Fonctions complètement multiplicatives de somme nulle

Let f be a function from N∗ to C not identically 0. We call it CMO if f(ab)=f(a)f(b) for all a and b and ∑n=1+∞f(n)=0. We give properties and examples of CMO functions. A basic example goes back to Euler, namely f(p)=−1∕p for every prime number p. We study how far from this example the CMO character is kept. The zeroes of Dirichlet series are implied in this study, as well as in other examples. The relation between CMO and the generalized Riemann hypothesis is pointed out at the end of the article.