Cancellation in sums over special sequences on $\mathrm {GL}_m$ and their applications

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 non-trivial bound for the additive twisted sums over primes on ${\mathrm {GL}}_m$ . The bound does not depend on the generalized Ramanujan conjecture or the non-existence 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.