On mean values of multivariable complex valued multiplicative functions and applications

Abstract

We study the asymptotic behavior of a density function $$ t \to N(f; \mathcal{N}; t) = \sum_{\{\mathbf{m}=(m_1, \dots, m_n) \in \mathbb{N}^n; \mathcal{N}(\mathbf{m}) \leq t\}} f(m_1, \dots, m_n) $$ when $f:\mathbb{N}^n \rightarrow \mathbb{C}$ is a suitable multivariable complex valued multiplicative function, and the family of generalized balls $\{\mathcal{N} \le t\} \cap [0, \infty)^n$ are determined by a positive definite form on $[0, \infty)^n$. Several arithmetic applications of this work are given.