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

Part of: Discontinuous groups and automorphic forms Additive number theory; partitions Exponential sums and character sums

Published online by Cambridge University Press:  20 August 2025

Qiang Ma  and
Rui Zhang
Qiang Ma*
Affiliation:
Institute for Advanced Study in Mathematics, https://ror.org/00a2xv884 Zhejiang University , Hangzhou, Zhejiang 310000, China
Rui Zhang
Affiliation:
https://ror.org/0207yh398 School of Science, Tianjin University of Technology, Tianjin, 300384, China e-mail: rzhang@mail.sdu.edu.cn
*
e-mail: qma829@zju.edu.cn
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Abstract

Let $a(n)$ be the nth 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.

Keywords

Exponential estimatesautomorphic representationsWaring–Goldbach problem

MSC classification

Primary: 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations
Secondary: 11L07: Estimates on exponential sums 11P55: Applications of the Hardy-Littlewood method

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Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 30
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

Q. M. was supported by the New Cornerstone Science Foundation and the National Postdoctoral Program for Innovative Talents (No. GZC20232335). R. Z. was supported in part by NSFC (No. 12031008 and No. 11871307) and the National Key R&D Program of China (No. 2021YFA1000701).

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