Exponential sums twisted by general arithmetic functions
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866910766197637120 |
|---|---|
| author | Dong, Anji Robles, Nicolas Zaharescu, Alexandru Zeindler, Dirk |
| author_facet | Dong, Anji Robles, Nicolas Zaharescu, Alexandru Zeindler, Dirk |
| contents | We examine exponential sums of the form $\sum_{n \le X} w(n) e^{2πiαn^k}$, for $k=1,2$, where $α$ satisfies a generalized Diophantine approximation and where $w$ are different arithmetic functions that might be multiplicative, additive, or neither. A strategy is shown on how to bound these sums for a wide class of functions $w$ belonging within the same ecosystem. Using this new technology we are able to improve current results on minor arcs that have recently appeared in the literature of the Hardy-Littlewood circle method. Lastly, we show how a bound on $\sum_{n \le X} |μ(n)| e^{2πiαn}$ can be used to study partitions asymptotics over squarefree parts and explain their connection to the zeros of the Riemann zeta-function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20101 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exponential sums twisted by general arithmetic functions Dong, Anji Robles, Nicolas Zaharescu, Alexandru Zeindler, Dirk Number Theory Primary: 11L03, 11L07, 11L20. Secondary: 11P55, 11P82 We examine exponential sums of the form $\sum_{n \le X} w(n) e^{2πiαn^k}$, for $k=1,2$, where $α$ satisfies a generalized Diophantine approximation and where $w$ are different arithmetic functions that might be multiplicative, additive, or neither. A strategy is shown on how to bound these sums for a wide class of functions $w$ belonging within the same ecosystem. Using this new technology we are able to improve current results on minor arcs that have recently appeared in the literature of the Hardy-Littlewood circle method. Lastly, we show how a bound on $\sum_{n \le X} |μ(n)| e^{2πiαn}$ can be used to study partitions asymptotics over squarefree parts and explain their connection to the zeros of the Riemann zeta-function. |
| title | Exponential sums twisted by general arithmetic functions |
| topic | Number Theory Primary: 11L03, 11L07, 11L20. Secondary: 11P55, 11P82 |
| url | https://arxiv.org/abs/2412.20101 |