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

Zhao Feng 1
  • 1 School of Mathematics and Information Science, North China University of Water Resources and Electric Power, Henan 450046, Zhengzhou, China
Zhao Feng
  • Corresponding author
  • School of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou, Henan 450046, China
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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 with |pi(N/j)1/3|N1/3δ+ε(1ij), for some 0<δ190. Furthermore, we give the quantitative relations between the length of short intervals and the size of exceptional sets.

1 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 sufficiently 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, 5≤ j ≤ 8, we could consider the exceptional sets of these problems. Denote by Ej(N) the set of integers n ∈ 𝓐j, not exceeding N with j = 5, 6, 7, 8 such that

np13+p23++pj3,

where

A5:={nN:n1(mod2),n0,±2(mod9),n0(mod7)},A6:={nN:n0(mod2),n±1(mod9)},A7:={nN:n1(mod2),n0(mod9)},A8:={nN:n0(mod2)}.

Hua [1] also proved that Ej(N) ≪ N(log N)A, where A > 0 is arbitrary. In 2000, Ren [2] improved E5(N) ≪ N152/153+ε for j = 5. In 2005, Kumchev [3] proved the following theorem in this realm.

Theorem 1.1

Let j = 5, 6, 7, 8, let 𝓐j be defined as in (2), and define θj by

θ5=79/84,θ6=31/35,θ7=17/28,θ8=23/84.

Then we have

Ej(N)Nθj.

In 2014, Zhao [4] improved the above to θ7 = 1/2+ε, θ8 = 1/6+ε for j = 7, 8. In this paper, if j = 7,8, we investigate this problem with pi taking values in short intervals, i.e.

n=p13+p23++pj3withpi(N/j)1/3y,1ij,

where y = o(N1/3) and pj are primes. Let Ej(N, y) denote the number of integers that NAj,NnN+N23y, which cannot be represented as in (3). In the case of j = 7, 8, our result is stated as follows.

Theorem 1.2

For y=N13δwith0<δ190, we have E8(N,y)N2923θ21,E7(N,y)N2916θ21, where 0 < θ ≤ 1, δ and θ satisfy 3δ+θ = 1.

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 final 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.

When j = 4, it is a conjecture, which is out of reach at present. For j = 9, Lü and Xu [8] proved that (3) holds unconditionally with δ = 1/198, 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 Ej(N, y) for j = 7, 8, 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., 3δ+θ = 1.
  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 fixed results for the number of prime variables, we actually obtain the wider range of short interval. Though the number of primes is different, the results are the same by the method in the paper.
  3. As the number of variables is becoming smaller, the more difficult the question is. For example, when j = 4, 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 = 5, 6, 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 notation AB means that c1ABc2 A, rR means R < r ≤ 2R. The letter ε denotes a positive constant, which is arbitrary small, but not the same at different occurrences.

2 Outline of the method

In this section, we give an outline of the proof of Theorem 1.2 (take j = 8 for example). In order to apply the circle method, for some δ > 0, we set

x=(N/8)1/3,y=N1/3δ+ε,

and

P=N1/21,P=N2/7,Q=N20/21,Q=N31/36+2ε.

By Dirichlet’s lemma ([16], Lemma 2.1), each α ∈ [1/Q, 1+1/Q] may be written in the form

α=a/q+λ,|λ|1/(qQ)

for some integers a, q with 1 ≤ aqQ and (a, q) = 1. Denote by 𝓜(a, q) the set of α satisfying (6), and define the major arcs 𝓜 as follows:

M:=1qP1aq(a,q)=1M(a,q).

Again by Dirichlet’s lemma, each number α ∈ [1/Q, 1+1/Q]∖𝓜 can be written as

α=a/q+λ,|λ|1/(qQ)

with (a, q) = 1, 1 ≤ aqQ. Define the minor arcs C(𝓜) to be the set of α ∈ [1/Q, 1+1/Q]∖ 𝓜 satisfying (8) with PqQ.

Obviously, 𝓜 and C(𝓜) are disjoint. Let 𝓡 be the complement of 𝓜 and C(𝓜) in [1/Q, 1+1/Q], so that

[1/Q,1+1/Q]=MC(M)R.

