Quantitative relations between short intervals and exceptional sets of cubic Waring-Goldbach problem

Abstract In this paper, we are able to prove that almost all integers n satisfying some necessary congruence conditions are the sum of j almost equal prime cubes with j = 7, 8, i.e., N=p13+…+pj3 $\begin{array}{} N=p_1^3+ \ldots +p_j^3 \end{array} $ with |pi−(N/j)1/3|≤N1/3−δ+ε(1≤i≤j), $\begin{array}{} |p_i-(N/j)^{1/3}|\leq N^{1/3- \delta +\varepsilon} (1\leq i\leq j), \end{array} $ for some 0<δ≤190. $\begin{array}{} 0 \lt \delta\leq\frac{1}{90}. \end{array} $ Furthermore, we give the quantitative relations between the length of short intervals and the size of exceptional sets.


Introduction
In the Waring-Goldbach problem, one studies the representation of positive integers by powers of j primes, where j is a positive integer. One of the most famous results of Hua [1] in 1938 states that each su ciently large odd integer n can be written as the sum of nine cubes of primes.
When the number of variables j is becoming smaller, such as, ≤ j ≤ , we could consider the exceptional sets of these problems. Denote by E j (N) the set of integers n ∈ A j , not exceeding N with j = , , , such that Hua [1] also proved that E j (N) N(log N) −A , where A > is arbitrary. In 2000, Ren [2] improved E (N) N / +ε for j = . In 2005, Kumchev [3] proved the following theorem in this realm. Theorem 1.1. Let j = , , , , let A j be de ned as in (2), and de ne θ j by θ = / , θ = / , θ = / , θ = / .

Z. Feng
Then we have E j (N) N θj .
In 2014, Zhao [4] improved the above to θ = / + ε, θ = / + ε for j = , . In this paper, if j = , , we investigate this problem with p i taking values in short intervals, i.e. n = p + p + . . . + p j with where y = o(N / ) and p j are primes. Let E j (N, y) denote the number of integers that N ∈ A j , N ≤ n ≤ N + N y, which cannot be represented as in (3). In the case of j = , , our result is stated as follows. , where < θ ≤ , δ and θ satisfy δ + θ = .
Theorem 1.2 is proved by the circle method. When treating the major arcs, we apply the iterative method of Liu [5] and a mean value theorem of Choi and Kumchev [6] to establish the asymptotic formula. It is known that the estimation of exponential sums plays an essential role in this problem. The upper bound of the exponential sums leads to the nal result directly. So the new estimate for exceptional sums over primes in short interval in Kumchev [7] plays an important role in treating the minor arcs.
For j = , Lü and Xu [8] proved that (3) holds unconditionally with δ = / , which is as strong as the result under the Generalized Riemann Hypothesis. In the direction of Waring-Goldbach problem in short intervals, there have been some developments in the last few years. Such as the recent work of Wei and Wooley [9], Huang [10] and Kumchev and Liu [11]. There are other similar problems (see [12,13] and their references).

Remarks:
1. In this series of problems, we could not only focus our attention to the size of y, but also concern with the cardinality of E j (N, y) for j = , , such as Theorem 2 in Liu and Sun [14]. In this paper, we give the exact relation formula about the length of short intervals and the size of exceptional sets. Compared with Theorem 2 in [14], a wider range of the length of short intervals is given. At the same time, quantitative relation between size of exceptional sets and length of short intervals is obtained, i.e., δ + θ = .
2. This paper is focusing on the quantitative relation of short intervals and the exceptional sets. In the reference [15], improved results in shorter intervals are given. Compared with their xed results for the number of prime variables, we actually obtain the wider range of short interval. Though the number of primes is di erent, the results are the same by the method in the paper.
3. As the number of variables is becoming smaller, the more di cult the question is. For example, when j = , it is a conjecture out of reach at present. Moreover, as the length of short intervals is becoming shorter, the size of exceptional sets is becoming larger. When j = , , this method does not work for this question, so we could not obtain similar results.
Notation. As usual, φ(n) and Λ(n) stand for the functions of Euler and von Mangoldt, respectively. The letter N is a large integer, and L = log N.
The letter ε denotes a positive constant, which is arbitrary small, but not the same at di erent occurrences.

Outline of the method
In this section, we give an outline of the proof of Theorem 1.2 (take j = for example). In order to apply the circle method, for some δ > , we set and By Dirichlet's lemma ( [16], Lemma 2.1), each α ∈ [ /Q * , + /Q * ] may be written in the form for some integers a, q with ≤ a ≤ q ≤ Q * and (a, q) = . Denote by M(a, q) the set of α satisfying (6), and de ne the major arcs M as follows: Again by Dirichlet's lemma, each number α ∈ [ /Q * , + /Q * ]\M can be written as Let N be a su ciently large integer and n ∈ A satisfying N ≤ n ≤ N + N y. Denote by where e(t) = e πit and Then we can write Clearly, in order to prove Theorem 1.2, it is su cient to show that r(n) > for almost all integers n ∈ A ∩ [N, N + N / y]. The tools that we need are an estimate for exponential sums over primes in short intervals of Liu, Lü and Zhan [17], Kumchev [7] and a mean value theorem of Choi and Kumchev [6], which are stated as follows.
Lemma 2.1. For integer k ≥ , let < y ≤ x and α = a/q + λ be a real number with ≤ a ≤ q and (a, q) = .
De ne Then for any xed ε > , we have where the implied constant depends on ε and k only.
where the implied constant is absolute.
Next we bound S(α) on C(M) ∪ R. We rst estimate S(α) on C(M), and this has been done in Kumchev [7] in which Theorem 1.2 states that

