On a mean value of a multiplicative function of two variables

Abstract

We prove the existence of a mean value of arithmetical functions of two variables : $\lim_{x, y \to \infty} (xy)^{-1} \sum_{m \le x, n \le y} f(m, n)$ under some conditions, and, when $f$ is a multiplicative function of two variables, express the mean value as an infinite product over all primes. Five examples are given which are not obtained by trivial generalizations of results on arithmetical functions of one variable.