Front. Math. China 2013, 8(6): 1407–1423 DOI 10.1007/s11464-013-0329-3

Waring-Goldbach problem for fourth powers in short intervals Hengcai TANG,

Feng ZHAO

1 School of Mathematics, Henan University, Kaifeng 475004, China 2 College of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450011, China c Higher Education Press and Springer-Verlag Berlin Heidelberg 2013 

Abstract We prove that almost all integers N satisfying some necessary congruence conditions are the sum of almost equal fourth prime powers. Keywords Circle method, exponential sum, short interval MSC 11P32, 11P05, 11N36, 11P55 1

Introduction

In the Waring-Goldbach problem, one studies the representation of positive integers by powers of primes, i.e., n = pk1 + pk2 + · · · + pks

(1)

is solvable or almost solvable in primes pj for all sufficiently large positive integers n satisfying some congruence conditions. Since the work of Hua [4] in 1938, many authors have considered the corresponding problems with restrictive conditions posed on the prime variables in (1). In 2008, Li and Wu got the following result. Theorem [7] Let j = 3 or 4. Suppose that the integer n satisfies some necessary congruence conditions. Then for any fixed ε > 0, the equation   n 1/2  1   n = p21 + · · · + p2j with pi −   n 2 −δ (1  i  j) j with δ =

9 80

− ε is solvable for almost all integers n.

In 2009, the second author considered the case k = 3, and obtained the following result. Received March 4, 2012; accepted July 8, 2013 Corresponding author: Feng ZHAO, E-mail: [email protected]

1408

Hengcai TANG, Feng ZHAO

Theorem [13] Let j = 5, 6, 7 or 8. Then for any fixed ε > 0, the equation n = p31 + · · · + p3j

with

with δj =

  n 1/3  1   pi −   n 3 −δj (1  i  j) j j −4 −ε 15(j − 2)

is solvable for almost all integers n, which satisfies some necessary congruence conditions. In this papaer, we consider the case k = 4 and give the following theorem. Theorem 1 Let 9  s  13 be an integer, and let N be a sufficiently large number. Denote N4,s := {n ∈ N : n ≡ s (mod 240)}, and define δs =

s−8 . 8(s + 88)

Then the equation ⎧ ⎨ n = p41 + · · · + p4s ,   1/4   ⎩ pi − n   U, s

1  i  s,

(2)

holds for almost all n ∈ N4,s satisfying N  n  N + N 3/4 U, where

1

U = N 4 −δs +ε

for any ε > 0. In Theorem 1, only the size of U is investigated. One can also consider the better estimates for the exception but with larger U. Our result is proved by circle method and it depends on the following ingredients: (i) a technique to get asymptotic formula for the enlarged major arcs developed by Liu [8]; (ii) estimates for exponential sums over primes in short intervals established by Liu et al. [9]; (iii) a new estimate for Dirichlet polynomials introduced by Choi and Kumchev [1]. Applying the same idea, one can also consider the cases when k is large. Notation As usual, ϕ(n), μ(n), and Λ(n) stand for the functions of Euler, M¨ obius, and Von Mangoldt, respectively. N is a large integer, L = log N and r ∼ R means R/2  r  2R. Throughout the paper, the letter ε denotes a

Waring-Goldbach problem for fourth powers in short intervals

1409

sufficiently small positive real number which may vary at different places. Any statement in which ε occurs holds for each positive ε, and any implied constant in such a statement is allowed to depend on ε.

2

Outline of proof

Let 9  s  13 be an integer and let δs = P = N a(s) , where a(s) =

96 δs , s−8

s−8 8(s+88) .

