Prime polynomials in short intervals and in arithmetic progressions

Efrat Bank, Lior Bary-Soroker, Lior Rosenzweig

Research output: Contribution to journalArticlepeer-review


In this paper we establish function field versions of two classical conjectures on rime numbers. The first says that the number of primes in intervals (x, x + ε) is about xε/log x. The second says that the number f primes p < x in the arithmetic progression p = a (mod d), for d <x1-δ, is about π(x)/ϕ(d), where ϕ is the Euler totient function. holds uniformly for all prime powers q, degree k monic polynomials f ε Fqt and ε0(f, q)≤ ε, where ε0 is either 1/k, or 2/k if p k(k - 1), or 3/k if further p = 2 and deg f' ≤ 1. Here I(f,ε)= {g ε Fq t | deg(f - g) ≤ ε deg f}, and πq(I(f, ε)) denotes the number of prime polynomials in I(f,ε). We show that this estimation ails in the neglected cases. holds uniformly for all relatively prime polynomials D, f ε Fqt satisfying ||D||≤ qk(1-δ0), where δ0 is either 3/k or 4/k 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 P= f (mod D). We also generalize these results to arbitrary factorization types.

Original languageEnglish
Pages (from-to)277-295
Number of pages19
JournalDuke Mathematical Journal
Issue number2
StatePublished - 2015

All Science Journal Classification (ASJC) codes

  • General Mathematics


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