On a Diophantine Inequality with s Primes

In [1], the author also obtained that H(c)≤ 5, for 1< c< (3/2). Later, the result was improved in [2–6]. Now, the best result for H(c)≤ 5 is 2< c< (52/25) by Li and Cai [3]. In [4], the authors first proved that H(c)≤ 4, for 1< c< (81/68). Later, the result was improved in [7]. Now, the best result for H(c)≤ 4 is 1< c< (1193/889) by Zhang and Li [8]. When s � 3 in inequality (2), Tolev [9] obtained the result 1< c< (27/26). Afterwards, the range of c was enlarged by several authors in [10–16]. Now, the best result for s � 3 is 1< c< (43/36) by Cai [12]. When s � 2, in inequality (2), Laporta obtained 1< c< (15/14) in [17]. Laporta’s result was improved by some authors in [4, 18, 19]. Now, the best result for r � 2 is 1< c< (59/44) by Li and Cai [19]. In this paper, we focus on the Diophantine inequality (2) and prove the following result.


Introduction
Suppose that k ≥ 1 is an integer and c > 1 is not an integer. Let ε be a small positive number.
e Waring-Goldbach problem is to study the solvability of the Diophantine equality: in prime numbers p 1 , . . . , p s . In [1], the author studied the analog of the Waring-Goldbach problem. For any sufficiently large real number N, let H(c) denote the smallest natural number s such that the Diophantine inequality, is solvable in prime numbers p 1 , . . . , p s . In [1], the author proved that In [1], the author also obtained that H(c) ≤ 5, for 1 < c < (3/2). Later, the result was improved in [2][3][4][5][6]. Now, the best result for H(c) ≤ 5 is 2 < c < (52/25) by Li and Cai [3].

Some Lemmas
In order to prove our theorem, we need the following lemmas. Lemma 1. Let r be a positive integer. ere exists a function ϕ(y) which is r � [log X] times continuously differentiable and satisfies and its Fourier transformation, satisfies Proof. is can be found in Piatetski-Shapiro [1]. □ Lemma 2. Let a(l) be a sequence of complex numbers; then, for L, Q ≥ 1, we have where z denotes the conjugate of the complex number z.
is is eorem 1 of Baker and Weingartner [6]. □ can be written in O(log 10 X) sums.
Proof. We can obtain Let F(c) � e(xc c ) in Lemma 9; we reduce the estimation of S * * to the estimations of type I sums: and type II sums: and estimate (34) follows from Lemmas 10 and 11. By the inverse Fourier transformation formula, we obtain where (42) en, we have

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.