On moments of the error term of the multivariable k-th divisor functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912113296932864 |
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| author | Guo, Zhen Li, Xin |
| author_facet | Guo, Zhen Li, Xin |
| contents | Suppose $k\geqslant3$ is an integer. Let $τ_k(n)$ be the number of ways $n$ can be written as a product of $k$ fixed factors. For any fixed integer $r\geqslant2$, we have the asymptotic formula \begin{equation*}
\sum_{n_1,\cdots,n_r\leqslant x}τ_k(n_1 \cdots n_r)=x^r\sum_{\ell=0}^{r(k-1)}d_{r,k,\ell}(\log x)^{\ell}+O(x^{r-1+α_k+\varepsilon}), \end{equation*} where $d_{r,k,\ell}$ and $0<α_k<1$ are computable constants. In this paper we study the mean square of $Δ_{r,k}(x)$ and give upper bounds for $k\geqslant4$ and an asymptotic formula for the mean square of $Δ_{r,3}(x)$. We also get an upper bound for the third power moment of $Δ_{r,3}(x)$. Moreover, we study the first power moment of $Δ_{r,3}(x)$ and then give a result for the sign changes of it. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_06656 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On moments of the error term of the multivariable k-th divisor functions Guo, Zhen Li, Xin Number Theory Suppose $k\geqslant3$ is an integer. Let $τ_k(n)$ be the number of ways $n$ can be written as a product of $k$ fixed factors. For any fixed integer $r\geqslant2$, we have the asymptotic formula \begin{equation*} \sum_{n_1,\cdots,n_r\leqslant x}τ_k(n_1 \cdots n_r)=x^r\sum_{\ell=0}^{r(k-1)}d_{r,k,\ell}(\log x)^{\ell}+O(x^{r-1+α_k+\varepsilon}), \end{equation*} where $d_{r,k,\ell}$ and $0<α_k<1$ are computable constants. In this paper we study the mean square of $Δ_{r,k}(x)$ and give upper bounds for $k\geqslant4$ and an asymptotic formula for the mean square of $Δ_{r,3}(x)$. We also get an upper bound for the third power moment of $Δ_{r,3}(x)$. Moreover, we study the first power moment of $Δ_{r,3}(x)$ and then give a result for the sign changes of it. |
| title | On moments of the error term of the multivariable k-th divisor functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2411.06656 |