On Incidences of $\varphi$ and $\sigma$ in the Function Field Setting

TL;DR

This paper investigates the analogues of a classical number theory conjecture involving Euler's totient and sum of divisors functions within polynomial rings over finite fields, revealing triviality for most fields and infinite solutions for specific cases.

Contribution

It extends the classical conjecture to finite fields, providing a complete characterization of solutions and demonstrating infinite solutions when the field size is 2 or 3.

Findings

Trivial solutions only occur when $q eq 2,3$.

Infinite solutions exist for $q=2$ or $3$.

Complete characterization of solutions in special cases.

Abstract

Erd\H{o}s first conjectured that infinitely often we have $\varphi(n) = \sigma(m)$, where $\varphi$ is the Euler totient function and $\sigma$ is the sum of divisor function. This was proven true by Ford, Luca and Pomerance in 2010. We ask the analogous question of whether infinitely often we have $\varphi(F) = \sigma(G)$ where $F$ and $G$ are polynomials over some finite field $\mathbb{F}_q$. We find that when $q\not=2$ or $3$, then this can only trivially happen when $F=G=1$. Moreover, we give a complete characterisation of the solutions in the case $q=2$ or $3$. In particular, we show that $\varphi(F) = \sigma(G)$ infinitely often when $q=2$ or $3$.