Article
| Title: | On certain sums of arithmetic functions involving the greatest common divisor (English) |
| Author: | Intarawong, Traiwat |
| Author: | Srichan, Teerapat |
| Author: | Yuttanan, Boonrod |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 76 |
| Issue: | 1 |
| Year: | 2026 |
| Pages: | 221-229 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | Let $x>1$ and $\alpha $, $\beta $ be positive integers such that $1< \alpha <\beta $. We consider sums of type $\sum _{m^\alpha n^\beta \leq x}f(\gcd (m, n)),$ taken over the region $\{ (m, n)\in \mathbb {N}^2\colon m^\alpha n^\beta \leq x\}$, where $f$ belongs to certain classes of arithmetic functions and $\gcd (m, n)$ denotes the greatest common divisor of the integers $m$, $n$. (English) |
| Keyword: | arithmetic function |
| Keyword: | greatest common divisor |
| MSC: | 11A05 |
| MSC: | 11A25 |
| MSC: | 11N37 |
| DOI: | 10.21136/CMJ.2026.0222-25 |
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| Date available: | 2026-03-13T09:32:38Z |
| Last updated: | 2026-03-16 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153569 |
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| Reference: | [1] Heyman, R.: A summation involving the divisor and GCD functions.J. Integer Seq. 23 (2000), Article ID 20.9.8, 6 pages. Zbl 1446.11182, MR 4167938 |
| Reference: | [2] Heyman, R., Tóth, L.: On certain sums of arithmetic functions involving the GCD and LCM of two positive integers.Result. Math. 76 (2021), Article ID 49, 23 pages. Zbl 1471.11007, MR 4221669, 10.1007/s00025-020-01337-7 |
| Reference: | [3] Heyman, R., Tóth, L.: Estimates for $k$-dimensional spherical summations of arithmetic functions of the GCD and LCM.Number Theory in Memory of Eduard Wirsing Springer, Cham (2023), 157-183. Zbl 1560.11143, MR 4647432, 10.1007/978-3-031-31617-3_11 |
| Reference: | [4] Heyman, R., Tóth, L.: Hyperbolic summation for functions of the GCD and LCM of several integers.Ramanujan J. 62 (2023), 273-290. Zbl 1523.11001, MR 4632218, 10.1007/s11139-022-00681-2 |
| Reference: | [5] Kiuchi, I., Eddin, S. S.: On sums of arithmetic functions involving the greatest common divisor.Available at https://arxiv.org/abs/2102.03714v1 (2021), 13 pages. 10.48550/arXiv.2102.03714 |
| Reference: | [6] Landau, E.: Au sujet d'une certaine expression asymptotique.Interméd. des math. 20 (1913), 155 French \99999JFM99999 44.0208.01 . |
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