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.

References in corpus (1)

• Notes de lecture de l'article "Partial sums of the Möbius function" de Kannan Soundararajan