Recurrent number theoretic functions of several variables (Q1359069)

Some results of \textit{C. Methfessel} [Arch. Math. 63, 321-328 (1994; Zbl 0812.11014)] on recurrent multiplicative arithmetical functions are generalized to the case of several variables. As \(\mathbb{C} [X_1, \dots, X_r]\) for \(r\geq 2\) is not a principal ideal domain, it is necessary to prove some elementary algebraic facts on the calculation of recurrent functions in several variables. The main tool is a vector space duality between \(\mathbb{C} [X_1, \dots, X_r]\) and the space of all functions \(f:\mathbb{N}^r \to\mathbb{C}\). This gives a correspondence between spaces of recurrent functions and ideals in \(\mathbb{C} [X_1, \dots, X_r]\). The practical calculation can be done using Gröbner bases. It is proved that a multiplicative function \(g\) in \(r\) variables which satisfies enough recurrences is of the form \[ g(n_1, \dots, n_r)= n^{l_1}_1 \cdots n^{l_r}_r h(n_1, \dots, n_r), \] with \(h\) multiplicitive and periodic in each variable.