Cancellation in sums over special sequences on ${\mathrm{G}\mathrm{L}}_{\mathbf{m}}$ and their applications

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 ${\mathrm{G}\mathrm{L}}_m\mathrm{.}$ 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.