$k$-additive uniqueness of the set of squares for multiplicative functions

TL;DR

This paper proves that the set of squares is a unique additive set for multiplicative functions when involving three or more squares, uniquely determining the identity function.

Contribution

It establishes that for any multiplicative function satisfying additive conditions on three or more squares, the function must be the identity, extending previous results for fewer squares.

Findings

Set of squares is a $k$-additive uniqueness set for $k extgreater 2$

Such functions are uniquely determined as the identity function

Extends previous results for two squares to more squares

Abstract

P. V. Chung showed that there are many multiplicative functions $f$ which satisfy $f(m^2+n^2) = f(m^2)+f(n^2)$ for all positive integers $m$ and $n$. In this article, we show that if more than $2$ squares in the additive condition are involved, then such $f$ is uniquely determined. That is, if a multiplicative function $f$ satisfies \[ f(a_1^2 + a_2^2 + \dotsb + a_k^2) = f(a_1^2) + f(a_2^2) + \dotsb + f(a_k^2) \] for arbitrary positive integers $a_i$, then $f$ is the identity function. In this sense, we call the set of all posotive squares a \emph{$k$-additive uniqueness set} for multiplicative functions.