Prime polynomials in short intervals and in arithmetic progressions

In this paper we establish function field versions of two classical conjectures on rime numbers. The first says that the number of primes in intervals (x, x + ^ε) is about x^ε/log x. The second says that the number f primes p < x in the arithmetic progression p = a (mod d), for d <x^1-δ, is about π(x)/ϕ(d), where ϕ is the Euler totient function. holds uniformly for all prime powers q, degree k monic polynomials f ε F_qt and ε₀(f, q)≤ ε, where ε₀ is either 1/k, or 2/k if p k(k - 1), or 3/k if further p = 2 and deg f' ≤ 1. Here I(f,ε)= {g ε F_q t | deg(f - g) ≤ ε deg f}, and π_q(I(f, ε)) denotes the number of prime polynomials in I(f,ε). We show that this estimation ails in the neglected cases. holds uniformly for all relatively prime polynomials D, f ε F_qt satisfying ||D||≤ q^k(1-δ₀), where δ₀ is either 3/k or 4/k if p = 2 and (f/D)' is a constant. Here π_q(k) is the number of degree k prime polynomials and π_q(k;D, f) is the number of such polynomials in the arithmetic progression P= f (mod D). We also generalize these results to arbitrary factorization types.