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The integer part of a nonlinear form with integer variables
Journal of Inequalities and Applications volume 2015, Article number: 357 (2015)
Abstract
Using the Davenport-Heilbronn method, we show that if \(\lambda_{1},\lambda_{2},\ldots,\lambda_{9}\) are positive real numbers, at least one of the ratios \(\lambda_{i}/\lambda_{j}\) (\(1\leq i< j\leq9\)) is irrational, then the integer parts of \(\lambda_{1}x_{1}^{3}+\lambda_{2}x_{2}^{3}+\lambda_{3}x_{3}^{4}+\lambda_{4}x_{4}^{4} +\lambda_{5}x_{5}^{5}+\cdots+\lambda_{9}x_{9}^{5}\) are prime infinitely often for natural numbers \(x_{1},x_{2},\ldots,x_{9}\).
1 Introduction
In 2010, Brüdern et al. [1] proved that if \(\lambda _{1},\ldots,\lambda_{s}\) are positive real numbers, \(\lambda_{1}/\lambda_{2}\) is irrational, all Dirichlet L-functions satisfy the Riemann hypothesis \(s\geq \frac{8}{3}k+2\), then the integer parts of
are prime infinitely often for natural numbers \(x_{j}\).
Motivated by [1], using the Davenport-Heilbronn method, we consider the integer part of a nonlinear form with integer variables and mixed powers 3, 4 and 5, and establish one result as follows.
Theorem 1.1
Let \(\lambda_{1},\lambda_{2},\ldots,\lambda_{9}\) be positive real numbers, at least one of the ratios \(\lambda_{i}/\lambda_{j}\) (\(1\leq i< j\leq9\)) is irrational. Then the integer parts of
are prime infinitely often for natural numbers \(x_{1},x_{2},\ldots,x_{9}\).
It is noted that Theorem 1.1 holds without the Riemann hypothesis.
2 Notation
Throughout, we use p to denote a prime number and \(x_{j}\) to denote a natural number. We denote by δ a sufficiently small positive number and by ε an arbitrarily small positive number. Constants, both explicit and implicit, in Landau or Vinogradov symbols may depend on \(\lambda_{1},\lambda_{2},\ldots,\lambda _{9}\). We write \(e(x)=\exp(2\pi i x)\). We use \([x]\) to denote the integer part of real variable x. We take X to be the basic parameter, a large real integer. Since at least one of the ratios \(\lambda_{i}/\lambda_{j}\) (\(1\leq i< j\leq9\)) is irrational, without loss of generality we may assume that \(\lambda_{1}/ \lambda_{2}\) is irrational. For the other cases, the only difference is in the following intermediate region, and we may deal with the same method in Section 4.
Since \(\lambda_{1}/ \lambda_{2}\) is irrational, then there are infinitely many pairs of integers q, a with \(|\lambda_{1}/\lambda _{2}-a/q|\leq q^{-2}\), \((a,q)=1\), \(q>0\) and \(a\neq 0\). We choose q to be large in terms of \(\lambda_{1},\lambda_{2},\ldots ,\lambda_{9}\) and make the following definitions.
Let ν be a positive real number, we define
It follows from (2.1) that
From (2.3) it is clear that
thus
To estimate J, we split the range of infinite integration into three sections, traditional named the neighborhood of the origin \(\frak{C}=\{\alpha\in{\mathbb{R}}:|\alpha|\leq\tau\}\), the intermediate region \(\frak{D}=\{\alpha\in{\mathbb{R}}:\tau<|\alpha |\leq P\}\) and the trivial region \(\frak{c}=\{\alpha\in{\mathbb{R}}:|\alpha|>P\}\).
