Sur l'exemple d'Euler d'une fonction complètement multiplicative à somme nulle

Abstract

Euler published a formula that now reads ∑1∞λ(n)n=0, λ being the completely multiplicative function equal to −1 on the prime numbers. Thus (λ(n)n) is an example of a CMO function (completely multiplicative with sum 0). We extend this formula by considering λ as defined on Beurling's generalized prime numbers and integers, according to Diamond's condition on generalized primes, which implies a regular distribution of the generalized integers (théorème 3). As an application, we show how to contruct a CMO function carried by a set of integers whose counting function is of the form Dxa(1+o(1))(x→∞), for any given a between 0 and 1 (théorème 1).