Lemma 2.3. Let k ≥ and θ be a real number with
in view of y = x θ , / < θ ≤ and our choice of P in (5). Now we estimate S(α) on R. To this end, also by Dirichlet's lemma on rational approximation, we further For α ∈ R , we have |λ| ≥ qQ * ≥ N / y , and therefore, If α ∈ R , then By Lemma 2.1, From (13) For the major arcs, we have the following asymptotic formula, which will be proved in Section 4.
Proposition 2.5. Let M be de ned as in (7). Then for any su ciently large n ∈ A ∩ [N, where C is a positive constant, φ(q) is the Euler function and In order to prove Theorem 1.2, we also need the following lemma, which can be viewed as a generalization of Hua's lemma ( [16], Lemma 2.5 ) in short intervals. Then for any ε > and ≤ s ≤ k, we have Proof. This is Lemma 4.1 in Li and Wu [18].
From (17) and (18), we deduce that One could nd that (see Lemma 6.2 in [19]) And Lemma 2.6 implies Collecting (16), (19), (20) and noting that x = (N/ ) / , we obtain If ρ is such that y x −ρ+ε , this leads to the bound In the case of j = , we obtain the following asymptotic formula on major arcs by similar argument as described in Section 4, For j = , estimations on C(M) ∪ R are also similar to the case of j = . The other treatment is quite similar, so we omit the details. This completes the proof of Theorem 1.2.
The following lemma is important for proving Proposition 2.5. Proof. It is similar to that of Lemma 7 in [20], so we omit the details.
Recall the de nition of x, y as in (4) where δχ = or 0 according as χ is principal or not. We also set De ne further where the sum * χ mod r denotes summation for all primitive characters modulo r. The proof of Proposition 2.5 depends on the following two lemmas, which will be proved in Section 5.

Lemma 3.2.
Let P * , Q * be as in (2.2). We have Lemma 3.3. Let P * , Q * be as in (2.2). We have Further if d = , the estimate can be improved to where A > is arbitrary .

Proof of Proposition 2.5
With Lemmas 3.2 and 3.3 known, we can use the iterative idea in Liu [5] to prove Proposition 2.5.
Proof of Proposition 2.5. Since q ≤ P * , we have (p, q) = for p ∈ (x − y, x + y]. Using the orthogonality relation, we can write where S (λ) and W(χ, λ) are as in (25). By (32), we can write where We will prove that I produces the main term, and the other I k ( ≤ k ≤ ) contribute to error term. The computation of I is standard, and we can prove where C and S (n) are de ned in Proposition 2.5. It remains to estimate I k ( ≤ k ≤ ) . We shall only treat I , the most complicated one. The treatment for I k ( ≤ k ≤ ) are similar.
Suppose that χ * k (mod r k ) with r k |q being the primitive character inducing χ k . Thus we may write χ k = χ * k χ , where χ is the principal character modulo q, r = [r , . . . , r ]. It is easy to see that W(χ k , λ) = W(χ * k , λ).
for any xed A > . Following a similar procedure to treat I , we can show that Now the required asymptotic formula follows from (33), (34), (35) and (36).

Estimation of K(d)
The proofs of Lemmas 3.2 and 3.3 are rather similar to those of Proposition 2.2 in [18]. In order to use Choi and Kumchev's mean value theorem e ectively, we need a preliminary lemma in [18] as follows. Then we have The implied constant is absolute.
For R ≥ and r ∼ R, we have δχ = . Thus, we can apply (37) to write Since Therefore, the contribution of the rst term of (41) to the left-hand side of (38) is Introducing The contribution of the second term of (41) to the left-hand side of (38) is Finally, the contribution of the last term of (41) to the left-hand side of (38) is Now the inequality (38) follows from (40), (42), (43) and (44). This completes the proof of Lemma 3.2.

Estimation of J(d)
In this section, we establish Lemma 3.3. The idea of the proof is similar to that of Lemma 3.2, but there are several di erences.

Estimation of J(d).
Replacing W(χ, λ) by W(χ, λ) as in § , we get that the resulting error is Hence, Lemma 3.3 is a consequence of the estimate where R ≤ P * and c > is some constant. The case R < contributes to d − +ε yL which is obviously acceptable. For R ≥ , we have δχ = . Thus, W(χ, λ) = By partial summation and Perron's summation formula, we get where < b < L − and T = ( + |λ|N)yL . Using Lemmas . , . of [22] and a trivial estimate, we have holds for R ≤ P * andT < T ≤ T * ; and holds for R ≤ P * and T * < T ≤ T. The estimates (49), (50) and (51) follow from Lemma 2.2 via an argument similar to that leading to (43), so we omitted the details. The rst part of Lemma 3.3 is proved.
Estimation of J( ). The result is the same as that of J(d) except for the saving of L −A on its right hand side. To order to get this saving, we have to distinguish two cases L C < R ≤ P and R ≤ L C , where C is a constant depending on A. The proof of the rst case is the same as that of J(d), so we omit the details. Now we prove the second case R ≤ L C . We use the well-known explicit formula where ρ = β + iγ is a non-trivial zero of the function L(s, χ), and ≤ T ≤ u is a parameter. Then by inserting (44) intoŴ(χ, λ), and applying partial summation formula, we get W(χ, λ) d − +ε yL −A provided that Q * = N / + ε . Then the lemma follows.