Prime polynomials in short intervals over large finite fields
Efrat Bank, Lior Bary‐Soroker, Lior Rosenzweig · arXiv (Cornell University) · 2013
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 \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 ° 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.