Local Fourier uniformity of higher divisor functions on average
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909245065134080 |
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| author | Wang, Mengdi |
| author_facet | Wang, Mengdi |
| contents | Let $τ_k$ be the $k$-fold divisor function. By constructing an approximant of $τ_k$, denoted as $τ_k^*$, which is a normalized truncation of the $k$-fold divisor function, we prove that when $\exp\left(C\log^{1/2}X(\log\log X)^{1/2}\right)\leq H\leq X$ and $C>0$ is sufficiently large, the following estimate holds for almost all $x\in[X,2X]$: \[ \Big|\sum_{x<n\leq x+H}(τ_k(n)-τ_k^*(n)) e(α_dn^d+\cdots+α_1n)\Big|=o(H\log^{k-1}X), \] where $α_1, \dots, α_d\in \mathbb{R}$ are arbitrary frequencies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_18342 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local Fourier uniformity of higher divisor functions on average Wang, Mengdi Number Theory Combinatorics Let $τ_k$ be the $k$-fold divisor function. By constructing an approximant of $τ_k$, denoted as $τ_k^*$, which is a normalized truncation of the $k$-fold divisor function, we prove that when $\exp\left(C\log^{1/2}X(\log\log X)^{1/2}\right)\leq H\leq X$ and $C>0$ is sufficiently large, the following estimate holds for almost all $x\in[X,2X]$: \[ \Big|\sum_{x<n\leq x+H}(τ_k(n)-τ_k^*(n)) e(α_dn^d+\cdots+α_1n)\Big|=o(H\log^{k-1}X), \] where $α_1, \dots, α_d\in \mathbb{R}$ are arbitrary frequencies. |
| title | Local Fourier uniformity of higher divisor functions on average |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2402.18342 |