On the multiplicative group generated by \Big\{{[\sqrt {2}n]\over n}~\mid~n\in\mathbb{N} \Big\}. V

Abstract

Let $f,g$ be completely multiplicative functions, $\vert f(n)\vert=\vert g(n)\vert =1 (n\in\mathbb{N})$. Assume that $${1\over {\log x}}\sum_{n\le x}{\vert g([\sqrt{2}n])-Cf(n)\vert\over n}\to 0 \quad (x\to\infty).$$ Then $$f(n)=g(n)=n^{i\tau},\quad C=(\sqrt{2})^{i\tau}, \tau\in \mathbb{R}.$$