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1 February 2015 Prime polynomials in short intervals and in arithmetic progressions
Efrat Bank, Lior Bary-Soroker, Lior Rosenzweig
Duke Math. J. 164(2): 277-295 (1 February 2015). DOI: 10.1215/00127094-2856728

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+xϵ] is about xϵ/logx. The second says that the number of primes p<x in the arithmetic progression pa(modd), for d<x1δ, is about π(x)ϕ(d), where ϕ is the Euler totient function.

More precisely, for short intervals we prove: Let k be a fixed integer. Then

πq(I(f,ϵ))#I(f,ϵ)k,q holds uniformly for all prime powers q, degree k monic polynomials fFq[t] and ϵ0(f,q)ϵ, where ϵ0 is either 1k, or 2k if pk(k1), or 3k if further p=2 and degf'1. Here I(f,ϵ)={gFq[t]deg(fg)ϵdegf}, and πq(I(f,ϵ)) denotes the number of prime polynomials in I(f,ϵ). We show that this estimation fails in the neglected cases.

For arithmetic progressions we prove: let k be a fixed integer. Then

πq(k;D,f)πq(k)ϕ(D),q, holds uniformly for all relatively prime polynomials D,fFq[t] satisfying Dqk(1δ0), where δ0 is either 3k or 4k if p=2 and (f/D)' is a constant. Here πq(k) is the number of degree k prime polynomials and πq(k;D,f) is the number of such polynomials in the arithmetic progression Pf(modd).

We also generalize these results to arbitrary factorization types.

Citation

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Efrat Bank. Lior Bary-Soroker. Lior Rosenzweig. "Prime polynomials in short intervals and in arithmetic progressions." Duke Math. J. 164 (2) 277 - 295, 1 February 2015. https://doi.org/10.1215/00127094-2856728

Information

Published: 1 February 2015
First available in Project Euclid: 30 January 2015

zbMATH: 06416949
MathSciNet: MR3306556
Digital Object Identifier: 10.1215/00127094-2856728

Subjects:
Primary: 11T06

Rights: Copyright © 2015 Duke University Press

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