| Title:
|
Hyperbolic summation involving the function $\Omega (n) $ and LCM (English) |
| Author:
|
Karras, Meselem |
| Author:
|
Bouderbala, Mihoub |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
61 |
| Issue:
|
5 |
| Year:
|
2025 |
| Pages:
|
167-173 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study the sum $\sum \limits _{abc\le x}\Omega \left( \left[ a,b,c\right] \right) $, where $\Omega (n)$ denotes the number of distinct prime divisors of $n\in \mathbb{Z}_{\ge 1}$ counted with multiplicity, and $\left[ a,b,c\right] =\operatorname{lcm}\left( a,b,c\right) $. An asymptotic formula is derived for this sum over the hyperbolic region $\left\rbrace \left( a,b,c\right) \in \mathbb{Z}_{\ge 1}^{3},\ abc\le x\right\lbrace $. (English) |
| Keyword:
|
prime divisors |
| Keyword:
|
hyperbolic summation |
| Keyword:
|
integer part |
| MSC:
|
11A05 |
| MSC:
|
11A25 |
| MSC:
|
11N37 |
| idZBL:
|
Zbl 08162262 |
| DOI:
|
10.5817/AM2025-5-167 |
| . |
| Date available:
|
2026-01-23T10:39:38Z |
| Last updated:
|
2026-03-16 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153280 |
| . |
| Reference:
|
[1] Heyman, R., Tóth, L.: On Certain Sums of Arithmetic Functions Involving the GCD and LCM of Two Positive Integers.Results Math. 76 (1) (2021), 22 pp., Paper No. 49. MR 4221669 |
| Reference:
|
[2] Heyman, R., Tóth, L.: Hyperbolic summation for functions of the GCD and LCM of several integers.Ramanujan J. 62 (1) (2022), 1–18. MR 4632218 |
| Reference:
|
[3] Iksanov, A., Marynych, A., Raschel, K.: Asymptotics of arithmetic functions of GCD and LCM of random integers in hyperbolic regions.preprint, 2021, arXiv: 2112.11892v1 [math.NT]. |
| Reference:
|
[4] Ivic, A.: Sums of products of certain arithmetical functions.Publ. Inst. Math. (Beograd) (N.S.) 41 (55) (1987), 31–41. |
| Reference:
|
[5] Krätzel, E., Nowak, W.G., Tóth, L.: On certain arithmetic functions involving the greatest common divisor.Cent. Eur. J. Math. 10 (2012), 761–774. MR 2886571 |
| Reference:
|
[6] Nathanson, M.B.: Elementary Methods in Number Theory.Grad. Texts in Math., vol. 195, Springer, 2000. Zbl 0953.11002 |
| . |
