Multiplicative functions with sum zero

TL;DR

This paper introduces and studies a new class of multiplicative functions called MO functions, which generalize CMO functions with sum zero, exploring their properties, examples, and connections to the Riemann hypothesis.

Contribution

It generalizes CMO functions to MO functions, providing properties, examples, and linking them to the Riemann hypothesis.

Findings

Defined MO functions as a generalization of CMO functions.

Provided properties and examples of MO functions.

Connected MO functions to the Riemann hypothesis.

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 Sa\"{i}as [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.