Set

Q = N b(s) ,

b(s) = 1 −

(3)

7s + 40 δs . s−8

By Dirichlet’s lemma on rational approximation, each α ∈ [ Q1 , 1 + be written in the form 1 a , α = + λ, |λ|  q qQ

1 Q]

can (4)

for some integer a, q with 1  a  q  Q and (a, q) = 1. We define n to be the subset of α = aq + λ with P  q  Q,

|λ| 

1 . qQ

To define majors arcs, let P∗ = N c(s) ,

43

Q∗ = N 48 +2ε ,

(5)

where

3 δs . s−8 Then the major arcs M are defined as the union of all intervals c(s) =

a q

−

1 a 1 + , qQ∗ q qQ∗

with 1  a  q  P∗ and (a, q) = 1. Obviously, M and n are mutually disjoint. 1 1 ,1 + Q ], so that Let L be the complement of M and n in [ Q 1 Q

,1 +

1 = M ∪ n ∪ L. Q

Obviously, M ∩ n = ∅. Define

L1 = α : 1  q  P∗ ,

1 1   |λ|  qQ∗ qQ

1410

Hengcai TANG, Feng ZHAO

and

1  . L2 ⊂ α : P∗  q  P, |λ|  qQ

Obviously, we have L = L1 ∪ L2,

L1 ∩ L2 = ∅.

Let N be a sufficiently large integer and n ∈ N4,s satisfying N  n  N + N 3/4 U. Let



R(n, U ) =

(log p1 ) · · · (log ps ),

n=p4 +···+p4 s 1 |pj −(n/s)1/4 |U

where pj (1  j  s) are primes. For N1 =

 N 1/4 s

− U,

N2 =

 N 1/4 s

+ U,

(6)

f s (α)e(−nα)dα.

(7)

we define the exponential sums f (α) =



(log p)e(p4 α),

N1 pN2

where e(x) := e2πix . Then we have

1



s

f (α)e(−nα)dα =

R(n, U ) =

+ M

0



n∪L

We shall establish the following asymptotic formula on the major arcs in the next sections. Lemma 1 Let M and s be defined as above. Then for N  n  N + N 3/4 U and any A > 0, we have

f s (α)e(−nα)dα = Cs Ss (n)P0 + O(U s−1 N −3/4 L−A ), (8) M

where Cs is a positive constant and (m1 · · · ms )−3/4 N −3/4 U s−1 . P0 = n=m1 +···+ms N 4 mj N 4 1 2

Here, Ss (n) is defined in (15) below, which satisfies Ss (n) 1 for n ∈ N4,s .

Waring-Goldbach problem for fourth powers in short intervals

1411

Next, we estimate f (α) on n ∪ L. We first estimate f (α) on n, and this has been recently done by Tang [11] which states that

(log p)e(p α)  y k

1+δ

1

x1/2 xK /(K+1) Qxk−1 1/K 2 + + + 2k−1 . P y yK y

xpx+y

2

Here, we take δ > 0 arbitrarily small, k ∈ Z+ , K = 2k−1 , and x = for k = 4, we have f (α)  U 1+δ

1 P

+

√ k

N . Then

ε 2s−13 N 1/8 N 16/9 QN 3/4 1/64 − 3 − 8(s−8) 2s−16 N 8(s−8) + +  U . (9) U U8 U7

To bound f (α) on L, we need an estimate for exponential sums over primes in short intervals established by Liu et al. [9] which states that Lemma 2 Let

k  1,

and define

2  y  x,

α=

a + λ, q

Ξ = |λ|xk + x2 y −2 .

Then

(log p)e(pk α)

x
 (qx)ε

q 1/2 yΞ1/2 x1/2

+ q 1/2 x1/2 Ξ1/6 + y 1/2 x3/10 +

 x4/5 x + . Ξ1/6 q 1/2 Ξ1/2

Now, we apply Lemma 2 to estimate f (α) on L. For α ∈ L1, we have |λ|  and therefore,

1  N −1/2 U −2 , qQ∗

Ξ |λ|N + N 1/2 U −2 |λ|N.

Lemma 2 gives, for α ∈ L1,   U q|λ|N + N 1/8 q 1/2 (|λ|N )1/6 f (α)  N N 1/8  N 1/5 N 1/4 3/40 1/2 +N U + + (|λ|N )1/6 q|λ|N ε/4

2s−13

 U 2s−16 N

3 ε − 8(s−8) − 8(s−8)

If α=

.

a + λ ∈ L2, q

(10)

1412

Hengcai TANG, Feng ZHAO

then we have Ξ N 1/2 U −2 ,

P∗ < q  P,

qΞ  N Q−1 + P N 1/2 U −2 .

