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 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.

Changes in the introduction, accepted to Duke

Cited by in corpus (15)

• Squarefree polynomials and Mobius values in short intervals and arithmetic progressions
• On the Bateman-Horn conjecture for polynomials over large finite fields
• Higher moments of arithmetic functions in short intervals: a geometric perspective
• Some problems in analytic number theory for polynomials over a finite field
• Mean values of arithmetic functions in short intervals and in arithmetic progressions in the large-degree limit
• Chebotarev density theorem in short intervals for extensions of ${\mathbb{F}}_q(T)$
• Irreducible polynomials over ${\mathbb{F}}_{2^r}$ with three prescribed coefficients
• Prime and Möbius correlations for very short intervals in ${\mathbb{F}}_q[x]$
• Equidistribution of Kloosterman sums over function fields
• Number of rational points of symmetric complete intersections over a finite field and applications
• The Arithmetic of Consecutive Polynomial Sequences over Finite Fields
• Variance of sums in short intervals and $L$-functions in ${\mathbb{F}}_q[t]$
• Factorization type probabilities of polynomials with prescribed coefficients over a finite field
• Polynomial values modulo primes on average and sharpness of the larger sieve
• Factorization patterns on nonlinear families of univariate polynomials over a finite field