Multiplicative functions which are additive on triangular numbers

TL;DR

This paper proves that any multiplicative function satisfying an additive property on sums of positive triangular numbers must be the identity function, extending previous results for the case when k=2.

Contribution

It generalizes the characterization of multiplicative functions that are additive on sums of triangular numbers to any fixed k ≥ 3.

Findings

The only such multiplicative function is the identity function.

Extension of previous work from k=2 to arbitrary k ≥ 3.

Provides a new functional equation characterization for multiplicative functions.

Abstract

Fix $k \ge 3$. If a multiplicative function $f$ satisfies \[ f(x_1+x_2+\dots+x_k) = f(x_1) + f(x_2) + \dots + f(x_k) \] for arbitrary positive triangular numbers $x_1, x_2, \dots, x_k$, then $f$ is the identity function. This extends Chung and Phong's work for $k=2$.