The integer part of nonlinear forms with prime variables

1.   Introduction

Let $[x]$ be the greatest integer not exceeding $x$. For any natural number $k$, an interesting question is whether there exist $s = s(k)$ and infinitely many primes $p_1, \cdots, p_s, p$ such that

where $\lambda_1, \cdots, \lambda_s$ are real non-zero numbers and at least one of $\lambda_i/\lambda_j (1\leq i < j\leq s)$ is irrational. Following the work of Danicic ^[1] for the linear case $k = 1$ with s = 2, Li and Wang ^[2] made progress for the quadratic case $k = 2$ with $s = 3$, and Li and Su ^[3] for the cubic case $k = 3$ with $s = 5$.

In 1988, Srinivasan ^[4] established one result being of form $[\lambda_1 p_1+\lambda_2 p_2^k] = p$. Inspired by Srinivasan's conclusion, in this paper, we prove two more general and sharper results as follows.

Theorem 1.1. Suppose that $\lambda_1, \lambda_2, \lambda_3$ are positive real numbers, at least one of $\lambda_i/\lambda_j(1\leq i < j\leq 3)$ is irrational, and positive integer $k\geq 2$, then there exist infinitely many primes $p_1, p_2, p_3, p$ such that

Theorem 1.2. Suppose that $\lambda_1, \cdots, \lambda_5$ are positive real numbers, at least one of $\lambda_i/\lambda_j(1\leq i < j\leq 5)$ is irrational, and positive integer $k\geq 3$, then there exist infinitely many primes $p_1, \cdots, p_5, p$ such that

Here we only give the proof of Theorem 1.2, since Theorem 1.1 can be proved similarly. In Section 2, we give the outline of the proof of Theorem 1.2. In Sections 3 and 4, we restrict our attention to the neighbourhood of the origin and the intermediate region, respectively. In Section 5, we consider the trivial region and complete the proof of Theorem 1.2.

Throughout the paper, we use standard notations in number theory. In particular, $\delta$ stands for a sufficiently small positive number, $\varepsilon$ is an arbitrarily small positive number, $\nu$ is positive real number, and $N$ is a sufficiently large real number.

2.   Outline of the method

The basic method builds on the modification of the Hardy-Littlewood circle method first introduced by Davenport and Heilbronn. Denote

for $\nu > 0$ and $\alpha\neq 0$. By continuity, we define $K_{\nu}(0) = \nu$. Then we have

Since at least one of the ratios $\lambda_i/\lambda_j(1\leq i < j\leq 5)$ is irrational, we may assume that $\lambda_1/ \lambda_2$ is irrational, and for other cases, one may deal with them similarly. For $\lambda_1/ \lambda_2$ is irrational, there are infinitely many pairs of integers $q, a$ with

We choose $q$ to be large in terms of $\lambda_1, \cdots, \lambda_5$, and make the following definitions.

By (2.2),

Thus it suffices to establish a positive lower bound for $J({\mathbb{R}})$. In order to estimate $J({\mathbb{R}})$, we split $(-\infty, +\infty)$ into three parts $\frak{C} = \{\alpha\in {\mathbb{R}}:|\alpha|\leq \tau\}, \; \frak{D} = \{\alpha\in {\mathbb{R}}:\tau < |\alpha|\leq P\}, \; \frak{c} = \{\alpha\in {\mathbb{R}}:|\alpha| > P\}$, traditionally named the neighbourhood of the origin, the intermediate region, and the trivial region.

Thus

In the following sections, we compute the integrals in the neighbourhood of the origin, the intermediate region, and the trivial region, respectively.

3.   The neighbourhood of the origin

In this section, we evaluate the contribution from the neighbourhood of the origin and give a low bound.

Let $\rho = \beta+i\gamma$ be the zeros of the Riemann zeta function and $C$ be a positive constant. By Lemma 5 of ^[5], one has

Lemma 3.1. We have

Proof. These can be deduced from Lemmas 6–8 of ^[5].

From Lemma 8 of ^[6], we can deduce the following Lemmas 3.2 and 3.3.

Lemma 3.2. For $i = 1, \cdots, 4$, we have

Lemma 3.3. We have

Lemma 3.4. We have

Proof. Obviously,

By Lemmas 3.1–3.3, we have

The argument for other terms are similar, and the proof of Lemma 3.4 is concluded.

Lemma 3.5. We have

and

Proof. For $\alpha\neq 0$, we have

thus the left hand of (3.2)

The proof of (3.3) is similar to (36) in ^[5], we omit the details.

Combining (3.1), (3.2) and (3.3), we get

4.   The intermediate region

The goal of this section is to estimate the integral $J(\frak{D})$.

Lemma 4.1. We have

Proof. By (2.1) and Hua's inequality,

The proofs of (4.1) and (4.3) are similar.

Lemma 4.2. Suppose $\varepsilon > 0$ is given. Let $f(x)$ be a real valued polynomial in $x$ of degree $k\geq 2$. Suppose $\alpha$ is the leading coefficient of $f$ and there are integers $a, q$ such that $|q\alpha-a| < q^{-1}$ with $(a, q) = 1$. Then we have

Proof. This is Theorem 1 of ^[7].

Lemma 4.3. For every real number $\alpha \in \frak{D}$, let $W(\alpha) = \min(|S_1(\alpha)|, |S_2(\alpha)|)$, then

Proof. The proof is similar to Lemma 9 in ^[3]. In $\alpha \in \frak{D}$, we know that at least one $j$, $P < q_j\ll Q$, and Lemma 4.3 can be established.

Using Hölder's inequality, we have

Hence, by Lemmas 4.1 and 4.3, we have

5.   The trivial region and completion of the proof

In this section, we consider the contribution from the trivial region, and then establish Theorem 1.2.

Lemma 5.1. Let $V(\alpha) = \sum e(\alpha f(x_1, \cdots, x_m))$, where $f$ is any real function and the summation is over any finite set of values of $x_1, \cdots, x_m$. Then, for any $A > 4$, we have

Proof. This is Lemma 2 of ^[8].

By Lemmas 5.1, 4.1 and Schwarz's inequality,

Thus, we have

Combining (2.3), (3.4), (4.4) and (5.1), we get

i.e., under the conditions of Theorem 1.2,

has infinitely many primes solutions $p_1, \cdots, p_5, p$.

By (5.2), we have

and

This proves Theorem 1.2.

6.   Conclusions

In this work, using the circle method, we have established two theorems that integer part of nonlinear forms with prime variables represent primes infinitely. The results presented in this article are new and improve the author's earlier results.

Acknowledgments

The first author is supported by the National Natural Science Foundation of China (Grant No.12071132) and the Natural Science Foundation of Henan Province (Grant No.202300410031). The second author is supported by the Natural Science Foundation of Henan Province (Grant No.212300410195).