Multiplicative functions with sum zero

Abstract

$CMO$ functions are completely multiplicative functions $f$ for which $\sum_{n=1}^\infty f(n)$ $=0$. These functions were first introduced and studied by Kahane and Saïas [5]. The main purpose of this paper is to generalise such functions to multiplicative functions and we shall call them $MO$ functions. More precisely, we define $MO$ functions to be multiplicative functions for which $\sum_{n=1}^\infty f(n) = 0$ and $\sum_{k=0}^\infty f(p^k)$ $\ne 0$ for all $p \in \mathbb{P}$. We give some properties and find examples of $MO$ functions, as well as pointing out the connection between these functions and the Riemann hypothesis at the end of the paper.