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Prime polynomial values of linear functions in short intervals. (English) Zbl 1318.11150

The paper is mainly related with the prime number theorem and the Hardy-Littlewood prime tuple conjecture. For the first, it is conjectured that if \(I\) is an interval of length \(x^{\varepsilon}\), \(\varepsilon >0\), for \(x\) large enough, the number of primes is \(\sim \int_I\frac{dt} {\log t}\sim \frac{x^{\varepsilon}}{\log x}\). The analogue for function fields is that \(\sum_{f\in I} {\mathbf 1}(f)=\frac{\# I}{k}(1+O_k(q^{-1/2}))\) where \[ {\mathbf 1}(f)=\left\{ \begin{matrix} 1 &\text{if \(f\) is monic irreducible}\cr 0 & \text{otherwise}\end{matrix}\right. \] and we let \(\|f\|=q^{\deg f}\), \(I=I(f_0,\varepsilon)=\{f\in{\mathbb F}_q[t] : \|f-f_0\|\leq \|f_0\|^{\varepsilon}\}\), and \(f_0\) is a monic polynomial of degree \(k\), \(\frac{3}{k}\leq \varepsilon <1\). L. Rosenzweig and the authors proved this result [Duke Math. J. 164, No. 2, 277–295 (2015; Zbl 1395.11132)].
The Hardy-Littlewood prime tuple conjecture asserts that \[ {\mathbf 1}(h+a_1)\cdots {\mathbf 1}(h+a_n)\sim {\mathcal G}(a_1,\ldots,a_n) \frac{x}{(\log x)^n},\quad x\to \infty, \] where \({\mathcal G}(a_1,\ldots,a_n)= \prod_p \frac{1-\nu(p)p^{-1}}{(1-p^{-1})^n}\) and \(\nu(p) =\# \{h\bmod p: (h+a_1)\cdots(h+a_n)\equiv 0 \bmod p\}\).
For the function field case, the second author showed in [Int. Math. Res. Not. 2014, No. 2, 568–575 (2014; Zbl 1296.11165)] that for any \(k>0\) \[ \sum_{f\in{\mathbb F}_q[t] \text{\;monic}\atop \deg f=k} {\mathbf 1}(f+a_1)\cdots {\mathbf 1}(f+a_n)=\frac{q^k}{k^n} (1+O_{k,n}(q^{-1/2})) \] uniformly on all \(a_1,\ldots, a_n\in {\mathbb F}_q[t]\) with \(\deg(a_i)<k\) and \(q\) odd. The main result in this work is the following: Let \(B>0\) and \(0<\varepsilon<1\). Then the asymptotic formula \[ \sum_{f\in I(f_0,\varepsilon)} {\mathbf 1}(L_1(f))\cdots {\mathbf 1}(L_n(f)) = \frac{\# I(f_0,\varepsilon)}{\prod_{i=1}^n \deg (L_i(f_0))} (1+O_B(q^{-1/2})) \] holds uniformly for \(q\) odd, \(1\leq n\leq B\), distinct primitive linear functions \(L_1(X),\ldots,L_n(X)\) defined over \({\mathbb F}_q(t)\) each of height at most \(B\), and \(f_0\) a monic polynomial such that \(\frac{2}{\varepsilon}\leq \deg f_0\leq B\). Here \(L_i(X)=f_i(t)+g_i(t)X\) with \(f_i,g_i\in{\mathbb F}_q[t]\), \(g_i\neq 0\) and \(\mathrm{height}(L_i):=\max\{\deg f_i,\deg g_i\}\).
The proof is based in the computation of the Galois groups of some polynomials. This is carried out in Section 2. In Section 3 a theorem more general than the main result is proved. The proof is in the spirit of several similar results once one has the Galois group. In the last section it is presented a discussion on primes in short intervals and also on arithmetic progressions are considered.

MSC:

11R58 Arithmetic theory of algebraic function fields
11P32 Goldbach-type theorems; other additive questions involving primes
11P55 Applications of the Hardy-Littlewood method
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References:

[1] Andrade, Julio C.; Bary-Soroker, Lior; Rudnick, Zeev, Shifted convolution and the Titchmarsh divisor problem over \(F_q [t]\), Philos. Trans. R. Soc. A (2014), in press
[2] Bank, Efrat; Bary-Soroker, Lior; Rosenzweig, Lior, Prime polynomials in short intervals and in arithmetic progressions, Duke Math. J., 164, 2, 277-295 (2015) · Zbl 1395.11132
[3] Bary-Soroker, Lior, Irreducible values of polynomials, Adv. Math., 229, 2, 854-874 (2012) · Zbl 1271.11114
[4] Bary-Soroker, Lior, Hardy-Littlewood tuple conjecture over large finite fields, Int. Math. Res. Not. IMRN, 2, 568-575 (2014) · Zbl 1296.11165
[5] Bender, Andreas O., Decompositions into sums of two irreducibles in \(F_q [t]\), C. R. Math. Acad. Sci. Paris, 346, 17-18, 931-934 (2008) · Zbl 1213.11192
[6] Bender, Andreas O.; Pollack, Paul, On quantitative analogues of the Goldbach and twin prime conjectures over \(F_q [t] (2009)\)
[7] Carmon, Dan, The autocorrelation of the Mobius function and Chowla’s conjecture for the rational function field in characteristic 2, Philos. Trans. R. Soc. A (2015), in press · Zbl 1397.11158
[8] Carmon, Dan; Rudnick, Zeév, The autocorrelation of the Möbius function and Chowla’s conjecture for the rational function field, Q. J. Math., 65, 1, 53-61 (2014) · Zbl 1302.11073
[9] Entin, Alexei, On the Bateman-Horn conjecture for polynomials over large finite fields (2014) · Zbl 1371.11153
[10] Granville, Andrew, Unexpected irregularities in the distribution of prime numbers, (Proceedings of the International Congress of Mathematicians, vols. 1, 2. Proceedings of the International Congress of Mathematicians, vols. 1, 2, Zürich, 1994 (1995), Birkhäuser: Birkhäuser Basel), 388-399 · Zbl 0843.11043
[11] Keating, Jonathan P.; Rudnick, Zeév, The variance of the number of prime polynomials in short intervals and in residue classes, Int. Math. Res. Not. IMRN, 1, 259-288 (2014) · Zbl 1319.11084
[12] Pollack, Paul, Simultaneous prime specializations of polynomials over finite fields, Proc. Lond. Math. Soc. (3), 97, 3, 545-567 (2008) · Zbl 1221.11239
[13] Rodgers, Brad, The covariance of almost-primes in \(F_q [t]\), Int. Math. Res. Not. IMRN, 2014, 29 (2014) · Zbl 1351.11080
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