Multiplicative functions with small increment I.

Abstract. We prove that if \(f\) is a completely multiplicative function and $$ \sum_{n\le x}{{\vert f(n+1)-f(n)\vert}\over n} =O(\log x), $$ then either $$ \sum_{n\le x}{{\vert f(n)\vert}\over n} =O(\log x)\quad\text{or}\quad f(n)=n^{\sigma+it}\quad 0<\sigma\le 1, t\in{\mathbb R} $$