# Multiplicative functions which are additive on sums of two nonzero squares

## TL;DR

This paper characterizes multiplicative functions that satisfy a specific additive property on sums of two squares, showing they are either the identity or mostly zero with specific exceptions.

## Contribution

It provides a complete classification of multiplicative functions satisfying an additive condition on sums of two squares, identifying when they are the identity or trivial.

## Key findings

- If $f(3)f(11)
e 0$, then $f$ is the identity function.
- Otherwise, $f(n)=0$ for all $n
e 3,9,11$.

## Abstract

Let $f$ be a multiplicative function which satisfies \[ f(a^2+b^2+c^2+d^2) = f(a^2+b^2)+f(c^2+d^2) \] for positive integers $a$, $b$, $c$, and $d$. We show that $f$ is the identity function provided that $f(3)\,f(11) \ne 0$. Otherwise, $f(n)=0$ for all $n \ge 2$ except for $n=3,9,11$.