Let N be a sufficiently large integer and n ∈ 𝓐8 satisfying NnN+N23y. Denote by

r(n):=xypjx+yp13++p83=n(logp1)(logp8)=1/Q1+1/QS8(α)e(αn)dα,

where e(t) = e2πit and

S(α):=xypx+y(logp)e(αp3).

Then we can write

r(n)=1/Q1+1/QS8(α)e(αn)dα=M+C(M)+R.

Clearly, in order to prove Theorem 1.2, it is sufficient to show that r(n) > 0 for almost all integers n ∈ 𝓐8 ∩ [N,N+N2/3y]. 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 ≥ 1, let 2 < yx and α = a/q+ λ be a real number with 1 ≤ aq and (a, q) = 1. Define

Ξ:=|λ|xk+x2y2.

Then for any fixed ε > 0, we have

xypx+y(logp)e(αpk)(qx)εq12yΞ12x12+q12x12Ξ16+y12x310+x45Ξ16+xq12Ξ12,

where the implied constant depends on ε and k only.

Lemma 2.2

Let l be a positive integer, R ≥ 1, T ≥ 1, X ≥ 1 and κ = 1/log X. Then there is an absolute positive constant c such that

rRl|rχmodrTTX<n2XΛ(n)χ(n)nκ+iτdτ(l1R2TX11/20+X)(logRTX)c,

where the implied constant is absolute.

Next we bound S(α) on C(𝓜) ∪ 𝓡. We first 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 ≥ 3 and θ be a real number with (2k+2)/(2k+3) < θ ≤ 1. Suppose that 0 < ρρk(θ), where

ρk(θ)=min(114(2θ1),16(9θ8))if k=3,min(σk6(3θ1),16((2k+3)θ2k2))if k4,

with σk defined by σk1=min(2k1,2k(k2)). Then, for any fixed ε > 0,

sup|x<nx+xθΛ(n)e(αnk)|xθρ+ε+xθ+εP1/2.

Therefore, by Lemma 2.3 for αC(𝓜), we have

S(α)yxρ+ε,

in view of y = xθ, 8/9 < θ ≤1 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 write 𝓡 = 𝓡1 ∪ 𝓡2, where

R1=α:1qP,1qQ<|λ|1qQ,

and

R2α:P<qP,|λ|1qQ.

For α ∈ 𝓡1, we have |λ|1qQ1N1/3y2, and therefore,

Ξ|λ|N+x2y2|λ|N.

Lemma 2.1 gives, for α ∈ 𝓡1,

S(α)Nεyq|λ|NN16+N16q12(|λ|N)16+y12N110+N415(|λ|N)16+N13q|λ|Nyxρ+ε.

If α ∈ 𝓡2, then

P<qP,ΞN2/3y2,qΞNQ1+PN2/3y2.

By Lemma 2.1,

S(α)Nεy(qΞ)12N16+N16q13(qΞ)16+y12N110+N415Ξ16+N13(PΞ)12yxρ+ε.

From (13)-(15), we get

Proposition 2.4

Let C(𝓜) and 𝓡 be defined as the above, then we have

maxαC(M)RS(α)yxρ+ε.

For the major arcs, we have the following asymptotic formula, which will be proved in Section 4.

Proposition 2.5

Let 𝓜 be defined as in (7). Then for any sufficiently large n ∈ 𝓐8 ∩ [N,N+N2/3y], we have

MS8(α)e(αn)dαC8S8(n)N23y7,

where C8 is a positive constant, φ(q) is the Euler function and

S8(n):=q=11φ8(q)a=1(a,q)=1qh=1qeah3q8eanq.

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.

Lemma 2.4

Let k be a positive integer, XY ≥ 2 and

Sk(α):=XYnX+Ye(αnk).

Then for any ε > 0 and 1 ≤ sk, we have

01|Sk(α)|2sdαεXεY2ss.

Proof

This is Lemma 4.1 in Li and Wu [18]. □

Proof of Theorem 1.2

To prove Theorem 1.2, we apply the method introduced by Wooley [19]. We only give detailed proof for j = 8. Denote by E8(N, y) the set of integers n ∈ 𝓐8 ∩ [N,N+N2/3y] such that

np13++p83 with pi(N/8)1/3N1/3δ+ε(1i8).

Introduce the function

Z(α):=NE8(N,y)e(αn).

Clearly, we have

01S8(α)Z(α)dα=0,

and write Card( E8(N, y)) = Z

01|Z(α)|2dα=|E8(N,y)|=Z.

