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arXiv · 1302.0625

Prime polynomials in short intervals and in arithmetic progressions

Efrat Bank·Lior Bary-Soroker·Lior Rosenzweig
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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 \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.

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Efrat Bank, Lior Bary-Soroker, Lior Rosenzweig. 2013-02-04. Prime polynomials in short intervals and in arithmetic progressions. https://doi.org/10.1215/00127094-2856728

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