Multiplicative functions commutable with sums of squares

TL;DR

This paper proves that any multiplicative function commuting with sums of squares for integers greater than or equal to 4 must be the identity function, revealing a unique structural property of such functions.

Contribution

It establishes a rigidity result for multiplicative functions satisfying a sum-of-squares functional equation for all positive integers.

Findings

Any such function is the identity function.

The result holds for all integers k ≥ 4.

It characterizes the structure of multiplicative functions with this property.

Abstract

Let $k$ be an integer greater than or equal $4$. We show that if a multiplicative function $f$ satisfies \[ f(x_1^2 + x_2^2 + \dots + x_k^2) = f(x_1)^2 + f(x_2)^2 + \dots + f(x_k)^2 \] for all positive integers $x_i$'s, then $f$ is the identity function.