On multiplicative functions which are additive on positive cubes

Let \(k \ge 3\) $k\ge 3$ . If a multiplicative function f satisfies \(\begin{aligned} f(a_1^3 + a_2^3 + \cdots + a_k^3) = f(a_1^3) + f(a_2^3) + \cdots + f(a_k^3) \end{aligned}\) $$\begin{array}{c}\hfill f(a_1^3+a_2^3+\cdots +a_k^3)=f(a_1^3)+f(a_2^3)+\cdots +f(a_k^3)\end{array}$$ for all \(a_1, a_2, \ldots , a_k \in {\mathbb {N}}\) $a_1,a_2,\dots ,a_k\in \mathbb{N}$ , then f is the identity function. The set of positive cubes is said to be a k-additive uniqueness set for multiplicative functions. But, the condition \(k=2\) $k=2$ can be satisfied by infinitely many multiplicative functions. In additon, if \(k \ge 3\) $k\ge 3$ and a multiplicative function g satisfies \(\begin{aligned} g(a_1^3 + a_2^3 + \cdots + a_k^3) = g(a_1)^3 + g(a_2)^3 + \cdots + g(a_k)^3 \end{aligned}\) $$\begin{array}{c}\hfill g(a_1^3+a_2^3+\cdots +a_k^3)=g{(a_1)}^3+g{(a_2)}^3+\cdots +g{(a_k)}^3\end{array}$$ for all \(a_1, a_2, \ldots , a_k \in {\mathbb {N}}\) $a_1,a_2,\dots ,a_k\in \mathbb{N}$ , then g is the identity function. However, when \(k=2\) $k=2$ , there exist three different types of multiplicative functions.