Prime polynomials in short intervals and in arithmetic progressions. (English) Zbl 1395.11132

Bank, Efrat; Bary-Soroker, Lior; Rosenzweig, Lior

Duke Math. J. 164, No. 2, 277-295 (2015).

Summary: 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 \((x,x+x^\varepsilon]\) is about \(x^\varepsilon/\log x\). The second says that the number of primes \(p<x\) in the arithmetic progression \(p\equiv a\pmod d\), for \(d<x^{1-\delta}\), is about \(\frac{\pi(x)}{\varphi(d)}\), where \(\varphi\) is the Euler totient function.
More precisely, for short intervals we prove: Let \(k\) be a fixed integer. Then
\[ \pi_q(I(f,\varepsilon))\sim\frac{\#I(f,\varepsilon)}{k}, \quad q\to\infty \]
holds uniformly for all prime powers \(q\), degree \(k\) monic polynomials \(f\in \mathbb F_q[t]\) and \(\varepsilon_0(f,q)\leq\varepsilon\), where \(\varepsilon_0\) is either \(\frac{1}{k}\), or \(\frac{2}{k}\) if \(p\mid k(k-1)\), or \(\frac{3}{k}\) if further \(p=2\) and deg\(f'\leq1\). Here \(I(f,\varepsilon)=\{g\in \mathbb F_q[t]\mid \mathrm{deg}(f-g)\le \varepsilon\, \mathrm{deg}f\}\), and \(\pi_q(I(f,\varepsilon))\) denotes the number of prime polynomials in \(I(f,\varepsilon)\). We show that this estimation fails in the neglected cases.
For arithmetic progressions we prove: let \(k\) be a fixed integer. Then
\[ \pi_q(k;D,f )\sim \frac{\pi_q(k)}{\varphi(D)},\quad q\to\infty, \]
holds uniformly for all relatively prime polynomials \(D,f\in\mathbb F_q[t]\) satisfying \(\|D\|\le q^{k(1-\delta_0)}\), where \(\delta_0\) is either \(\frac3k\) or \(\frac4k\) if \(p=2\) and \((f/D)'\) is a constant. Here \(\pi_q(k)\) is the number of degree \(k\) prime polynomials and \(\pi_q(k;D,f )\) is the number of such polynomials in the arithmetic progression \(P\equiv f \pmod d\).
We also generalize these results to arbitrary factorization types.

MSC:

11R58	Arithmetic theory of algebraic function fields
11T06	Polynomials over finite fields
11B25	Arithmetic progressions

Keywords:

arithmetic progressions; prime polynomials

References:

[1]	L. Bary-Soroker, Dirichlet’s theorem for polynomial rings , Proc. Amer. Math. Soc. 137 (2009), 73-83. · Zbl 1259.12002
[2]	L. Bary-Soroker, Irreducible values of polynomials , Adv. Math. 229 (2012), 854-874. · Zbl 1271.11114
[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. · Zbl 1302.11073
[4]	S. D. Cohen, Uniform distribution of polynomials over finite fields , J. London Math. Soc. 6 (1972), 93-102. · Zbl 0253.12019
[5]	S. D. Cohen, The Galois group of a polynomial with two indeterminate coefficients , Pacific J. Math. 90 (1980), 63-76. · Zbl 0408.12011
[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. · Zbl 0843.11043
[7]	A. Granville, Different approaches to the distribution of primes , Milan J. Math. 78 (2010), 65-84. · Zbl 1275.11129
[8]	D. R. Heath-Brown, The number of primes in a short interval , J. Reine Angew. Math. 389 (1988), 22-63. · Zbl 0646.10032
[9]	D. R. Heath-Brown and D. A. Goldston, A note on the differences between consecutive primes , Math. Ann. 266 (1984), 317-320. · Zbl 0514.10031
[10]	M. N. Huxley, On the difference between consecutive primes , Invent. Math. 15 (1972), 164-170. · Zbl 0241.10026
[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. · Zbl 1319.11084
[12]	S. Lang and A. Weil, Number of points of varieties in finite fields , Amer. J. Math. 76 (1954), 819-827. · Zbl 0058.27202
[13]	H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory, I: Classical Theory , Cambridge Stud. Adv. Math. 97 , Cambridge Univ. Press, Cambridge, 2007. · Zbl 1142.11001
[14]	R. A. Rankin, The difference between consecutive prime numbers , J. London Math. Soc. 13 (1938), 242-247. · Zbl 0019.39403
[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. · Zbl 0063.06869
[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. · Zbl 0211.37104