On some generalizations of mean value theorems for arithmetic functions of two variables

TL;DR

This paper investigates the asymptotic behavior of sums of two-variable arithmetic functions, establishing conditions for the existence of limits and expressing them as infinite products for multiplicative functions, extending classical mean value theorems.

Contribution

It generalizes mean value theorems for arithmetic functions to two variables and provides explicit limit formulas as infinite products for multiplicative functions.

Findings

Established the existence of the limit for sums of two-variable functions.

Derived an explicit infinite product formula for the limit when the function is multiplicative.

Extended classical results to the two-variable case.

Abstract

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.