Now, Lemma 2 gives f (α)  N ε/4

U (qΞ)1/2

+ N 1/8 q 1/3 (qΞ)1/6

N 1/8

N 1/5 N 1/4  + (Ξ)1/6 (P∗ Ξ)1/2

+ N 3/40 U 1/2 + 2s−13

 U 2s−16 N

3 ε − 8(s−8) − 8(s−8)

.

(11)

From (9)–(11), we obtain the following result. Lemma 3 Let n and L be as above. Then we have 2s−13

sup |f (α)|  U 2s−16 N

3 ε − 8(s−8) − 8(s−8)

.

(12)

α∈n∪L

In order to prove Theorem 1, we also need the following lemma proved by Li and Wu [7], which can be viewed as a generalization of Hua’s lemma in short intervals. Lemma 4 Let X  Y  2, k ∈ N, and Sk∗ (α) :=

e(αnk ).

X−Y nX+Y

Then for any ε > 0 and 1  l  k, we have

1 l l |Sk∗ (α)|2 dα  X ε/8 Y 2 −l . 0

Proof of Theorem 1 Beginning with (7) and by Bessel’s inequality, we have

2    s f (α)e(−nα)dα  |f (α)|2s dα  N nN +N 3/4 U

n∪L

n∪L

1

 sup |f (α)|2s−16 α∈n∪L

U

2s−1

|f (α)|16 dα

0

N

− 34 − 8ε

,

where we have used Lemmas 3 and 4. Therefore, for all large integer n satisfying N  n  N + N 3/4 U,

U = N 1−δs +ε , 3

ε

except for the subset of cardinality at most O(U N 4 − 16 ), we have   3 ε   f s (α)e(−nα)dα  U s−1 N − 4 − 32 .  n∪L

This completes the proof.



Waring-Goldbach problem for fourth powers in short intervals

3

1413

Proof of Lemma 1

We first introduce some notations and lemmas. χ mod q, define C(χ, a) =

q h=1

 ah4  , χ(h)e q

For a Dirichlet character

C(q, a) = C(χ0 , a).

Let χ1 , . . . , χs be characters modulo q, where 9  s  13 is an integer. Then we write  an  C(χ1 , a) · · · C(χs , a), e − (13) B(n, q, χ1 , . . . , χs ) = q 1aq (a,q)=1

B(n, q) = B(n, q, χ0 , χ0 , . . . , χ0 ), and Ss (n) =

∞ B(n, q) q=1

ϕs (q)

(14)

.

(15)

The following lemma is very important when we use the iterative method to prove Lemma 1. Lemma 5 Let χj mod rj , j = 1, . . . , s, be primitive characters, r0 = [r1 , . . . , rs ], and let χ0 be the principal character mod q. Then we have qx, r0 |q

Proof

B(n, q, χ0 χ1 , . . . , χ0 χs ) − s−2 +ε 2  r logc x. 0 s ϕ (q)

The proof is standard now. See, for example, Leung and Liu [6].

Recall N1 , N2 as in (6), and define V (λ) =



e(m4 λ),

N1 mN2



W (χ, λ) =

(log p)χ(p)e(p4 λ) − δχ

N1 pN2



e(m4 λ),

(16)

N1 mN2

where δχ = 1 or 0 whether χ is principal or not. Define further ∗ s−2 [g, r]− 2 +ε max W (χ, λ) J(g) = rP∗

and K(g) =

rP∗

[g, r]−

χ mod r

s−2 +ε 2

χ mod r

∗



|λ|1/(rQ∗ )

1/(rQ∗ )

−1/(rQ∗ )

1/2

|W (χ, λ)|2 dλ

,

1414

Hengcai TANG, Feng ZHAO

 where ∗ is over all primitive characters. Lemma 1 depends on the following lemmas. Lemma 6 Let P∗ and Q∗ satisfy (5). Then we have K(g)  g−

s−2 +ε 2

U 1/2 N −3/8 Lc .

Lemma 7 Let P∗ and Q∗ satisfy (5). Then we have J(g)  g−

s−2 +ε 2

U Lc .

