On the Asymptotic Density of $k$-tuples of Positive Integers Satisfying Arbitrary GCD Conditions
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912367208562688 |
|---|---|
| author | Kuan, Chan Ieong |
| author_facet | Kuan, Chan Ieong |
| contents | We consider the problem of counting $k$-tuples of positive integers satisfying any arbitrary set of gcd conditions, where every integer is not larger than $x$. We first establish the conditions to guarantee the existence of such tuples, and then obtain asymptotic formulae for the count of such tuples with the help of a multivariable Dirichlet series. Part of this work can be viewed as a generalization of Tóth's work, where the conditions are pairwise. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_05630 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Asymptotic Density of $k$-tuples of Positive Integers Satisfying Arbitrary GCD Conditions Kuan, Chan Ieong Number Theory Combinatorics We consider the problem of counting $k$-tuples of positive integers satisfying any arbitrary set of gcd conditions, where every integer is not larger than $x$. We first establish the conditions to guarantee the existence of such tuples, and then obtain asymptotic formulae for the count of such tuples with the help of a multivariable Dirichlet series. Part of this work can be viewed as a generalization of Tóth's work, where the conditions are pairwise. |
| title | On the Asymptotic Density of $k$-tuples of Positive Integers Satisfying Arbitrary GCD Conditions |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2505.05630 |