Prime polynomials in short intervals and in arithmetic progressions

Abstract

In this paper we establish function field versions of two classical conjectures on prime numbers. The first says that the number of primes in intervals (x,x+xϵ] is about xϵ/logx. The second says that the number of primes p<x in the arithmetic progression p≡a(modd), for d<x1−δ, is about π(x)ϕ(d), where ϕ is the Euler totient function. More precisely, for short intervals we prove: Let k be a fixed integer. Then πq(I(f,ϵ))∼#I(f,ϵ)k,q→∞ holds uniformly for all prime powers q, degree k monic polynomials f∈Fq[t] and ϵ0(f,q)≤ϵ, where ϵ0 is either 1k, or 2k if p∣k(k−1), or 3k if further p=2 and degf'≤1. Here I(f,ϵ)={g∈Fq[t]∣deg(f−g)≤ϵdegf}, and πq(I(f,ϵ)) denotes the number of prime polynomials in I(f,ϵ). We show that this estimation fails in the neglected cases. For arithmetic progressions we prove: let k be a fixed integer. Then πq(k;D,f)∼πq(k)ϕ(D),q→∞, holds uniformly for all relatively prime polynomials D,f∈Fq[t] satisfying ‖D‖≤qk(1−δ0), where δ0 is either 3k or 4k if p=2 and (f/D)' is a constant. Here πq(k) is the number of degree k prime polynomials and πq(k;D,f) is the number of such polynomials in the arithmetic progression P≡f(modd). We also generalize these results to arbitrary factorization types.