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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.07132 |
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| _version_ | 1866908640800145408 |
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| author | Cai, Yi Li, Jinjiang Sui, Yankun Xue, Fei Zhang, Min |
| author_facet | Cai, Yi Li, Jinjiang Sui, Yankun Xue, Fei Zhang, Min |
| contents | Let $-1/2<a<0$ be a fixed real number and
\begin{equation*} Δ_{a}(x)=\sideset{}{'}\sum_{n\leq x} σ_a(n)-ζ(1-a)x-\frac{ζ(1+a)}{1+a}x^{1+a}+\frac{1}{2}ζ(-a). \end{equation*} In this paper, we investigate the higher--power moments of $Δ_a(x)$ and give the corresponding asymptotic formula for the integral $\int_{1}^{T}Δ_a^k(x)\mathrm{d}x$, which constitutes an improvement upon the previous result of Zhai [9] for $k=3,4,5$ and an enlargement of the upper bound of $k$ to $7$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_07132 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Higher-Power Moments of $ Δ_a(x) $ for $-1/2<a<0$ Cai, Yi Li, Jinjiang Sui, Yankun Xue, Fei Zhang, Min Number Theory Let $-1/2<a<0$ be a fixed real number and \begin{equation*} Δ_{a}(x)=\sideset{}{'}\sum_{n\leq x} σ_a(n)-ζ(1-a)x-\frac{ζ(1+a)}{1+a}x^{1+a}+\frac{1}{2}ζ(-a). \end{equation*} In this paper, we investigate the higher--power moments of $Δ_a(x)$ and give the corresponding asymptotic formula for the integral $\int_{1}^{T}Δ_a^k(x)\mathrm{d}x$, which constitutes an improvement upon the previous result of Zhai [9] for $k=3,4,5$ and an enlargement of the upper bound of $k$ to $7$. |
| title | On Higher-Power Moments of $ Δ_a(x) $ for $-1/2<a<0$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2511.07132 |