Exceptional Sets for Sums of Prime Cubes in Short Interval

'eWaring–Goldbach problem is to study the representation of positive integers as sums of powers of prime numbers. In this paper, we shall focus on the cubic Waring–Goldbach problem.'is topic can be traced back to the work of Hua [1]. He proved that almost all integers satisfying certain congruence conditions can be written as s cubes of primes, where s � 5, 6, 7, 8, and the abovementioned congruence conditions are


Introduction
e Waring-Goldbach problem is to study the representation of positive integers as sums of powers of prime numbers. In this paper, we shall focus on the cubic Waring-Goldbach problem. is topic can be traced back to the work of Hua [1]. He proved that almost all integers satisfying certain congruence conditions can be written as s cubes of primes, where s � 5, 6, 7, 8, and the abovementioned congruence conditions are N 5 � n ∈ N: n≢1 mod2, n≢0, ± 2 mod9, n ≢ 0mod7 { }, Let E s (N) denote the number of positive numbers n ∈ N s not exceeding N and cannot be represented as s cubes of primes; Hua showed that, for any A > 0, (2) roughout, we assume that N is a large natural number, and X < N. In this paper, we consider the exceptional set of even integers n in the short interval N − X ≤ n ≤ N, which cannot be represented as sum of eight cubes of primes. Precisely, let ξ(N, X) denote the set of natural numbers n ∈ N 8 with N − X ≤ n ≤ N, such that n cannot be written as the following expression: where p i are primes. Moreover, we set e purpose of this paper is to obtain E(N, X) � o(X) with X as small as possible. Zhao [2] proved that . e main result in the paper is as follows. Theorem 1. Let θ > 1/36 and E(N, X) be defined as above. For X > N θ , we have We shall prove eorem 1 by means of the Hardy-Littlewood method. e treatment of the integrals on the major arcs is standard, and we will focus on the treatment of the integrals on the minor arcs.
In this paper, we make use of the method of Vaughan [3] to deal with the equation x 3 1 + y 3 1 + y 3 2 + z 1 � x 3 2 + y 3 3 + y 3 4 + z 2 (see Lemma 3). We also make use of the result of Zhao [2] on the 10 th moment of the Weyl sums over cubes of primes restricted on minor arcs.
Notation. roughout the paper, ε denotes a sufficiently small positive number, and let A, c denote a positive constant. We need to point out that A, c and ε are allowed to change at different occurrences. With or without subscript, p denotes a prime number. Denote by d(n) the number of divisors of n, and as usual, we write e(x) for e 2πix .

Preliminaries
Before we prove eorem 1, we introduce the following theorem.

Theorem 2.
Let E � E(N, N 5/6 ) be defined in Section 1. en, we have We can get eorems 1 from 2 immediately. erefore, our task is to prove eorem 2. Let Write Let R(n) be the weighted number of solutions of n � p 3 1 + · · · + p 3 6 + p 3 7 + p 3 8 with U ≤ p 1 , . . . , p 6 ≤ 2U and U 5/6 ≤ p 7 , p 8 ≤ 2U 5/6 . By orthogonality, one has We write We define the set of major arcs M as the union of the intervals with 1 ≤ a ≤ q ≤ P and (a, q) � 1. We then denote the corresponding set of minor arcs by m � (Q − 1 , 1 + Q − 1 ]\M. So, we can get the lemma.
We will prove Lemma 1 in Section 4.

Lemma 3.
Let S be the number of solutions of with U ≤ x i ≤ 2U, U 5/6 ≤ y i ≤ 2U 5/6 , and z i ∈ ξ(N, N 5/6 ). We have We will prove Lemma 3 in Section 3.
Proof of eorem 2. On recalling the set ξ(N, N 5/6 ) defined in Section 1 and by means of an argument of Wooley (see [4]), we have where So, one has With the help of Lemma 1, we can get With an application of the Cauchy-Schwarz inequality, one has By Lemma 2, we have By considering the underlying Diophantine equations and Lemma 3, we get From (20) to (23), we have erefore, we conclude that 2 Journal of Mathematics is completes the proof of eorem 2.

Proof of Lemma 3
We write S 1 as the number of solutions of (14) with x 3 1 � x 3 2 , and let S 2 denote the number of solutions of (14) with It is easy to find that Lemma 4. We denote by T(k) the number of solutions of the Diophantine equation with 1 ≤ y i ≤ U 5/6 and k ≠ 0. en, we have e proof of Lemma 1 can follow from an argument of Hooley [5] (see the proof of Lemma 1 of Parsell [6] for a sketch of the necessary adjustments to Hooley's argument).

Lemma 5.
Suppose that |α − a/q| ≤ q − 2 and (a, q) Moreover, we can get the following lemma. Lemma 6. Let N be denoted as above. en, we have e proof can be also found in Vaughan [3].
Proof of Lemma 3. When y 3 1 + y 3 2 + z 1 � y 3 3 + y 3 4 + z 2 , the number R of solutions of it satisfies Given any one of the O(E 2 ) possible choices for z 1 and z 2 with z 1 ≠ z 2 , it follows from Lemma 4 that the number of permissible choices for y i is O(U 55/36+ε ), and thus the contribution arising from this class of solutions is O(U 55/36+ε E 2 ). When z 1 � z 2 , on the other hand, the variables y i satisfy the equation y 3 1 + y 3 2 � y 3 3 + y 3 4 ; the number of the solutions is O(U 5/3+ε ). us, we conclude that the number of solutions of this type is O(U 5/3+ε E). So, we can get (37) en, we have We consider n. By Dirichlet's theorem on Diophantine approximation, given α ∈ n, we may choose a, q with (a, q) � 1, |α − a/q| ≤ q − 1 U − 3/2 and q ≤ U 3/2 . It is easily verified that 1 ≤ a ≤ q. Moreover, since α ∉ N, we have q > U. erefore, by Lemma 5, we have Hence, we have When α ∈ N, by Lemma 6, we can get By (40) and (41), In view of (38) and (42), we complete the proof of Lemma 3.

Proof of Lemma 1
Let L � (log U) A for A > 0. We write We define the set of major arcs M ′ as the union of the intervals with 1 ≤ a ≤ q ≤ P and (a · q) � 1. It is easy to find that M ′ ⊆M. We then denote the corresponding set of minor arcs

Lemma 7.
Suppose that the integer n satisfies N/2 ≤ n ≤ N and n ∈ N 6 . en, one has e proof can be found in ( [3], Lemma 3.1). We define the multiplicative function w(q) by taking We have erefore, we have the following result.
Lemma 8. Let c be a constant. When m ≥ 2, one has for some A constant.
is is due to Zhao ([2], Lemma 1). Before we deal with the integrals on the minor arcs, we should give an upper bound of g(α). We also quote the following estimate proved by Ren [7].

(50)
Moreover, one has n − p 3 ∈ N 6 when n ∈ N 7 , and when N − 2N 5/6 ≤ n ≤ N, we have N/2 ≤ n − p 3 ≤ N, and by Lemma 7, we have By same measure, when N − N 5/6 ≤ n ≤ N and n ∈ N 8 , we can get When α ∈ m ′ ∩ M, by Lemma 9, we have Since q < U 5/6 , we have for some absolute constant A.
In view of (52) and (57), we complete the proof of Lemma 1.

Data Availability
Except references, no data were used to support this study.

Conflicts of Interest
e author declares no conflicts of interest.