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

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.