Some sums over irreducible polynomials

TL;DR

This paper proves conjectures about sums over irreducible polynomials in finite fields, showing they often converge to rational functions and revealing specific behaviors for certain sums.

Contribution

It establishes the convergence of sums over irreducible polynomials to rational functions, confirming several conjectures by Dinesh Thakur.

Findings

Sum over irreducible polynomials converges to rational functions.

In $ extbf{F}_2[T]$, the sum $ extstylerac{1}{P^k - 1}$ converges and equals zero for $k=1$.

Several conjectures by Thakur are proven.

Abstract

We prove a number of conjectures due to Dinesh Thakur concerning sums of the form $\sum_P h(P)$ where the sum is over monic irreducible polynomials $P$ in $\mathbb{F}_q[T]$, 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 $\mathbb{F}_2[T]$, the sum $\sum_P \tfrac{1}{P^k - 1}$ always converges to a rational function, and is $0$ for $k=1$.