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

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