For g = 1, it can be improved to be J(1)  U L−A , where A > 0 is arbitrary. The proofs of Lemmas 6 and 7 will be given in Sections 4 and 5, respectively. With Lemmas 5–7 known, we can use the iterative method in Liu [8] to prove Lemma 1. For q  P∗ and N1  p  N2 , we have (p, q) = 1. Therefore, we can rewrite the exponential sums f (α) as f

 C(q, a) 1 +λ = V (λ) + C(χ, a)W (χ, λ) =: a1 + a2 , q ϕ(q) ϕ(q)

a

χ mod q

where V (χ) and W (χ, λ) are as in (16). Thus,

s f (α)e(−nα)dα = (a1 + a2 )s e(−nα)dα M

M

=

 s

M

=

s

 i Csi as−i 1 a2

e(−nα)dα

i=0

Csi Ii ,

i=0

where Ii =

qP∗

×

 an  1 s−i (C(q, a)) e − ϕs (q) 1aq q (a,q)=1

1/(qQ∗ )

−1/(qQ∗ )

 i V s−i (λ) C(χ, a)W (χ, λ) e(−nλ)dλ. χ mod q

The computation of I0 is standard, and we can get I0 = Cs Ss (n)P0 + O(U s−1 N −3/4 L−A ),

Waring-Goldbach problem for fourth powers in short intervals

1415

where Ss (n) is defined by (15) which obviously satisfies Ss (n) 1. For the other terms, we only treat the most complicated one, Is . The treatment of other terms is similar, so that we do not want to go into the details. Reducing the characters in Is into primitive characters, we have  1  · · · B(n, q, χ1 , . . . , χs ) |Is | =  ϕs (q) qP∗ χ1 mod q χs mod q

1/(qQ∗ )   W (χ1 , λ)W (χ2 , λ) · · · W (χs , λ)e(−N λ)dλ × −1/(qQ∗ )





···





∗



···

∗



|B(n, q, χ1 χ0 , . . . , χs χ0 )| ϕs (q)

r1 P∗ rs P∗ χ1 mod r1 χs mod rs qP∗ , r0 |q

1/(qQ∗ ) |W (χ1 χ0 , λ)| |W (χ2 χ0 , λ)| · · · |W (χs χ0 , λ)|dλ, × ∗ −1/(qQ )

(17)

where χ0 is the principal character modulo q, r0 = [r1 , r2 , ..., rs ] depending on r1 , r2 , ..., rs . For q  P∗ and N1  p  N2 , we have (q, p) = 1. Using this and (16), we have W (χj χ0 , λ) = W (χj , λ) for the primitive characters χj above. Thus, by Lemma 5, we obtain ∗ ∗ 1/(r0 Q∗ ) ··· ··· |W (χ1 χ0 , λ)| Is  r1 P∗

rs P∗ χ1 mod r1

χs mod rs





···

r1 P∗

rs P∗



|B(n, q, χ1 χ0 , . . . , χs χ0 )| ϕs (q) qP∗ , r0 |q ∗ 1/(r0 Q∗ ) ∗ ··· |W (χ1 , λ)|

× |W (χ2 χ0 , λ)| · · · |W (χs χ0 , λ)|dλ  Lc

−1/(r0 Q∗ )

− s−2 +ε 2

r0

χ1 mod r1

χs mod rs

−1/(r0 Q∗ )

× |W (χ2 , λ)| · · · |W (χs , λ)|dλ.

(18)

In the last integral, we take out |W (χj , λ)| (j = 1, . . . , s − 2), and then use Cauchy’s inequality to get ∗ max |W (χ1 , λ)| × · · · |Is |  Lc r1 P∗ χ1 mod r1

×

|λ|1/(r1 Q∗ )





∗

×





max

∗



rs P∗

− s−2 +ε 2

r0

χs mod rs

1/(rs−1 Q∗ )

−1/(rs−1 Q∗ )

rs−1 P∗ χs−1 mod rs−1

×

|W (χs−2 , λ)|

|λ|1/(rs−2 Q∗ )

rs−2 P∗ χs−2 mod rs−2

∗



1/(rs Q∗ )

−1/(rs Q∗ )

1/2

|W (χs−1 , λ)| dλ 2

1/2

|W (χs , λ)|2 dλ

.

