Estimates for $k$-dimensional spherical summations of arithmetic functions of the GCD and LCM
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866914627397353472 |
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| author | Heyman, Randell Tóth, László |
| author_facet | Heyman, Randell Tóth, László |
| contents | Let $k\ge 2$ be a fixed integer. We consider sums of type $\sum_{n_1^2+\cdots+ n_k^2\le x} F(n_1,\ldots,n_k)$, taken over the $k$-dimensional spherical region $\{(n_1,\ldots,n_k)\in {\Bbb Z}^k: n_1^2+\cdots+ n_k^2\le x\}$, where $F:{\Bbb Z}^k\to {\Bbb C}$ is a given function. In particular, we deduce asymptotic formulas with remainder terms for the spherical summations $\sum_{n_1^2+\cdots+ n_k^2\le x} f((n_1,\ldots,n_k))$ and $\sum_{n_1^2+\cdots+ n_k^2\le x} f([n_1,\ldots,n_k])$, involving the GCD and LCM of the integers $n_1,\ldots,n_k$, where $f:{\Bbb N}\to {\Bbb C}$ belongs to certain classes of functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_10074 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Estimates for $k$-dimensional spherical summations of arithmetic functions of the GCD and LCM Heyman, Randell Tóth, László Number Theory Combinatorics 11A05, 11A25, 11N37 Let $k\ge 2$ be a fixed integer. We consider sums of type $\sum_{n_1^2+\cdots+ n_k^2\le x} F(n_1,\ldots,n_k)$, taken over the $k$-dimensional spherical region $\{(n_1,\ldots,n_k)\in {\Bbb Z}^k: n_1^2+\cdots+ n_k^2\le x\}$, where $F:{\Bbb Z}^k\to {\Bbb C}$ is a given function. In particular, we deduce asymptotic formulas with remainder terms for the spherical summations $\sum_{n_1^2+\cdots+ n_k^2\le x} f((n_1,\ldots,n_k))$ and $\sum_{n_1^2+\cdots+ n_k^2\le x} f([n_1,\ldots,n_k])$, involving the GCD and LCM of the integers $n_1,\ldots,n_k$, where $f:{\Bbb N}\to {\Bbb C}$ belongs to certain classes of functions. |
| title | Estimates for $k$-dimensional spherical summations of arithmetic functions of the GCD and LCM |
| topic | Number Theory Combinatorics 11A05, 11A25, 11N37 |
| url | https://arxiv.org/abs/2204.10074 |