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

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