Duke Mathematical Journal

Prime polynomials in short intervals and in arithmetic progressions

Efrat Bank, Lior Bary-Soroker, and 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+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

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

Dates
First available in Project Euclid: 30 January 2015

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1422627049

Digital Object Identifier
doi:10.1215/00127094-2856728

Mathematical Reviews number (MathSciNet)
MR3306556

Zentralblatt MATH identifier
06416949

Subjects
Primary: 11T06: Polynomials

Citation

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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References

  • [1] L. Bary-Soroker, Dirichlet’s theorem for polynomial rings, Proc. Amer. Math. Soc. 137 (2009), 73–83.
  • [2] L. Bary-Soroker, Irreducible values of polynomials, Adv. Math. 229 (2012), 854–874.
  • [3] D. Carmon and Z. Rudnick, The autocorrelation of the Möbius function and Chowla’s conjecture for the rational function field, Q. J. Math. 65 (2014), 53–61.
  • [4] S. D. Cohen, Uniform distribution of polynomials over finite fields, J. London Math. Soc. 6 (1972), 93–102.
  • [5] S. D. Cohen, The Galois group of a polynomial with two indeterminate coefficients, Pacific J. Math. 90 (1980), 63–76.
  • [6] A. Granville, “Unexpected irregularities in the distribution of prime numbers” in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, 388–399.
  • [7] A. Granville, Different approaches to the distribution of primes, Milan J. Math. 78 (2010), 65–84.
  • [8] D. R. Heath-Brown, The number of primes in a short interval, J. Reine Angew. Math. 389 (1988), 22–63.
  • [9] D. R. Heath-Brown and D. A. Goldston, A note on the differences between consecutive primes, Math. Ann. 266 (1984), 317–320.
  • [10] M. N. Huxley, On the difference between consecutive primes, Invent. Math. 15 (1972), 164–170.
  • [11] J. P. Keating and Z. Rudnick, The variance of the number of prime polynomials in short intervals and in residue classes, Int. Math. Res. Not. IMRN 1 (2012), 259–288.
  • [12] S. Lang and A. Weil, Number of points of varieties in finite fields, Amer. J. Math. 76 (1954), 819–827.
  • [13] H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory, I: Classical Theory, Cambridge Stud. Adv. Math.97, Cambridge Univ. Press, Cambridge, 2007.
  • [14] R. A. Rankin, The difference between consecutive prime numbers, J. London Math. Soc. 13 (1938), 242–247.
  • [15] A. Selberg, On the normal density of primes in small intervals, and the difference between consecutive primes, Arch. Math. Naturvid. 47 (1943), 87–105.
  • [16] J.-P. Serre, Topics in Galois Theory, ed. 2, Res. Notes in Math. 1, A. K. Peters, Ltd., Wellesley, Mass., 2008.
  • [17] K. Uchida. Galois group of an equation $X^{n}-aX+b=0$. Tohoku Math. J. 22 (1970), no. 4, 670–678.