Duke Mathematical Journal

Prime polynomials in short intervals and in arithmetic progressions

Efrat Bank, Lior Bary-Soroker, and Lior Rosenzweig

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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.

Article information

Duke Math. J., Volume 164, Number 2 (2015), 277-295.

First available in Project Euclid: 30 January 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11T06: Polynomials


Bank, Efrat; Bary-Soroker, Lior; Rosenzweig, Lior. Prime polynomials in short intervals and in arithmetic progressions. Duke Math. J. 164 (2015), no. 2, 277--295. doi:10.1215/00127094-2856728. https://projecteuclid.org/euclid.dmj/1422627049

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