On multiplicative functions which are small on average and zero free regions for the Riemann zeta function

Abstract

In this short note we prove the following result: If a completely multiplicative function $f:\mathbb{N}\to[-1,1]$ is small on average in the sense that $\sum_{n\leq x}f(n)\ll x^{1-δ}$, for some $δ>0$, and if the Dirichlet series of $f$, say $F(s)$, is such that $F(1)=0$, then we obtain that for any $ε>0$, $\sum_{p\leq x}(1+f(p))\log p\ll x^{1-δ+ε}$. Moreover, a necessary condition for the existence of such $f$ is that the Riemann zeta function $ζ(s)$ has no zeros in the half plane $Re(s)>1-δ$.