Abstract
Some results of C. Methfessel (Multiplicative and additive recurrent sequences, Arch. Math. 63, 321–328 (1994)) on recurrent multiplicative arithmetical functions are generalized to the case of several variables. As ℂ[X 1,…, X r ] for r ≧ 2 is no 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 ℂ[X 1,…, X r ] and the space of all functions f : ℕr → ℂ. This gives a correspondence between spaces of recurrent functions and ideals in ℂ[X 1,…, X r ]. The practical calculation can be done using Groebner bases. It is proved that a multiplicative function g in r variables which satisfies enough recurrences is of the form g (n 1…, n r )=n 1 … n h (n 1 …, n r ), with h multiplicative and periodic in each variable.
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