Some multiplicative functions over $\mathbb{F}_2$

TL;DR

This paper explores multiplicative functions over the finite field , deriving conditions for odd binary polynomials to be perfect, inspired by classical number theory concepts and previous works on odd perfect numbers.

Contribution

It introduces adaptations of multiplicative functions, Dirichlet convolution, and inverses over , providing new necessary conditions for odd perfect binary polynomials.

Findings

Derived necessary conditions for odd perfect binary polynomials

Extended multiplicative function concepts to 

Connected polynomial properties with classical number theory

Abstract

We adapt (over $\mathbb{F}_2$) the general notions of multiplicative function, Dirichlet convolution and Inverse. We get some interesting results, namely necessary conditions for an odd binary polynomial to be perfect. Note that we are inspired by the "analogous" works in \cite{Gall-Rahav-newcongr} and \cite{Touchard}, about odd perfect numbers.