Some multiple Dirichlet series of completely multiplicative arithmetic functions

Abstract

Let f_r: \mathbb{N}^r \longrightarrow \mathbb{C} be an arithmetic function of r variables, where r\geq 2. We study multiple Dirichlet series defined by \begin{equation*} D(f_r,s_1,\ldots,s_r)=\sum\limits_{\substack{n_1,\ldots,n_r=1 \\ (n_1,\ldots,n_r)=1}}^{+\infty}\frac{f_r(n_1,\ldots,n_r)}{n_1^{s_1}\cdots n_r^{s_r}}, \end{equation*} where f_r(n_1,\ldots,n_r)=f(n_1)\cdots f(n_r) and f is a completely multiplicative or a specially multiplicative arithmetic function of a single variable. We obtain formulas for these series expressed by infinite products over the primes. We also consider the cases of certain particular completely multiplicative and specially multiplicative functions.