(19)

1416

Hengcai TANG, Feng ZHAO

Now, we introduce an iterative procedure to bound the above sums over rs , . . . , r1 consecutively. We first estimate the above sum over rs in (19) via Lemma 6. Since r0 = [r1 , . . . , rs ] = [[r1 , . . . , rs−1 ], rs ], the sum over rs in (19) 





− s−2 +ε 2

[[r1 , . . . , rs−1 ], rs ]

rs P∗

∗



1/(rs Q∗ ) −1/(rs Q∗ )

χs mod rs

1/2

|W (χs , λ)| dλ 2

= K([r1 , . . . , rs−1 ])  [r1 , . . . , rs−1 ]−

s−2 +ε 2

U 1/2 N −3/8 Lc .

This contributes to the sum over rs−1 of (19) in amount s−2 [[r1 , . . . , rs−2 ], rs−1 ]− 2 +ε  U 1/2 N −3/8 Lc rs−1 P∗



×

∗



1/2

1/(rs−1 Q∗ ) −1/(rs−1 Q∗ )

χs−1 mod rs−1

|W (χs−1 , λ)|2 dλ

= U 1/2 N −3/8 Lc K([r1 , . . . , rs−2 ])  [r1 , . . . , rs−2 ]−

s−2 +ε 2

U N −3/4 Lc ,

where we have used Lemma 6 again. Inserting this last bound into (19), we can bound the sum over rs−2 as s−2 [[r1 , . . . , rs−3 ], rs−2 ]− 2 +ε  U N −3/4 Lc ×



rs−2 P∗ ∗

max

|λ|1/(qQ∗ )

χs−2 mod rs−2

|W (χs−2 , λ)|

= U N −3/4 Lc J([r1 , . . . , rs−3 ])  [r1 , . . . , rs−3 ]−

s−2 +ε 2

U 2 N −3/4 Lc .

Similarly, we can bound the sums over rs−3 , . . . , r1 and find that |Is |  U s−2 N −3/4 Lc

r1 P∗

− s−2 +ε 2

r1

χ1 mod r1

∗

max

|W (χ1 , λ)|

|λ|1/(r1 Q∗ )

= U s−2 N −3/4 Lc J(1)  U s−1 N −3/4 L−A , where we have used Lemma 7 consecutively.



Waring-Goldbach problem for fourth powers in short intervals

4

1417

Proof of Lemma 6

To begin with, we introduce the mean value theorem of Dirichlet polynomial which is quoted from Choi and Kumchev [1]. Lemma 8 Let R  1, X  2, and T  2. Then we have  ∗ 2T   Λ(n)χ(n)n−it dt  r∼R, d|r χ mod r

T

Xn2X

 (d−1 R2 T X 11/20 + X) logc (RT X). We approximate the W (χ, λ) in (16) by  (χ, λ) = (Λ(m)χ(m) − δχ )e(m4 λ). W N1 mN2

Since we have Λ(n) instead of the log p in W (χ, λ), the error term is  (χ, λ)  N 1/8 . W (χ, λ) − W Since [g, r](g, r) = gr, we have

− s−2 +ε 2

[g, r]

rP∗



∗



χ mod r

 N 1/8



[g, r]−

s−2 +ε 2

rP∗

1/(rQ∗ )

−1/(rQ∗ )

1/2

 (χ, λ)|2 dλ |W (χ, λ) − W

r 1/2 (Q∗ )1/2 

r −3+ε 1/2 r (g, r) rP∗ s−2 5  g− 2 +ε N 1/8 (Q∗ )−1/2 d3−ε r − 2 +ε  g−

s−2 +ε 2

N 1/8 (Q∗ )−1/2

d|g, dP∗

g

− s−2 +ε 2

N

g

− s−2 +ε 2

U 1/2 N −3/8 Lc .

Note that  1/(rQ∗ ) −1/(rQ∗ )



 

rP∗ , d|r

P∗3 (Q∗ )−1/2

1/2 |W (χ, λ)| dλ

1/(rQ∗ )

−1/(rQ∗ )



1/8

1/(rQ∗ )

−1/(rQ∗ )

2

1/2

 (χ, λ)|2 + |W (χ, λ) − W  (χ, λ)|2 dλ |W 1/2   (χ, λ)|2 dλ |W +

1/(rQ∗ ) −1/(rQ∗ )

1/2

 (χ, λ)|2 dλ |W (χ, λ) − W

.

