On multiplicative functions which are additive on positive cubes

TL;DR

The paper proves that for multiplicative functions, being additive on sums of positive cubes with at least three terms uniquely determines the function as the identity, unlike the case with fewer terms.

Contribution

It establishes the uniqueness of the identity function for multiplicative functions additive on sums of positive cubes when the number of terms is three or more.

Findings

For k ≥ 3, such functions are necessarily the identity.

For k=2, infinitely many multiplicative functions satisfy the conditions.

The set of positive cubes is a k-additive uniqueness set for multiplicative functions.

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.