Abstract
Let . In this study, for prime numbers and a sufficiently large real number , we prove the Diophantine inequality , where and . When , this result improves a previous result.
1. Introduction
Suppose that is an integer and is not an integer. Let be a small positive number. The Waring–Goldbach problem is to study the solvability of the Diophantine equality:in prime numbers . In [1], the author studied the analog of the Waring–Goldbach problem. For any sufficiently large real number , let denote the smallest natural number such that the Diophantine inequality,is solvable in prime numbers . In [1], the author proved that
In [1], the author also obtained that , for . Later, the result was improved in [2–6]. Now, the best result for is by Li and Cai [3].
In [4], the authors first proved that , for . Later, the result was improved in [7]. Now, the best result for is by Zhang and Li [8]. When in inequality (2), Tolev [9] obtained the result . Afterwards, the range of was enlarged by several authors in [10–16]. Now, the best result for is by Cai [12]. When , in inequality (2), Laporta obtained in [17]. Laporta’s result was improved by some authors in [4, 18, 19]. Now, the best result for is by Li and Cai [19].
In this paper, we focus on the Diophantine inequality (2) and prove the following result.
Theorem 1. Let . For any sufficiently large real number , let and let denote the number of solutions of the Diophantine inequality:in prime numbers , where and . We can obtain
For in Theorem 1, we can get better result than Li and Cai [3].
Corollary 2. Under the notations of Theorem 1, for , we haveFor comparison, .
Notation 1. In this paper, let and , . We also assumeLet denote von Mangoldt’s function. We write if the range of is . means that .
2. Some Lemmas
In order to prove our theorem, we need the following lemmas.
Lemma 1. Let be a positive integer. There exists a function which is times continuously differentiable and satisfiesand its Fourier transformation,satisfies
Proof. This can be found in Piatetski-Shapiro [1].
Lemma 2. Let be a sequence of complex numbers; then, for , we havewhere denotes the conjugate of the complex number .
Proof. This is Lemma 2 of Fouvry and Iwaniec [20].
Lemma 3. We have
Proof. Similar to Lemma 6 of [16], we havewhere means that , and and satisfyWe haveby the mean-value theorem, where . Therefore, we can obtainwhich proves Lemma 3.
Lemma 4. (i)(ii)
Proof. This is Lemma 7 of Tolev [16].
Lemma 5. Let and let . For any exponent pair and , we have
Proof. This is Lemma 3 of Li and Cai [3].
Lemma 6. For , we have
Proof. This is Lemma 14 of Tolev [16].
Lemma 7. For , we have
Proof. This is (50) of Zhai and Cao [5].
Lemma 8. Let be real numbers such that . Setwhere , and . Then, we have
Proof. This is Theorem 1 of Baker and Weingartner [6].
Lemma 9. Let . Suppose that . Assume further that is a complex-valued function such that . Then, the sum,can be written in sums.
Proof. This is Lemma 3 of Heath-Brown [21].
3. The Estimate of
In this section, we draw our attention to the estimate of exponential sums, which also has lots of applications (e.g., see [22–35]). Suppose that and ; then, we estimate the exponential sums in the following two forms. Type I:and Type II:
Lemma 10. For complex number sequences and , suppose that , , ; then, we have
Proof. By Cauchy’s inequality and Lemma 2 with , we obtainwherewith . By Lemma 5, we choose the exponent pair . Then, we can obtainNow, Lemma 10 follows from (26) and (28).
Lemma 11. Let be a sequence of complex numbers. Suppose that , , and ; then, we have
Proof. If , by Lemma 5, we choose the exponent pair . Then, we obtainIf , then, by Lemma 10, we haveIf , by Lemma 8 with or , we haveNow, the proof of Lemma 11 is completed.
Lemma 12. For , we have
Proof. We can obtainLetLet in Lemma 9; we reduce the estimation of to the estimations of type I sums:and type II sums:and estimate (34) follows from Lemmas 10 and 11.
4. Proof of the theorem
Let
By the definition of in Lemma 1, we have
By the inverse Fourier transformation formula, we obtainwhere
Let
Then, we have
From Lemma 10 in [36], we have . Thus, by Lemma 1, we have
It follows from Lemmas 1, 4, and 6 that
By Lemma 3 and (43)–(45), we obtain
Let
We have
It follows from (48) and Cauchy’s inequality that
By Lemma 5 with the exponent pair (see in [37]), we obtain
It follows from (50) thatwhere Lemma 12 is used.
Now, by (49) and (51), we obtainwhere Lemma 7 is used. From (52), we obtain
It follows from Lemma 1 that
By (40), (46), (53), and (54),
It follows from (39) and (55) that
Now, by (56), the proof of the theorem is completed.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.