An $L^2$-bound for the Barban-Vehov weights
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911314834620416 |
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| author | Ramaré, Olivier Alterman, Sebastian Zuniga |
| author_facet | Ramaré, Olivier Alterman, Sebastian Zuniga |
| contents | Let $λ$ the Barban--Vehov weights, defined in $(1)$. Let $X\ge z_1\ge100$ and $z_2=z_1^τ$ for some $τ>1$. We prove that
\begin{equation*}
\sum_{n\le
X}\frac{1}{n}\Bigl(\sum_{\substack{d|n}}λ_d\Bigr)^2
\le
f(τ)\frac{\log X}{\log (z_2/z_1)},
\end{equation*}
for a completely determined function $f:(1,\infty)\to\mathbb{R}_{>0}$.
In particular, we may take $f(2)=30$, saving more than a factor of $5$ on what was the best known result for $τ=2$. Two related estimates are also provided for general $τ>1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_12662 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An $L^2$-bound for the Barban-Vehov weights Ramaré, Olivier Alterman, Sebastian Zuniga Number Theory 11A25, 11N36 Let $λ$ the Barban--Vehov weights, defined in $(1)$. Let $X\ge z_1\ge100$ and $z_2=z_1^τ$ for some $τ>1$. We prove that \begin{equation*} \sum_{n\le X}\frac{1}{n}\Bigl(\sum_{\substack{d|n}}λ_d\Bigr)^2 \le f(τ)\frac{\log X}{\log (z_2/z_1)}, \end{equation*} for a completely determined function $f:(1,\infty)\to\mathbb{R}_{>0}$. In particular, we may take $f(2)=30$, saving more than a factor of $5$ on what was the best known result for $τ=2$. Two related estimates are also provided for general $τ>1$. |
| title | An $L^2$-bound for the Barban-Vehov weights |
| topic | Number Theory 11A25, 11N36 |
| url | https://arxiv.org/abs/2405.12662 |