Multiplicative functions which are additive on triangular numbers

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 $k=2$.

5 pages