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

2019/11/13 by Zihan, Zhang, Dongchun, Han
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1911.05295

Abstract

Let \mathbbF_q be a finite field with q elements and \mathbbF_q[x] the ring of polynomials over \mathbbF_q. Let l(x), k(x) be coprime polynomials in \mathbbF_q[x] and Φ(k) the Euler function in \mathbbF_q[x]. Let π(l, k; n) be the number of monic irreducible polynomials of degree n in \mathbbF_q[x] which are congruent to l(x) module k(x). For any positive integer n, we denote by Ω(n) the least prime divisor of n. In this paper, we show that π(l, k; n)=(1)/(Φ(k))\fracqⁿn+O(n^α)+O(\fracq^\fracnΩ(n)n), where α only depends on the choice of k(x)∈\Fq. Note that the above error term improves the one implied by Weil's conjecture. Our approach is completely elementary.

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