By Proposition 2.5, we have

C(M)RS8(α)Z(α)dα=MS8(α)Z(α)dα=NE8(N,y)MS8(α)e(αn)dαZN23y7.

From (17) and (18), we deduce that

|E8(z)|N23y7C(M)R|S8(α)Z(α)|dαmaxαC(M)R|S(α)|I112I212,

where I1=01|S(α)|6|Z(α)|2dα,I2=01|S(α)|8dα. One could find that (see Lemma 6.2 in [19])

I1yε(y3Z2+y4Z).

And Lemma 2.6 implies

01|S(α)|8dα01|S3(α)|8dαNεy5.

Collecting (16), (19), (20) and noting that x = (N/8)1/3, we obtain

Zy7x2xρ+εy5(Z+y1/2Z1/2)

If ρ is such that y2x2−ρ+ε, this leads to the bound

ZN(42ρ)/3+εy3

which implies ZN2923θ21, by the choice of ρ in Lemma 2.3.

In the case of j = 7, we obtain the following asymptotic formula on major arcs by similar argument as described in Section 4,

MS7(α)e(αn)dαC7S7(n)N23y6.

For j = 7, estimations on C(𝓜) ∪ 𝓡 are also similar to the case of j = 8. The other treatment is quite similar, so we omit the details. This completes the proof of Theorem 1.2. □

3 Preliminaries for Proposition 2.5

For χ mod q, define

C(χ,a):=h=1qχ¯(h)eah3q,C(q,a):=C(χ0,a).

If χ1, χ2,…, χ8 are characters mod q, then we write

B8(n,q;χ1,,χ8)=a=1(a,q)=1qeanqC(χ1,a)C(χ2,a)C(χ8,a),

and

B8(n,q)=B8(n,q;χ0,,χ0).

The following lemma is important for proving Proposition 2.5.

Lemma 3.1

Let χi mod ri with i = 1,…, 8 be primitive characters, r0 = [r1,…, r8], and χ0 be the principal character mod q. Then

qzr0|q1φ8(q)B8(n,q;χ1χ0,,χ8χ0)r03+εlogcz.

Proof

It is similar to that of Lemma 7 in [20], so we omit the details. □

Recall the definition of x, y as in (4), and define

S0(λ):=xynx+ye(λn3),

and

W(χ,λ):=xypx+y(logp)χ(p)e(λp3)δχS0(λ),

where δχ = 1 or 0 according as χ is principal or not. We also set

Wχ:=max|λ|1/(rQ)|W(χ,λ)|,Wχ2:=1/rQ1/rQ|W(χ,λ)|2dλ1/2.

Define further

J(d)=rP[d,r]3+εχmodrWχ,
K(d)=rP[d,r]3+εχmodrWχ2,

where the sum χmodr 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

K(d)d3+εy1/2N1/3Lc.

Lemma 3.3

Let P, Q be as in (2.2). We have

J(d)d3+εyLc.

Further if d = 1, the estimate can be improved to

J(1)yLA,

where A > 0 is arbitrary.

4 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 qP*, we have (p, q) = 1 for p ∈ (xy, x + y]. Using the orthogonality relation, we can write

Saq+λ=C(q,a)φ(q)S0(λ)+1φ(q)χmodqC(χ,a)W(χ,λ),

where S0(λ) and W(χ, λ) are as in (25). By (32), we can write

MS8(α)e(αN)dα=0k8C8kIk,

where

Ik:=1qP1φ8(q)a=1(a,q)=1qC8k(q,a)eaNq1/rQ1/rQS08k(λ)χmodqC(χ,a)W(χ,λ)ke(λN)dλ.

We will prove that I0 produces the main term, and the other Ik (1 ≤ k ≤ 8) contribute to error term.

The computation of I0 is standard, and we can prove

I0=C8S8(n)N23y7{1+o(1)},

where C8 and 𝔖8(n) are defined in Proposition 2.5.

It remains to estimate Ik (1 ≤ k ≤ 8). We shall only treat I8, the most complicated one. The treatment for Ik (1 ≤ k ≤ 7) are similar.