1418

Hengcai TANG, Feng ZHAO

Therefore, we only need to show that

− s−2 +ε 2

[g, r]

r∼R



∗



χ mod r

1/2

1/(rQ∗ ) −1/(rQ∗ )

 (χ, λ)|2 dλ |W

 g−

s−2 +ε 2

U 1/2 N −3/8 Lc , (20)

where R  P∗ . By Gallagher’s lemma (see [3, Lemma 1]), we get

1/(rQ∗ )

 (χ, λ)|2 dλ |W  1 2 ∞     RQ∗ −∞ 4

−1/(rQ∗ )

2  (Λ(m)χ(m) − δχ ) dv

v
  1 2 N24    ∗ RQ N14 −rQ∗

v
  1 2 N24    ∗ RQ N14 −rQ∗ where X = max{v 1/4 , N1 },

2  (Λ(m)χ(m) − δχ ) dv





2  (Λ(m)χ(m) − δχ ) dv,

(21)

XmX+Y

X + Y = min{(v + rQ∗ )1/4 , N2 }.

In the case R < 1, the quantity in the last sum in (21)  N −3/4 Q∗ L. This contributes to (20) as  g−

s−2 +ε 2

 1 s−2 Q∗2 2 1/2 N L  g− 2 +ε N −1/4 L. 2 3/2 ∗ N Q

For R  1, we have χ = χ0 , and hence, δχ = 0. Thus, we have

1/(rQ∗ ) −1/(rQ∗ )

 (χ, λ)|2 dλ  |W

  1 2 N24   4 RQ∗ ∗ N1 −rQ



2  Λ(m)χ(m) dv. (22)

XmX+Y

Applying Perron’s summation formula, we see that the inner sum of (22) can be written as

b+iT (X + Y )s − X s 1 dt + O(L2 ) F (s, χ) S= 2πi b−iT s for T = N 1/4 and 0 < b < L−1 , with F (s, χ) = XmX+Y

Λ(m)χ(m)m−s .

Waring-Goldbach problem for fourth powers in short intervals

1419

Using trivial estimates, we see that for 0 < b < L−1 , (X + Y )s − X s  min(T0−1 , (|t| + b)−1 ) s for T0 = 16πN (RQ∗ )−1 . Therefore, for b → 0, we have

dt −1 + L2 . |F (it, χ)|dt + |F (it, χ)| S  T0 |t| |t|T0 T0 |t|T And consequently, (22) becomes 

1/(rQ∗ )  (χ, λ)|2 dλ  U N −5/4 max |W −1/(rQ∗ )

v∼N/s



|t|T0

× max

v∼N/s

T0 <|t|T

|F (it, χ)|dt

dt |F (it, χ)| |t|

2 +

2 +

(23)

N 3/4 U (RQ∗ )2

N 3/4 U L4 . (RQ∗ )2

The last term above contributes to the left-hand side of (20) in amount 



[g, r]−

r∼R

 g−

N 3/8 U 1/2 L2 RQ∗

s−2 +ε 2

χ mod r

s−2 +ε 2

N 3/8 U 1/2 L2 Q∗

 r − s−2 +ε 2 (g, r)

r∼R

g

− s−2 +ε 2

U

g

− s−2 +ε 2

U 1/2 N −3/8 Lc ,

1/2

N

3/8

Q

−1 2

L

and therefore, the left-hand side of (20)  U 1/2 N −5/8 max

v∼N/s

+

N 3/8 U 1/2

+ g−

RQ∗ s−2 +ε 2



[g, r]−

s−2 +ε 2

r∼R

max



v∼N/s



∗

χ mod r − s−2 +ε 2

[g, r]

r∼R

χ mod r

|t|T0 ∗

|F (it, χ)|dt

|F (it, χ)| T0 <|t|T

dt |t|

U 1/2 N −3/8 Lc .

