On the distribution of k-full lattice points in Z<sup>2</sup>
Let $ \mathbb{Z}^2 $ be the two-dimensional integer lattice. For an integer $ k\geq 2 $, we say a non-zero lattice point in $ \mathbb{Z}^2 $ is $ k $-full if the greatest common divisor of its coordinates is a $ k $-full number. In this paper, we first prove that the density of $ k $-full lattice po...
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| Format: | Article |
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AIMS Press
2022-03-01
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| Series: | AIMS Mathematics |
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| Online Access: | http://www.aimspress.com/article/doi/10.3934/math.2022591?viewType=HTML |
| _version_ | 1811272479176392704 |
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| author | Shunqi Ma |
| author_facet | Shunqi Ma |
| author_sort | Shunqi Ma |
| collection | DOAJ |
| description | Let $ \mathbb{Z}^2 $ be the two-dimensional integer lattice. For an integer $ k\geq 2 $, we say a non-zero lattice point in $ \mathbb{Z}^2 $ is $ k $-full if the greatest common divisor of its coordinates is a $ k $-full number. In this paper, we first prove that the density of $ k $-full lattice points in $ \mathbb{Z}^2 $ is $ c_k = \prod_{p}(1-p^{-2}+p^{-2k}) $, where the product runs over all primes. Then we show that the density of $ k $-full lattice points on a path of an $ \alpha $-random walk in $ \mathbb{Z}^2 $ is almost surely $ c_k $, which is independent on $ \alpha $.
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| first_indexed | 2024-04-12T22:41:02Z |
| format | Article |
| id | doaj.art-f690a65cabf14e58aa462834829a39bc |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-04-12T22:41:02Z |
| publishDate | 2022-03-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-f690a65cabf14e58aa462834829a39bc2022-12-22T03:13:43ZengAIMS PressAIMS Mathematics2473-69882022-03-0176105961060810.3934/math.2022591On the distribution of k-full lattice points in Z<sup>2</sup>Shunqi Ma 0School of Mathematics and Statistics, Qingdao University, 308 Ningxia Road, Shinan District, Qingdao, Shandong, ChinaLet $ \mathbb{Z}^2 $ be the two-dimensional integer lattice. For an integer $ k\geq 2 $, we say a non-zero lattice point in $ \mathbb{Z}^2 $ is $ k $-full if the greatest common divisor of its coordinates is a $ k $-full number. In this paper, we first prove that the density of $ k $-full lattice points in $ \mathbb{Z}^2 $ is $ c_k = \prod_{p}(1-p^{-2}+p^{-2k}) $, where the product runs over all primes. Then we show that the density of $ k $-full lattice points on a path of an $ \alpha $-random walk in $ \mathbb{Z}^2 $ is almost surely $ c_k $, which is independent on $ \alpha $. http://www.aimspress.com/article/doi/10.3934/math.2022591?viewType=HTMLk-full lattice pointsk-full numberdensityrandom walktwo-dimensional integer lattice |
| spellingShingle | Shunqi Ma On the distribution of k-full lattice points in Z<sup>2</sup> AIMS Mathematics k-full lattice points k-full number density random walk two-dimensional integer lattice |
| title | On the distribution of k-full lattice points in Z<sup>2</sup> |
| title_full | On the distribution of k-full lattice points in Z<sup>2</sup> |
| title_fullStr | On the distribution of k-full lattice points in Z<sup>2</sup> |
| title_full_unstemmed | On the distribution of k-full lattice points in Z<sup>2</sup> |
| title_short | On the distribution of k-full lattice points in Z<sup>2</sup> |
| title_sort | on the distribution of k full lattice points in z sup 2 sup |
| topic | k-full lattice points k-full number density random walk two-dimensional integer lattice |
| url | http://www.aimspress.com/article/doi/10.3934/math.2022591?viewType=HTML |
| work_keys_str_mv | AT shunqima onthedistributionofkfulllatticepointsinzsup2sup |