Rekurrente zahlentheoretische Funktionen in mehreren Variablen

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.