Thus, to prove (20), it suffices to show that the estimate ∗ 2T1 s−2 − s−2 +ε 2 [g, r] |F (it, χ)|dt  g− 2 +ε N 1/4 Lc r∼R

χ mod r

holds for R  P∗ and 0 < T1  T0 , and ∗ 2T2 s−2 s−2 [g, r]− 2 +ε |F (it, χ)|dt  g− 2 +ε RQ∗ N −3/4 T2 Lc r∼R

χ mod r

T2

(24)

T1

(25)

1420

Hengcai TANG, Feng ZHAO

holds for R  P∗ and T0 < T2  T. To get estimate (24), we note that [g, r](g, r) = gr. Then the left-hand side of (24) is g



− s−2 +ε 2

− s−2 +ε 2

(R/d)





∗

r∼R, d|r χ mod r

d|g, dR

2T1

|F (it, χ)|dt.

(26)

T1

By Lemma 8, the above quantity can be estimated as  g−

  R − 72 +ε  R2 T1 N 11/80 + N 1/4 Lc d d

s−2 +ε 2

d|g, dR

 g−

s−2 +ε 2

N 91/80  1/4 τ (g) + N Lc Q∗

 g−

s−2 +ε 2

N 1/8 Lc

provided that

71

Q∗ N 80 +ε . Similarly, we can prove (25) by taking T = T2 in Lemma 8. Lemma 6 now follows from (24) and (25). 

5

Proof of Lemma 7

In this section, we prove Lemma 7. The idea of the proof is similar to that of Lemma 6, but there are several differences.  (χ, λ) as in §4, we get that the Estimation of J(g) Replacing W (χ, λ) by W resulting error s−2 [g, r]− 2 +ε rN 1/8  rP∗



s−2 +ε 2

N 1/8

g

− s−2 +ε 2

N

1/8

g

− s−2 +ε 2

U Lc .

 g−

d3−ε

d|g, dP∗



5

r − 2 +ε

rP∗ , d|r

P∗3

Hence, Lemma 7 is a consequence of the estimate rP∗

[g, r]−

s−2 +ε 2

χ mod r

∗

max

|λ|1/(rQ∗ )

where R  P∗ and c > 0 is some constant.

+ε  (χ, λ)  g− s−2 2 U Lc , W

(27)

Waring-Goldbach problem for fourth powers in short intervals

1421

s−2

The case R < 1 contributes to g− 2 +ε U L which is obviously acceptable. For R  1, we have δχ = 0. Thus,  (χ, λ) = Λ(m)χ(m)e(m4 λ). W N1 mN2

Define H(s, χ) =



Λ(m)χ(m)m

−s

,

N2

V (s, λ) =

ws−1 e(λw4 )dw.

N1

N1 mN2

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

b+iT 1  W (χ, λ) = H(s, χ)V (s, λ)ds + O(1), 2πi b−iT where

0 < b < L−1 ,

T = (1 + |λ|N )U L2 .

Using [12, Lemmas 4.3, 4.5] and a trivial estimate, we have 

U 1 1  , max . , V (s, λ)  N b/4 min N 1/4 |t| + 1 N1 wN2 |t + 8πλw4 | Take

N 1/2 , T = U2

T∗ =

Thus, it suffices to show that the estimates ∗ 2T1 s−2 − s−2 +ε [g, r] 2 |H(it, χ)|dt  g− 2 +ε N 1/4 Lc χ mod r

χ mod r

χ mod r

(31)

(32)

T2

holds for R  P∗ and T < T2  T∗ ; and ∗ 2T3 s−2 s−2 [g, r]− 2 +ε |H(it, χ)|dt  g− 2 +ε N (RQ∗ )−1 U Lc r∼R

(30)

T1

holds for R  P∗ and 0 < T1  T; ∗ 2T2 s−2 s−2 [g, r]− 2 +ε |H(it, χ)|dt  g− 2 +ε U (T2 + 1)1/2 Lc r∼R

(29)

16πN . RQ∗

 (χ, λ) is bounded by Then for b → 0, W

dt U  |H(it, χ)|dt + |H(it, χ)|  W (χ, λ)  1/4 N |t| + 1 |t|T T |t|T∗

∗ RQ |H(it, χ)|dt + O(1) + N T∗ |t|T

r∼R

(28)

