cusp forms by very short Dirichlet polynomials. The authors combine an approximate functional equation with a variant of the Kuznetsov trace formula and mollification to study averaged mollified moments over spectral parameters and forms. They prove that the averaged quantity $\sum_j^{*}\int_T^{2T} |L_{F_j}(\tfrac12+\theta+it)^{-1} - M(\tfrac12+\theta+it,F_j)|^2 dt$ matches a main term up to $O(U^{-eta})$, under explicit growth conditions on the mollifier lengths, and they derive corollaries allowing even shorter mollifiers. The results deepen our understanding of short Dirichlet polynomial approximations for high-rank $L$-functions and have potential implications for mollified moment methods and non-vanishing results in the automorphic setting.

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.

Theorems & Definitions (19)

• Theorem 1.1
• proof
• Theorem 1.2
• Corollary 1.3
• proof
• Remark 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• Remark 2.2