A characterization of arithmetic functions satisfying $f(u^2+k{v}^2)=f^2(u)+k{f}^2(v)$

Abstract

In this paper, we mainly discuss the characterization of a class of arithmetic functions $f:N\to C$ such that $f(u^2+k{v}^2)=f^2(u)+k{f}^2(v)$ $(k,u,v\in N)$. We obtain a characterization with given condition, propose a conjecture and show the result holds for $k\in \{2,3,4,5\}$.