Multiplicative Arithmetic Functions of Several Variables: A Survey

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.

Estimates for $k$-dimensional spherical summations of arithmetic functions of the GCD and LCM

“…, which is the inverse of the k-variable constant 1 function under convolution (3.1). See the survey [20] on properties of (multiplicative) arithmetic functions of several variables. Our first result is the following.…”

“…, n k ≤ x have been given by Essouabri et al [6], de la Bretèche [7], Ushiroya [22]. Also see the survey paper by the second author [20]. Results for sums of type (1.1), more generally for sums over n 1 , .…”

Expansions of arithmetic functions of several variables with respect to certain modified unitary Ramanujan sums

“…Similar to the one dimensional case, the unitary convolution (2.6) preserves the multiplicativity of functions. See our paper [14], which is a survey on (multiplicative) arithmetic functions of several variables.…”

The multiple Dirichlet product and the multiple Dirichlet series

“…The properties were studied by some mathematicians. In [10], Tóth described the details of such studies. In particular, if we define an addition as (f + g)(n) := f (n) + g(n), he mentioned that (Ω k , +, * ) is an integral domain with the identity function I which is defined by…”