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 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 π_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'-π_q(k)/ϕ(D)|\leq c(k)π_q(k)q^{-1/2}/ϕ(D), as long as 1\leq °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.