I8=1qP1φ8(q)a=1(a,q)=1qeanq1/rQ1/rQχmodqC(χ,a)W(χ,λ)8e(λn)dλ=1qPχ1modqχ8modq1φ8(q)a=1(a,q)=1qC(χ1,a)C(χ8,a)eanq×1/rQ1/rQW(χ1,λ)W(χ8,λ)e(λn)dλ.

Suppose that χk (mod rk) with rk|q being the primitive character inducing χk. Thus we may write χk = χk χ0, where χ0 is the principal character modulo q, r0 = [r1, …, r8]. It is easy to see that W(χk, λ) = W(χk, λ). Since r0 = [r1, …, r8] = [[r1, …, r7], r8], by Lemma 3.1 and Cauchy’s inequality, we have

|I8|Lcr1Pχ1modr1Wχ1r6Pχ6modr6Wχ6×r7Pχ7modr7Wχ72r8Pr03+εχ8modr8Wχ82.

Now we introduce an iterative procedure to bound the above sums over r8, …, r1 consecutively. Since r0 = [r1, …, r8] = [[r1, …, r7], r8], we use (29) two times, (30) five times and (31) once to get

|I8|Lcy1/2N1/3r1Pχ1modr1Wχ1r6Pχ6modr6Wχ6×r7P[r1,,r7]3+εχ7modr7Wχ72LcN2/3y6r1Pr13+εχ1modr1Wχ1LAN2/3y7,

for any fixed A > 0.

Following a similar procedure to treat I8, we can show that

|Ik|LAN2/3y7,(1k7).

Now the required asymptotic formula follows from (33), (34), (35) and (36).□

5 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 effectively, we need a preliminary lemma in [18] as follows.

Lemma 5.1

Let χ be a Dirichlet character modulo r. Let 2 ≤ X < Y ≤ 2X, T0 = (log(Y/X))−1, T = X6 and κ = 1/log X. Define

F(s,χ):=Xn2XΛ(n)χ(n)ns.

Then we have

XnYΛ(n)χ(n)logYX|τ|T0|F(κ+iτ,χ)|dτ+T0<|τ|T|F(κ+iτ,χ)||τ|dτ+1.

The implied constant is absolute.

Proof of Lemma 3.2

Introduce

W~(χ,λ):=xynx+yΛ(n)χ(n)e(λn3)δχS0(λ).

Then we have

W~(χ,λ)W(χ,λ)N1/6,

which implies

Wχ2≪∥W~χ2+N1/6rQ1/2.

The contribution of O(N1/6(r/Q*)1/2) to the left-hand side of (29 ) is

N1/6rP[d,r]3+εr1/2(Q)1/2d3+εN1/6Q1/2rPrl3+εr1/2d3+εN1/6Q1/2l|dlPl3+εrPl|rr5/2+εd3+εy1/2N1/3Lc,

where we have used [d, r](d, r) = dr, l = (d, r), (4) and (5). Thus in order to prove (29), it suffices to show that

rR[d,r]3+εχmodrW~χ2d3+εy1/2N1/3Lc

for any RP*.

By Gallagher’s lemma ([21], Lemma 1), we have

W~χ21RQ+|vRQ/3n3v+RQ/3xy<nx+yΛ(n)χ(n)δχ|2dv1/21RQ(xy)3RQ/3(x+y)3+RQ/3|XnYΛ(n)χ(n)δχ|2dv1/2,

where

X:=max{(vRQ/3)1/3,xy},Y:=min{(v+RQ/3)1/3,x+y}.

If R = 1, we have

|XnYΛ(n)χ(n)δχ|=|XnY(Λ(n)1)|(YX)L{(v+Q/3)1/3(xy)}LQN2/3L,

which implies, in view of Q* < x2y,

d3+εW~χ2d3+εQ1(QN2/3L)2(x2y+Q)1/2d3+εy1/2N1/3Lc.

For R ≥ 2 and rR, we have δχ = 0. Thus, we can apply (37) to write

W~χ2yx41/2|τ|T0|F(κ+iτ,χ)|dτ+(x2y)1/2RQT0<|τ|T|F(κ+iτ,χ)||τ|dτ+(x2y)1/2RQ.

Since

T01=log(Y/X)RQv1RQx3,

and

(x+y)3+RQ/3(xy)3+RQ/3x2y.

Therefore, the contribution of the first term of (41) to the left-hand side of (38) is

d3+ε(x4y)1/2l|dlRRl3+ε(l1R2T0x11/20+x)d3+εN1/3y1/2(N17/20Q1+1)Lcd3+εN1/3y1/2Lc.