3 The neighborhood of the origin
Lemma 3.1
If \(\alpha=a/q+\beta\), where \((a,q)=1\), then
Proof
This is Theorem 4.1 of [2]. □
If \(|\alpha|\in\frak{C}\), by Lemma 3.1, taking \(a=0\), \(q=1\), then
Lemma 3.2
Let \(\rho=\beta+i\gamma\) be a typical zero of the Riemann zeta function, C be a positive constant,
then
Proof
Equations (3.2), (3.3), (3.4) can be seen from Lemma 5, (29) and (33) given by Vaughan [3]. □
Lemma 3.3
We have
Proof
These results are from Lemma 5 of [3]. □
Lemma 3.4
We have
Proof
It is obvious that
Then by (3.1), Lemmas 3.2 and 3.3, we have
The other cases are similar, and the proof of Lemma 3.4 is completed. □
Lemma 3.5
We have
Proof
It follows from Vaughan [2] that for \(\alpha\neq0\),
Thus
 □
Lemma 3.6
We have
Proof
From (2.3) one has
Let \(|\sum_{i=1}^{9}\lambda_{i} x_{i}-x-\frac{1}{2}|\leq\frac{1}{4}\), then \(\sum_{i=1}^{9}\lambda_{i} x_{i}-\frac{3}{4}\leq x\leq \sum_{i=1}^{9}\lambda_{i} x_{i}-\frac{1}{4}\). Based on
one may take
hence
This completes the proof of Lemma 3.6. □
4 The intermediate region
Lemma 4.1
We have
Proof
By (2.2) and Hua’s inequality, for \(i=1,2\), we have
The proofs of (4.2)-(4.4) are similar to (4.1). □
Lemma 4.2
Suppose that \((a,q)=1\), \(|\alpha-a/q|\leq q^{-2}\), \(\phi (x)=\alpha x^{k}+\alpha_{1}x^{k-1}+\cdots+\alpha_{k-1}x+\alpha_{k}\), then
Proof
This is Lemma 2.4 (Weyl’s inequality) of Vaughan [2]. □
Lemma 4.3
For every real number \(\alpha\in\frak{D}\), let \(W(\alpha)=\min(|F_{1}(\lambda_{1}\alpha)|,|F_{2}(\lambda_{2}\alpha)|)\), then
Proof
For \(\alpha\in\frak{D}\) and \(i=1,2\), we choose \(a_{i}\), \(q_{i}\) such that
with \((a_{i},q_{i})=1\) and \(1\leq q_{i}\leq Q\).
Firstly, we note that \(a_{1}a_{2}\neq0\). Secondly, if \(q_{1},q_{2}\leq P\), then
We recall that q was chosen as the denominator of a convergent to the continued fraction for \(\lambda_{1}/\lambda_{2}\). Thus, by Legendre’s law of best approximation, we have \(|q'\frac{\lambda_{1}}{\lambda_{2}}-a'|>\frac{1}{2q}\) for all integers \(a'\), \(q'\) with \(1\leq q'< q\), thus \(|a_{2}q_{1}|\geq q=[N^{1-8\delta}]\). However, from (4.5) we have \(|a_{2}q_{1}|\ll q_{1}q_{2}P \ll N^{18\delta}\), this is a contradiction. We have thus established that for at least one i, \(P< q_{i}\ll Q\). Hence Lemma 4.2 gives the desired inequality for \(W(\alpha)\). □
Lemma 4.4
We have
Proof
By Lemmas 4.1, 4.3 and Hölder’s inequality, we have
 □
5 The trivial region
Lemma 5.1
(Lemma 2 of [4])
Let \(V(\alpha)=\sum e(\alpha f(x_{1},\ldots,x_{m}))\), where f is any real function and the summation is over any finite set of values of \(x_{1},\ldots,x_{m}\). Then, for any \(A>4\), we have
Lemma 5.2
We have
Proof
By Lemmas 5.1, 4.1 and Schwarz’s inequality, we have
 □
6 The proof of Theorem 1.1
From Lemmas 3.4, 3.5 and 3.6 we conclude that \(J({\frak{C}})\gg X^{\frac {13}{6}}\). From Lemma 4.4 it follows that \(J({\frak{D}})=o(X^{\frac{13}{6}})\). From Lemma 5.2 we have \(J({\frak{c}})=o(X^{\frac{13}{6}})\). Thus
namely, under conditions of Theorem 1.1,
has infinitely many solutions in positive integers \(x_{1},x_{2},\ldots,x_{9}\) and prime p. It is evident from (6.1) that
and hence
The proof of Theorem 1.1 is complete.
References
Brüdern, J, Kawada, K, Wooley, TD: Additive representation in thin sequences, VIII: Diophantine inequalities in review. In: Number Theory: Dreaming in Dreams. Series on Number Theory and Its Applications, vol. 6, pp. 20-79 (2010)
Vaughan, R: The Hardy-Littlewood Method, 2nd edn. Cambridge Tracts in Mathematics, vol. 125. Cambridge University Press, Cambridge (1997)
Vaughan, R: Diophantine approximation by prime numbers I. Proc. Lond. Math. Soc. 28, 373-384 (1974)
Davenport, H, Roth, KF: The solubility of certain Diophantine inequalities. Mathematika 2, 81-96 (1955)
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Lai, K. The integer part of a nonlinear form with integer variables. J Inequal Appl 2015, 357 (2015). https://doi.org/10.1186/s13660-015-0874-2
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DOI: https://doi.org/10.1186/s13660-015-0874-2