Multiplicative functions with small increment II.

Abstract. We prove that if \(f\) is a multiplicative function satisfying the relations $$ \varlimsup_{n\to\infty}{1\over{\log x}}\sum_{n\le x}{\vert f(n)\vert\over n}=\infty,\quad \varlimsup_{n\to\infty}{1\over{\log x}}\sum_{n\le x} {\vert f(n+K)-f(n)\vert\over n} 0<\infty, $$ then there are real numbers \(\sigma,t\) with \((0< \sigma\le 1)\) and a Dirichlet character \(\chi \pmod K\) such that \(f(n)=n^{\sigma+it}\chi(n)\).