Multiplicative functions commutable with binary quadratic forms $x^2 \pm xy + y^2$

TL;DR

This paper characterizes multiplicative functions that commute with specific binary quadratic forms, showing they are either the identity, constant, or indicator functions, depending on the form.

Contribution

It provides a complete classification of multiplicative functions commuting with two particular quadratic forms, revealing their structural constraints.

Findings

Functions commuting with $x^2+xy+y^2$ are the identity.

Functions commuting with $x^2-xy+y^2$ are identity, constant, or indicator functions.

The results clarify the algebraic structure of such multiplicative functions.

Abstract

If a multiplicative function $f$ is commutable with a quadratic form $x^2+xy+y^2$, i.e., \[ f(x^2+xy+y^2) = f(x)^2 + f(x)\,f(y) + f(y)^2, \] then $f$ is the identity function. In other hand, if $f$ is commutable with a quadratic form $x^2-xy+y^2$, then $f$ is one of three kinds of functions: the identity function, the constant function, and an indicator function for $\mathbb{N}\setminus p\mathbb{N}$ with a prime $p$.