On multiplicative functions which are additive on positive cubes

Abstract

Let \(k \ge 3\). If a multiplicative function f satisfies

$$\begin{aligned} f(a_1^3 + a_2^3 + \cdots + a_k^3) = f(a_1^3) + f(a_2^3) + \cdots + f(a_k^3) \end{aligned}$$

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 \(k=2\) can be satisfied by infinitely many multiplicative functions. In additon, if \(k \ge 3\) and a multiplicative function g satisfies

$$\begin{aligned} g(a_1^3 + a_2^3 + \cdots + a_k^3) = g(a_1)^3 + g(a_2)^3 + \cdots + g(a_k)^3 \end{aligned}$$

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.