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

preprint2016arXivOpen access

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 $k \in \{2, 3, 4, 5 \}$ .

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

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A characterization of arithmetic functions satisfying f(u2+kv2)=f2(u)+kf2(v)f(u^{2}+kv^{2})=f^{2}(u)+kf^{2}(v)f(u2+kv2)=f2(u)+kf2(v)

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