Fonctions compl\`etement multiplicatives de somme nulle

TL;DR

This paper characterizes completely multiplicative functions with sum zero, providing necessary and sufficient conditions, especially for bounded functions, and explores their invariance and specific cases like Euler's example.

Contribution

It offers new criteria for identifying completely multiplicative functions with zero sum, including invariance properties and detailed conditions for bounded functions.

Findings

Conditions on prime values for CMO functions are established.

Necessary and sufficient criteria for bounded functions are provided.

Invariance of CMO property under small modifications is demonstrated.

Abstract

Completely multiplicative functions whose sum is zero ($CMO$).The paper deals with $CMO$, meaning completely multiplicative ($CM$) functions $f$ such that $f(1)=1$ and $\sum\limits\_1^\infty f(n)=0$. $CM$ means $f(ab)=f(a)f(b)$ for all $(a,b)\in \N^{*2}$, therefore $f$ is well defined by the $f(p)$, $p$ prime. Assuming that $f$ is $CM$, give conditions on the $f(p)$, either necessary or sufficient, both is possible, for $f$ being $CMO$ : that is the general purpose of the authors.The $CMO$ character of $f$ is invariant under slight modifications of the sequence $(f(p))$ (theorem 3). The same idea applies also in a more general context (theorem 4).After general statements of that sort, including examples of $CMO$ (theorem 5), the paper is devoted to "small" functions, that is, functions of the form $\frac{f(n)}{n}$, where the $f(n)$ are bounded. Here is a typical result : if $|f(p)|\le 1$ and $Re\, f(p)\le0$ for all $p$, a necessary and sufficient condition for $\big(\frac{f(n)}{n}\big)$ to be $CMO$ is $\sum \, Re\, f(p)/p=-\infty$ (theorem 8). Another necessary and sufficient condition is given under the assumption that $|1+f(p)|\le 1$ and $f(2)\not=-2$ (theorem 7). A third result gives only a sufficient condition (theorem 9). The three results apply to the particular case $f(p)=-1$, the historical example of Euler.Theorems 7 and 8 need auxiliary results, coming either from the existing literature (Hal\'asz, Montgomery--Vaughan), or from improved versions of classical results (Ingham, Ska{\l} ba) about $f(n)$ under assumptions on the $f*1(n)$, * denoting the multiplicative convolution (theorems 10 and 11).