T3

(33)

1422

Hengcai TANG, Feng ZHAO

holds for R  P∗ and T∗ < T3  T. Estimates (31)–(33) follow from Lemma 8 via an argument similar to that leading to (24), so we omit the details here. The first part of Lemma 7 is proved. Estimation of J(1) The result is the same as that of J(g) except for the saving of L−A on its right-hand side. To order to get this saving, we have to distinguish two cases LC < R  P and R  LC , where C is a constant depending on A. The proof of the first case is the same as that of J(g), so we omit the details. Now, we prove the second case R  LC . We use the well-known explicit formula

 u   uρ 2 +O + 1 log (ruT ) , Λ(m)χ(m) = δχ u − (34) ρ T |γ|T

mu

where ρ = β + iγ is a non-trivial zero of the function L(s, χ), and T ∈ [2, u] is  (χ, λ), we get a parameter. Then inserting (34) into W  

N2 4  e(u λ)d (Λ(m)χ(m) − δχ ) W (χ, λ) =

N1 N2

=

nu

e(u4 λ)

N1



U



uρ−1 du + O((1 + |λ|N )U T −1 L2 )

|γ|T

N (β−1)/4 + O(N U Q∗ −1 T −1 L2 ).

|γ|T

Now, let

η(T ) = c2 log−4/5 T.



By [10], χ mod r 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 [2, § 21], the Siegel zero does not exist in the present situation, since r ∼ R  LC . Thus, by the large-sieve type zero-density estimates for Dirichlet L-functions [5], we have

1−η(T ) ∗ (β−1)/4 c N L T 12(1−α)/5 N (α−1)/4 dα 0

r∼R χ mod r |γ|T

L

c

1−η(T )

N −12(1−α)ε/5 dα

0 −12η(T )ε/5

LN c

 exp(−c3 εL1/5 ) 5

provided that T = N 48 −ε . Consequently, ∗ s−2 +ε  (χ, λ)  g− s−2 2 [g, r]− 2 +ε max W U L−A rP∗

χ mod r 43

|λ|1/(rQ∗ )

provided that Q∗ = N 48 +2ε . Then the lemma follows.



Waring-Goldbach problem for fourth powers in short intervals

1423

Acknowledgements The first author was supported by the National Natural Science Foundation of China (Grant No. 11301142) and the Scientific Foundation of Henan University (Grant No. 2012YBZR030). The second author was supported by the National Natural Science Foundation of China (Grant No. 11126325) and the Innovation Scientists and Technicians Troop Construction Projects of Henan Province.

References 1. Choi S K K, Kumchev A V. Mean values of Dirichlet polynomials and applications to linear equations with prime variables. Acta Arith, 2006, 123: 125–142 2. Davenport H. Multiplicative Number Theory. Berlin: Springer, 1980 3. Gallagher P X. A large sieve density estimate near σ = 1. Invent Math, 1970, 11: 329– 339 4. Hua L K. Some results in the additive prime number theory. Quart J Math, 1938, 9: 68–80 5. Huxley M N. Large values of Dirichlet polynomials III. Acta Arith, 1974/75, 26: 435– 444 6. Leung M C, Liu M C. On generalized quadratic equations in three prime variables. Monatsh Math, 1993, 115: 133–169 7. Li H Z, Wu J. Sums of almost equal prime squares. Funct Approx Comment Math, 2008, 38(1): 49–65 8. Liu J Y. On Lagrange’s theorem with prime variables. Quart J Math, 2003, 54: 454–462 9. Liu J Y, L¨ u G S, Zhan T. Exceptional sums over prime variables in short intervals. Sci China Ser A, 2006, 49(4): 448–457 10. Prachar K. Primzahlverteilung. Berlin: Springer, 1957 11. Tang H C. A note on some results of Hua in short intervals. Lith Math J, 2011, 51: 75–81 12. Titchmarsh E C. The Theory of the Riemann Zeta-Function. New York: Oxford University press, 1986 13. Zhao F. Sums of almost equal prime cubes. Acta Math Hungar, 2010, 126(3): 279–293