Introducing

M(l,R,T,x):=rRl|rχmodrT2T|F(κ+iτ,χ)|dτ.

The contribution of the second term of (41) to the left-hand side of (38) is

d3+ε(x2y)1/2(RQ)1l|dlRRl3+εmaxT0TTT1M(l,R,T,x)d3+ε(x2y)1/2(RQ)1l|dlRRl3+ε(l1R2x11/20+T01x)Lcd3+εN1/3y1/2(N17/20Q1+1)Lcd3+εN1/3y1/2Lc.

Finally, the contribution of the last term of (41) to the left-hand side of (38) is

d3+ε(x2y)1/2(RQ)1l|dlRrRl|rrl3+εd3+εN1/3y1/2Q1N2/3d3+εN1/3y1/2Lc.

Now the inequality (38) follows from (40), (42), (43) and (44). This completes the proof of Lemma 3.2.□

6 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 differences.

Estimation of J(d). Replacing W(χ, λ) by (χ, λ) as in §5, we get that the resulting error is

rP[d,r]3+εrN1/6d3+εN1/6l|dlPr3εrPl|rl3+εd3+εN1/6P3d3+εyLc.

Hence, Lemma 3.3 is a consequence of the estimate

rP[d,r]3+εχmodrmax|λ|1/(rQ)W~(χ,λ)d3+εyLc,

where RP* and c > 0 is some constant.

The case R < 1 contributes to d−3+εyL which is obviously acceptable. For R ≥ 1, we have δχ = 0. Thus,

W~(χ,λ)=xymx+yΛ(m)χ(m)e(m3λ).

Define

H(s,χ)=xymx+yΛ(m)χ(m)ms,V(s,λ)=xyx+yws1e(λw3)dw.

By partial summation and Perron’s summation formula, we get

W~(χ,λ)=12πibiTb+iTH(s,χ)V(s,λ)ds+O(1),

where 0 < b < L−1 and T = (1 + | λ | N)yL2. Using Lemmas 4.3, 4.5 of [22] and a trivial estimate, we have

V(s,λ)Nb/3minyN1/3,1|t|+1,maxxywx+y1|t+6πλw3|.

Take

T^=N2/3y2,T=12πNRQ.

Then for b → 0, (χ,λ) is bounded by

W~(χ,λ)yN1/3|t|T^|H(it,χ)|dt+T^|t|T|H(it,χ)|dt|t|+1+RQNT|t|T|H(it,χ)|dt+O(1)

Thus, it suffices to show that the estimates

rR[d,r]3+εχmodrT12T1|H(it,χ)|dtd3+εN1/3Lc

holds for RP* and 0 < T1;

rR[d,r]3+εχmodrT22T2|H(it,χ)|dtd3+εU(T2+1)1/2Lc

holds for RP* and < T2T*; and

rR[d,r]3+εχmodrT32T3|H(it,χ)|dtd3+εN(RQ)1ULc

holds for RP* and T* < T3T.

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 first part of Lemma 3.3 is proved.

Estimation of J(1). The result is the same as that of J(d) except for the saving of LA on its right hand side. To order to get this saving, we have to distinguish two cases LC < RP and RLC, where C is a constant depending on A. The proof of the first case is the same as that of J(d), so we omit the details.

Now we prove the second case RLC. We use the well-known explicit formula

muΛ(m)χ(m)=δχu|γ|Tuρρ+O(uT+1)log2(ruT),

where ρ = β + iγ is a non-trivial zero of the function L(s, χ), and 2 ≤ Tu is a parameter. Then by inserting (44) into (χ, λ), and applying partial summation formula, we get

W^(χ,λ)=xyx+ye(u3λ)d{xy<nuΛ(m)χ(m)δχ}=N1N2e(u3λ)|γ|Tuρ1du+O(1+|λ|N)yT1L2y|γ|TNβ13+O(NyQ1T1L2).

