On multiplicative functions which are additive on positive cubes

Abstract

Let $k \geq 3$. If a multiplicative function $f$ satisfies \[ f(a_1^3 + a_2^3 + \cdots + a_k^3) = f(a_1^3) + f(a_2^3) + \cdots + f(a_k^3) \] for all $a_1, a_2, \ldots, a_k \in \mathbb{N}$, 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 for $k=2$ can be satisfied by infinitely many multiplicative functions. Besides, if $k \geq 3$ and a multiplicative function $g$ satisfies \[ g(a_1^3 + a_2^3 + \cdots + a_k^3) = g(a_1)^3 + g(a_2)^3 + \cdots + g(a_k)^3 \] for all $a_1, a_2, \ldots, a_k \in \mathbb{N}$, then $g$ is the identity function. However, when $k=2$, there exist three different types of multiplicative functions.