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: $$\displaystyle \lim_{x \to \infty} \frac{1}{x^2 (\log x)^{k-1}} \sum_{n_1 , n_2 \le x} f (n_1, n_2)$$ where $k$ is a fixed positive integer. Moreover, we express this limit as an infinite product over all prime numbers in the case that $f$ is a multiplicative function of two variables. This study is a generalization of Cohen-van der Corput's results to the case of two variables.