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arXiv · 2411.06978

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

Qiang Ma·Rui Zhang
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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.

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Qiang Ma, Rui Zhang. 2024-11-11. Cancellation in sums over special sequences on $\mathbf{{\rm{GL}}_{m}}$ and their applications. https://arxiv.org/abs/2411.06978

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