Cancellation in sums over special sequences on $\mathbf{\rm{GL}_{m}}$ and their applications
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915257542246400 |
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| author | Ma, Qiang Zhang, Rui |
| author_facet | Ma, Qiang Zhang, Rui |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_06978 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cancellation in sums over special sequences on $\mathbf{\rm{GL}_{m}}$ and their applications Ma, Qiang Zhang, Rui Number Theory 11L07, 11F66, 11P55 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. |
| title | Cancellation in sums over special sequences on $\mathbf{\rm{GL}_{m}}$ and their applications |
| topic | Number Theory 11L07, 11F66, 11P55 |
| url | https://arxiv.org/abs/2411.06978 |