Now let η(T) = c2 log−4/5T. By Prachar [23], χmodr L(s, χ) is zero-free in the region σ ≥ 1 − η(T), |t| ≤ T except for the possible Siegel zero. But by Siegel’s theorem (see [24], section 21), the Siegel zero does not exist in the present situation, since rRLC. Thus by the large-sieve type zero-density estimates for Dirichlet L-functions (see [25]), we have

rRχmodr|γ|TNβ13Lc01η(T)T12(1α)5Nα13dαLc01η(T)N12(1α)ε5dαLcN12η(T)ε5exp(c3εL1/5)

provided that T=N536ε. Consequently,

rP[d,r]3+εχmodrmax|λ|1/(rQ)W~(χ,λ)d3+εyLA

provided that Q* = N31/36+2ε. Then the lemma follows.

Acknowledgement

This work is supported by the National Natural Science Foundation of China (Grant No. 11371122, 11471112). The author would like to express thanks to the referees for reading the manuscript carefully and giving suggestions professionally. Theorem 1.2 is actually improved by the valuable advices. The author also express thanks to Doctor Ge Wenxu, because of the helpful discussion during writing the paper.

References

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    Hua L.K., Some results in the additive prime number theory, Quart. J. Math. (Oxford) 1938, 9, 68-80.

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    Ren X.M., The Waring-Goldbach problem for cubes, Acta Arith, 2000, 94, 287-301.

    • Crossref
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    Kumchev A.V., On the Waring-Goldbach problem: exceptionalal sets for cubes and higher powers, Cana. J. Math., 2005, Vol. 57 (2), 298-327.

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    Zhao L.L., On the Waring-Goldbach problem for fourth and sixth powers, Proc London Math Soc., 2014, 108, 1593-1622.

    • Crossref
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    Liu J.Y., On Lagrange’s theorem with prime varibles, Quart. J. Math. (Oxford) 2003, 54, 454-462.

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    Choi S.K.K., Kumchev A.V., Mean valus of Dirichlet polynimials and applications to linear equations with prime varibles, Acta Arith. 2006, 123, 125-142.

    • Crossref
    • Export Citation
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    Kumchev A.V., On Weyl sums over primes in short intervals,"Number Theory: Arithmetic in Shangri-La", World Scientific, 2013, 116-131.

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    Wei B., Wooley T.D., On sums of powers of almost equal primes, Proc. London Math. Soc., 2015, 111 (3), 1130-1162.

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    • Crossref
    • Export Citation
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    Liu H.F., Kumchev A.V., On sums of powers of almost equal primes, J. Number Theory, 2017, 176, 344-364.

    • Crossref
    • Export Citation
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    Lü G.S., On sums of a prime and four prime squares in short intervals, J. Number Theory, 2008, 128, 805-819.

    • Crossref
    • Export Citation
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    Lü G.S., On sums of nine almost equal prime cubes, Acta Math. Sin., 2006, 49, 195-204 (in Chinese).

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    Liu Z.X., Sun Q.F., Sums of cubes of primes in short intervals, The Ramanujan Journal, 2012, 28 (3), 309-321.

    • Crossref
    • Export Citation
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    Ren X.M., Yao Y.J., Exceptional set for sums of almost equal prime cubes (in Chinese), Sic Sin Math, 2015, 45(1), 23-30.

    • Crossref
    • Export Citation
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    Vaughan R.C., The Hardy-Littlewood Method, 2nd edition, Cambridge University Press, 1997.

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    Liu J.Y., Lü G.S., Zhan T., Exceptional sums over prime varibles in short intervals, Sci. China Ser. 2006, A 49, no. 4, 448-457.

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    Li H.Z., Wu J., Sums of almost equal squares, Funct. Approx. Comment. Math., 2008, 38 (1), 49-65.

    • Crossref
    • Export Citation
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    Wooley T.D., Slim exceptional sets for sums of cubes, Canad. J. Math., 2002, 54, (2), 417-448.

    • Crossref
    • Export Citation
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    Leung M.C., Liu M.C., On generalized quadratic equations in three prime varibles, Monatsh. Math., 1993, 115, 133-169.

    • Crossref
    • Export Citation
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    Gallagher P.X., A large sieve density estimate near σ =1, Invent. Math., 1970, 11, 329-339.

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    Titchmarsh E.C., The Theory of the Riemann zeta-function. New York: Oxford University press, 1986.

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    Prachar K., Primzahlverteilung. Springer, Berlin, 1957.

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    Davenport H., Multiplicative Number Theory. Springer, Berlin, 1980.

