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

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 $3$.