Article
| Title: | Coprimality of integers in Piatetski-Shapiro sequences (English) |
| Author: | Pimsert, Watcharapon |
| Author: | Srichan, Teerapat |
| Author: | Tangsupphathawat, Pinthira |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 73 |
| Issue: | 1 |
| Year: | 2023 |
| Pages: | 197-212 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We use the estimation of the number of integers $n$ such that $\lfloor n^c \rfloor $ belongs to an arithmetic progression to study the coprimality of integers in $\mathbb {N}^c=\{ \lfloor n^c \rfloor \}_{n\in \mathbb {N}}$, $c>1$, $c\notin \mathbb {N}$. (English) |
| Keyword: | greatest common divisor |
| Keyword: | natural density |
| Keyword: | Piatetski-Shapiro sequence |
| MSC: | 11A05 |
| MSC: | 11K06 |
| idZBL: | Zbl 07655763 |
| idMR: | MR4541097 |
| DOI: | 10.21136/CMJ.2022.0044-22 |
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| Date available: | 2023-02-03T11:12:39Z |
| Last updated: | 2023-09-13 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151512 |
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| Reference: | [2] Cesàro, E.: Étude moyenne di plus grand commun diviseur de deux nombres.Ann. Mat. (2) 13 (1885), 235-250 French \99999JFM99999 17.0144.05. 10.1007/BF02420800 |
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| Reference: | [7] Diaconis, P., Erdős, P.: On the distribution of the greatest common divisor.A Festschrift for Herman Rubin Institute of Mathematical Statistics Lecture Notes - Monograph Series 45. Institute of Mathematical Statistics, Beachwood (2004), 56-61. Zbl 1268.11139, MR 2126886, 10.1214/lnms/1196285379 |
| Reference: | [8] Dirichlet, G. L.: Über die Bestimmung der mittleren Werthe in der Zahlentheorie.Abh. König. Preuss. Akad. Wiss. (1849), 69-83 German. 10.1017/CBO9781139237345.007 |
| Reference: | [9] Fernández, J. L., Fernández, P.: Asymptotic normality and greatest common divisors.Int. J. Number Theory 11 (2015), 89-126. Zbl 1322.11102, MR 3280945, 10.1142/S1793042115500062 |
| Reference: | [10] Hardy, G. H., Wright, E. M.: An Introduction to the Theory of Numbers.Oxford University Press, Oxford (2008). Zbl 1159.11001, MR 2445243 |
| Reference: | [11] Lambek, J., Moser, L.: On integers $n$ relatively prime to $f(n)$.Can. J. Math. 7 (1955), 155-158. Zbl 0064.27903, MR 0068563, 10.4153/CJM-1955-020-0 |
| Reference: | [12] Piatetski-Shapiro, I. I.: On the distribution of the prime numbers in sequences of the form $[f(n)]$.Mat. Sb., N. Ser. 33 (1953), 559-566. Zbl 0053.02702, MR 0059302 |
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