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    Huxley M.N., Large values of Dirichlet polynomials III, Acta Arith., 1974, 26, 435-444.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    Hua L.K., Some results in the additive prime number theory, Quart. J. Math. (Oxford) 1938, 9, 68-80.

  • [2]

    Ren X.M., The Waring-Goldbach problem for cubes, Acta Arith, 2000, 94, 287-301.

    • Crossref
    • Export Citation
  • [3]

    Kumchev A.V., On the Waring-Goldbach problem: exceptionalal sets for cubes and higher powers, Cana. J. Math., 2005, Vol. 57 (2), 298-327.

    • Crossref
    • Export Citation
  • [4]

    Zhao L.L., On the Waring-Goldbach problem for fourth and sixth powers, Proc London Math Soc., 2014, 108, 1593-1622.

    • Crossref
    • Export Citation
  • [5]

    Liu J.Y., On Lagrange’s theorem with prime varibles, Quart. J. Math. (Oxford) 2003, 54, 454-462.

  • [6]

    Choi S.K.K., Kumchev A.V., Mean valus of Dirichlet polynimials and applications to linear equations with prime varibles, Acta Arith. 2006, 123, 125-142.

    • Crossref
    • Export Citation
  • [7]

    Kumchev A.V., On Weyl sums over primes in short intervals,"Number Theory: Arithmetic in Shangri-La", World Scientific, 2013, 116-131.

  • [8]

    Lü G.S., Xu Y.F., Hua’s Theorem with nine almost equal prime variables, Acta Math. Hungar., 2007, 116(4), 309-326.

    • Crossref
    • Export Citation
  • [9]

    Wei B., Wooley T.D., On sums of powers of almost equal primes, Proc. London Math. Soc., 2015, 111 (3), 1130-1162.

  • [10]

    Huang B.R., Exponential sums over primes in short intervals and an application to the Waring-Goldbach Problem, Mathematika, 2016, 62, 508-523.

    • Crossref
    • Export Citation
  • [11]

    Liu H.F., Kumchev A.V., On sums of powers of almost equal primes, J. Number Theory, 2017, 176, 344-364.

    • Crossref
    • Export Citation
  • [12]

    Lü G.S., On sums of a prime and four prime squares in short intervals, J. Number Theory, 2008, 128, 805-819.

    • Crossref
    • Export Citation
  • [13]

    Lü G.S., On sums of nine almost equal prime cubes, Acta Math. Sin., 2006, 49, 195-204 (in Chinese).

  • [14]

    Liu Z.X., Sun Q.F., Sums of cubes of primes in short intervals, The Ramanujan Journal, 2012, 28 (3), 309-321.

    • Crossref
    • Export Citation
  • [15]

    Ren X.M., Yao Y.J., Exceptional set for sums of almost equal prime cubes (in Chinese), Sic Sin Math, 2015, 45(1), 23-30.

    • Crossref
    • Export Citation
  • [16]

    Vaughan R.C., The Hardy-Littlewood Method, 2nd edition, Cambridge University Press, 1997.

  • [17]

    Liu J.Y., Lü G.S., Zhan T., Exceptional sums over prime varibles in short intervals, Sci. China Ser. 2006, A 49, no. 4, 448-457.

  • [18]

    Li H.Z., Wu J., Sums of almost equal squares, Funct. Approx. Comment. Math., 2008, 38 (1), 49-65.

    • Crossref
    • Export Citation
  • [19]

    Wooley T.D., Slim exceptional sets for sums of cubes, Canad. J. Math., 2002, 54, (2), 417-448.

    • Crossref
    • Export Citation
  • [20]

    Leung M.C., Liu M.C., On generalized quadratic equations in three prime varibles, Monatsh. Math., 1993, 115, 133-169.

    • Crossref
    • Export Citation
  • [21]

    Gallagher P.X., A large sieve density estimate near σ =1, Invent. Math., 1970, 11, 329-339.

    • Crossref
    • Export Citation
  • [22]

    Titchmarsh E.C., The Theory of the Riemann zeta-function. New York: Oxford University press, 1986.

  • [23]

    Prachar K., Primzahlverteilung. Springer, Berlin, 1957.

  • [24]

    Davenport H., Multiplicative Number Theory. Springer, Berlin, 1980.

  • [25]

    Huxley M.N., Large values of Dirichlet polynomials III, Acta Arith., 1974, 26, 435-444.

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