On additive arithmetical functions with values in topological groups IV.

Abstract. We prove that if \(G\) is an additively written Abelian topological group with the translation invariant metric \(\rho\) and $$ {1\over {\log x}}\sum_{n\le x} {\rho(\psi(n+1),\varphi(n))\over n}\to 0\quad (x\to \infty), $$ where \(\psi, \varphi: {\mathbb N}\to G\) are completely additive functions, then \(\varphi(n)=\psi(n)\) \((\forall n\in {\mathbb N})\), and the extension \(\varphi :{\mathbb R}_x\to G\) is a continuous homomorphism, where \({\mathbb R}_x\) is the multiplicative group of positive real numbers.
We also prove that if $$ {1\over {\log x}}\sum_{n\le x} {\rho(\psi([\sqrt{2}n]), \varphi(n)+A)\over n}\to 0\quad (x\to \infty), $$ then \(\varphi(n)=\psi(n)\ (\forall n\in{\mathbb N})\), and the extension \(\varphi=\psi :{\mathbb R}_x\to G\) is a continuous homomorphism, with \(\psi(\sqrt{2})=A\).