Sur certaines équations fonctionnelles arithmétiques

Abstract

Let p k denote the kth prime number. A completely additive arithmetical function is defined on ℕ * by the values f(p k ) and the formula f(n)=∑ k≥1 f(p k )v p k (n) (n≥1), where v p stands for the p-adic valuation. We study a class ℰ of completely additive functions characterised by an approximate functional equation linking f(p k ) to f(k). The logarithm is a maximal element of ℰ, but a typical element is the Gutman–Ivić–Matula function, defined by the formula