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

TL;DR

This paper characterizes a class of arithmetic functions satisfying a quadratic form relation, proposes a conjecture, and verifies the result for specific values of k.

Contribution

It provides a new characterization of functions satisfying a quadratic form equation and extends the result to certain values of k.

Findings

Characterization of functions satisfying the relation for specific k values

Proposed a conjecture for the general case

Validated the characterization for k in {2, 3, 4, 5}

Abstract

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