Multiplicative Arithmetic Functions of Several Variables: A Survey

  title={Multiplicative Arithmetic Functions of Several Variables: A Survey},
  author={L{\'a}szl{\'o} Fejes T{\'o}th},
  journal={arXiv: Number Theory},
  • L. Tóth
  • Published 2014
  • Mathematics
  • arXiv: Number Theory
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… Expand
Ramanujan-Fourier series of certain arithmetic functions of two variables
We study Ramanujan-Fourier series of certain arithmetic functions of two variables. We generalize Delange's theorem to the case of arithmetic functions of two variables and give sufficient conditionsExpand
Expansions of arithmetic functions of several variables with respect to certain modified unitary Ramanujan sums
We introduce new analogues of the Ramanujan sums, denoted by $\widetilde{c}_q(n)$, associated with unitary divisors, and obtain results concerning the expansions of arithmetic functions of severalExpand
Derivation of arithmetical functions under the Dirichlet convolution
We present the group-theoretic structure of the classes of multiplicative and firmly multiplicative arithmetical functions of several variables under the Dirichlet convolution, and we giveExpand
Extended arithmetic functions
In this paper, we give an attempt to extend some arithmetic properties such as multiplicativity and convolution products to the setting of operator theory and we provide significant examples whichExpand
Formal power series in several variables
This work refers to a formal power series in n variables as an n-way array of complex or real numbers and investigates its algebraic properties without analytic tools. Expand
A non-commutative multiple Dirichlet power product and an application
Abstract We define a multiple Dirichlet r-product of ideal functions as one of the generalizations of the well-known Dirichlet product of arithmetic functions. This product is not commutative butExpand
Ramanujan expansions of arithmetic functions of several variables
We generalize certain recent results of Ushiroya concerning Ramanujan expansions of arithmetic functions of two variables. We also show that some properties on expansions of arithmetic functions ofExpand
An identical equation for arithmetic functions of several variables and applications
Abstract: An identical equation for arithmetic functions is proved generalizing the 2-variable case due to Venkataraman. It is then applied to characterize multiplicative functions which areExpand
On some generalizations of mean value theorems for arithmetic functions of two variables
Let $f: \mathbb{N}^2 \mapsto \mathbb{C}$ be an arithmetic function of two variables. We study the existence of the limit: \[\displaystyle \lim_{x \to \infty} \frac{1}{x^2 (\log x)^{k-1}} \sum_{n_1 ,Expand
On multivariable averages of divisor functions
Abstract We deduce asymptotic formulas for the sums ∑ n 1 , … , n r ≤ x f ( n 1 ⋯ n r ) and ∑ n 1 , … , n r ≤ x f ( [ n 1 , … , n r ] ) , where r ≥ 2 is a fixed integer, [ n 1 , … , n r ] stands forExpand


We study (A; +; ), the ring of arithmetical functions with uni- tary convolution, giving an isomorphism between (A; +; ) and a generalized power series ring on innitely many variables, similar to theExpand
Mean-Value Theorems for Multiplicative Arithmetic Functions of Several Variables
The Wintner theorem is generalized and the multiplicative case is considered by expressing the mean-value as an infinite product over all prime numbers by the Riemann zeta function. Expand
Arithmetical functions : an introduction to elementary and analytic properties of arithmetic functions and to some of their almost-periodic properties
Preface Acknowledgements Notation 1. Tools from number theory Photographs 2. Mean-value theorems and multiplicative functions, I 3. Related arithmetical functions 4. Uniformly almost-periodicExpand
A Multivariate Arithmetic Function of Combinatorial and Topological Significance
It is shown that the necessary and sufficient conditions for this function to vanish are equivalent to familiar Harvey's conditions that characterize possible branching data of finite cyclic automorphism groups of Riemann surfaces. Expand
Arithmetic of double series
Introduction. Two theories of numerical functions have received much attention. The first has for basis the Cauchy multiplication of power series and is appropriately used in considering functionsExpand
Groups of Arithmetical Functions
Here the sum is taken over all positive integer divisors d of n. This somewhat surprising choice of a product is quite fruitful, allowing one to obtain interesting numbertheoretic formulas fromExpand
Menon's identity and arithmetical sums representing functions of several variables
We generalize Menon's identity by considering sums representing arithmetical functions of several variables. As an application, we give a formula for the number of cyclic subgroups of the directExpand
On some arithmetic convolutions
0. Introduction. In this paper we first review some of the known arithmetical convolutions with particular reference to a class of convolutions which may be called Lehmer's ~-products. These productsExpand
The paper is devoted to the study of some properties of generalized arithmetical functions extended to the case of three variables. The convolution in this case is a convolution of the incidenceExpand
Introduction to Arithmetical Functions
1. Multiplicative Functions.- 2. Ramanujan Sums.- 3. Counting Solutions of Congruences.- 4. Generalizations of Dirichlet Convolution.- 5. Dirichlet Series and Generating Functions.- 6. AsymptoticExpand