On a class of arithmetic convolutions involving arbitrary sets of integers
| dc.creator | Tóth, László | |
| dc.date | 2006-10-19 | |
| dc.date.accessioned | 2026-07-07T07:29:13Z | |
| dc.date.available | 2026-07-07T07:29:13Z | |
| dc.description | Let $d,n$ be positive integers and $S$ be an arbitrary set of positive integers. We say that $d$ is an $S$-divisor of $n$ if $d|n$ and gcd $(d,n/d)\in S$. Consider the $S$-convolution of arithmetical functions given by (1.1), where the sum is extended over the $S$-divisors of $n$. We determine the sets $S$ such that the $S$-convolution is associative and preserves the multiplicativity of functions, respectively, and discuss other basic properties of it. We give asymptotic formulae with error terms for the functions $σ_S(n)$ and $τ_S(n)$, representing the sum and the number of $S$-divisors of $n$, respectively, for an arbitrary $S$. We improve the remainder terms of these formulae and find the maximal orders of $σ_S(n)$ and $τ_S(n)$ assuming additional properties of $S$. These results generalize, unify and sharpen previous ones. We also pose some problems concerning these topics. | |
| dc.identifier | https://arxiv.org/abs/math/0610581 | |
| dc.identifier | http://arxiv.org/abs/math/0610581 | |
| dc.identifier | Math. Pannonica, 13 (2002), 249-263 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/118018 | |
| dc.subject | Number Theory | |
| dc.subject | 11A25, 11N37 | |
| dc.title | On a class of arithmetic convolutions involving arbitrary sets of integers | |
| dc.type | text |