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

Abstract

Erdős first conjectured that infinitely often we have $\phi (n)=\sigma (m)$, where $\phi$ 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 $\phi (F)=\sigma (G)$ where $F$ and $G$ are polynomials over some finite field ${\mathbb{F}}_q$. We find that when $q\ne 2$ 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 $\phi (F)=\sigma (G)$ infinitely often when $q=2$ or $3$.