On k-additive uniqueness of the set of squares for multiplicative functions

Abstract

P. V. Chung showed that there are many multiplicative functions f which satisfy $$f(m^2+n^2) = f(m^2)+f(n^2)$$ for all positive integers m and n. In this article, we show that if more than 2 squares in the additive condition are involved, then such f is uniquely determined. That is, if a multiplicative function f satisfies $$\begin{aligned} f(a_1^2 + a_2^2 + \cdots + a_k^2) = f(a_1^2) + f(a_2^2) + \cdots + f(a_k^2) \end{aligned}$$ for arbitrary positive integers $$a_i$$ , then f is the identity function. In this sense, we call the set of all positive squares a k-additive uniqueness set for multiplicative functions.