Topic
Dirichlet convolution
About: Dirichlet convolution is a research topic. Over the lifetime, 78 publications have been published within this topic receiving 701 citations.
Papers published on a yearly basis
Papers
TL;DR: In this paper, general properties of multiplicative arithmetic functions of several variables and related convolutions, including the Dirichlet convolution and the unitary convolution, are surveyed.
Abstract: We survey general properties of multiplicative arithmetic functions of several variables and related convolutions, including the Dirichlet convolution and the unitary convolution. We introduce and investigate a new convolution, called gcd convolution. We define and study the convolutes of arithmetic functions of several variables, according to the different types of convolutions. We discuss the multiple Dirichlet series and Bell series and present certain arithmetic and asymptotic results of some special multiplicative functions arising from problems in number theory, group theory, and combinatorics. We give a new proof to obtain the asymptotic density of the set of ordered r-tuples of positive integers with pairwise relatively prime components and consider a similar question related to unitary divisors.
60 citations
35 citations
TL;DR: The Dirichlet convolution of two arithmetic functions f and g is defined by for all n∈Z+ for all positive integers m and n as discussed by the authors, where m, n denotes the greatest common divisor.
Abstract: An arithmetic function f is said to be multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1, where (m, n) denotes as usual the greatest common divisor of m and n. Furthermore an arithmetic function is said to be linear (or completely multiplicative) if f(1) = 1 and f(mn) = f(m)f(n) for all positive integers m and n.The Dirichlet convolution of two arithmetic functions f and g is defined by for all n∈Z+. Recall that the set of all multiplicative functions, denoted by M, with this operation is an abelian group.
24 citations
Journal Article•
TL;DR: In this paper, an asymptotic formula for the average order of the gcd-sum function was derived by using a new convolution identity, which was later used to prove the same result.
Abstract: We prove an asymptotic formula for the average order of the gcd-sum function by using a new convolution identity.
22 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2022 | 1 |
| 2021 | 4 |
| 2020 | 6 |
| 2019 | 4 |
| 2018 | 5 |
| 2017 | 3 |