Some sums over irreducible polynomials

Abstract

We prove a number of conjectures due to Dinesh Thakur concerning sums of the form ${\sum }_{P}h\left(P\right)$ where the sum is over monic irreducible polynomials $P$ in ${\mathbb{F}}_q\left[T\right]$, the function $h$ is a rational function and the sum is considered in the $T^{-1}$-adic topology. As an example of our results, in ${\mathbb{F}}_2\left[T\right]$, the sum ${\sum }_{P}1∕\left(P^k-1\right)$ always converges to a rational function, and is $0$ for $k=1$.