An improved asymptotic formula for the distribution of irreducible polynomials in arithmetic progressions over Fq

TL;DR

This paper derives an improved asymptotic formula for counting irreducible polynomials in arithmetic progressions over finite fields, refining previous error bounds using elementary methods.

Contribution

It presents a new asymptotic estimate with sharper error terms for the distribution of irreducible polynomials in arithmetic progressions over finite fields.

Findings

The main formula includes an improved error term compared to Weil's conjecture.

The approach is elementary and does not rely on advanced algebraic geometry.

Provides explicit bounds involving the least prime divisor of n.

Abstract

Let $\mathbb{F}_{q}$ be a finite field with $q$ elements and $\mathbb{F}_{q}[x]$ the ring of polynomials over $\mathbb{F}_{q}$. Let $l(x), k(x)$ be coprime polynomials in $\mathbb{F}_{q}[x]$ and $\Phi(k)$ the Euler function in $\mathbb{F}_{q}[x]$. Let $\pi(l, k; n)$ be the number of monic irreducible polynomials of degree $n$ in $\mathbb{F}_{q}[x]$ which are congruent to $l(x)$ module $k(x)$. For any positive integer $n$, we denote by $\Omega(n)$ the least prime divisor of $n$. In this paper, we show that $$\pi(l, k; n)=\frac{1}{\Phi(k)}\frac{q^{n}}{n}+O\left(n^{\alpha}\right)+O\left(\frac{q^{\frac{n}{\Omega{(n)}}}}{n}\right),$$ where $\alpha$ only depends on the choice of $k(x)\in\Fq$. Note that the above error term improves the one implied by Weil's conjecture. Our approach is completely elementary.