Cancellation in sums over special sequences on $\mathbf{{\rm{GL}}_{m}}$ and their applications

TL;DR

This paper establishes new bounds for sums of automorphic $L$-function coefficients over special sequences, with applications to the Sato--Tate and Goldbach conjectures, independent of some long-standing conjectures.

Contribution

It provides nontrivial bounds for additive twisted sums over primes on ${ m GL}_m$, advancing understanding of cancellation in automorphic coefficient sums.

Findings

Bound for additive twisted sums over primes on ${ m GL}_m$

Application to Sato--Tate conjecture

Proposal of a Goldbach conjecture on average

Abstract

Let $a(n)$ be the $n$-th Dirichlet coefficient of the automorphic $L$-function or the Rankin--Selberg $L$-function. We investigate the cancellation of $a(n)$ over sequences linked to the Waring--Goldbach problem, by establishing a nontrivial bound for the additive twisted sums over primes on ${\mathrm{GL}}_m .$ The bound does not depend on the generalized Ramanujan conjecture or the nonexistence of Landau--Siegel zeros. Furthermore, we present an application associated with the Sato--Tate conjecture and propose a conjecture about the Goldbach conjecture on average bound.