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+x^epsilon] is about x^epsilon/log x and the second says that the number of primes p<x that are congruent to a modulo d, for d^(1+delta)<x, is about pi(x)/phi(d). More precisely, we prove: Let 1\leq m<k be integers, let q be a prime power, and let f be a monic polynomial of degree k with coefficients in the finite field with q elements. Then there is a constant c(k) such that the number N of prime polynomials g=f+h with deg h \leq m satisfies |N-q^(m+1)/k|\leq c(k)q^(m+1/2). Here we assume m\geq 2 if \gcd(q,k(k-1))>1 and m\geq 3 if q is even and deg f' \leq 1. We show that this estimation fails in the neglected cases. Let \pi_q(k) be the number of monic prime polynomials of degree k with coefficients in the finite field with q elements \FF_q. For relatively prime polynomials f,D\in \FF_q[t] we prove that the number N' of monic prime polynomials g that are congruent to f modulo D and of degree k satisfies |N'-\pi_q(k)/\phi(D)|\leq c(k)\pi_q(k)q^{-1/2}/\phi(D), as long as 1\leq \deg D\leq k-3 (or \leq k-4 if p=2 and (f/D)' is constant). We also generalize these results to other factorization types.