On the exceptional set of Goldbach numbers in a short interval

LXXVII.3 (1996)

On the exceptional set of Goldbach numbers in a short interval

by

Chao Hua Jia (Beijing)

1. Introduction. An even number which can be written as a sum of two primes is called a Goldbach number .

In 1937, I. M. Vinogradov proved the famous three primes theorem. Soon after that, employing Vinogradov’s idea, Hua Loo Keng [3] proved that if B is a sufficiently large positive constant and N is sufficiently large, then the even numbers in (2, N ), except for O(N log

N ) values, are all Goldbach numbers.

In 1973, Ramachandra [18] obtained the result in a short interval. He showed that if A = N

with ϕ =

, then the even numbers in (N, N + A), except for O(A log

N ) values, are all Goldbach numbers. If the estimation for the zero density of Huxley [4] is applied, ϕ =

can be arrived at. In 1991, Jia Chao Hua [8] employed the sieve method combined with the circle method to get ϕ =

.

In 1993, Perelli and Pintz [17] developed a new technique in the circle method to make great progress in solving this problem. They showed ϕ =

. Mikawa [13] proved ϕ =

by the sieve method.

In 1994, Jia Chao Hua [10] proved ϕ =

. Li Hongze [12] improved it to ϕ =

. Jia Chao Hua [11] got further ϕ =

. In [10], the new tech- nique of [17], the sieve method and the mean value estimation for Dirichlet polynomials were applied.

In this paper, we shall develop further the technique of [10] to prove the following:

Theorem. Suppose that B is a sufficiently large positive constant, ε is a sufficiently small positive constant, N is sufficiently large and A = N

. Then the even numbers in (N, N + A), except for O(A log

N ) values, are all Goldbach numbers.

Project supported by the Tian Yuan Item in the National Natural Science Foundation of China.

[207]

For A = N

, we can also get the conclusion of Theorem 2 of [10].

Throughout this paper, we always suppose that B is a sufficiently large positive constant, ε is a sufficiently small positive constant and E = B

, δ = ε

. We also suppose that N is sufficiently large and that A = N

, Y = N

, Q =

√

A log

N . Let c, c

and c

denote positive constants which have different values at different places. m ∼ M means that there are positive constants c

and c

such that c

M < m ≤ c

M . We often use M (s, χ) (M may be another capital letter) to denote a Dirichlet polynomial of the form

M (s, χ) = X

m∼M

a(m)χ(m) m

,

where a(m) is a complex number with a(m) = O(1), and χ is a character mod q.

The author would like to thank Professors Wang Yuan and Pan Cheng- biao for their encouragement.

2. Some preliminary work. Let χ be a character mod q and let χ

be a principal character. Set E

= 1 if χ = χ

, and E

= 0 if χ 6= χ

.

We divide the characters mod q into two classes. We call χ a good char- acter if L(s, χ) has no zeros in the region

(1) 1 − 24E log log Y

ε log Y ≤ σ ≤ 1, |t| ≤ 6Y.

Otherwise, we call χ a bad character .

By Siegel’s theorem and the zero free region of the L-function (see pp. 255 and 257 of [15]), we know that L(s, χ) 6= 0 in the region

σ > 1 − c(µ)

max(q

, log

(|t| + 2)) , where µ (> 0) is arbitrary and c(µ) > 0.

Take µ = ε/(500E). If a bad character exists, then

(2) q ≥ log

N.

Moreover, by the estimation for the zero density (see [14] and [5]), X

χ (mod q)

N (σ, T, χ)  (qT )

log

qT,

we know that for any modulus q ≤ Q, the number of bad characters is O(log

N ). Let

I(a, q) =

 a

q − log

N qY , a

q + log

N qY



and let

E

= [

q≤log^EN

[

I(a, q), E

= [1/Q, 1 + 1/Q) − E

.

Lemma 1. Assume that r (≥ 2) and q are positive integers, d

(n) = P

n=n₁...n_r

1 and x

< y ≤ x. Then X

x−y<n≤x

d

(n)  y(log x)

.

P r o o f. See Theorem 2 of [20].

Let

(3) τ (χ) =

X

χ(r)e

 r q

 .

Lemma 2. τ (χ

) = µ(q). For any character χ mod q, |τ (χ)| ≤ √ q.

P r o o f. See pp. 24 and 28 of [15].

Lemma 3. For any complex numbers a(n),

η

\

−η

   X

n

a(n)e(nt)  

dt  η

∞

\

−∞

 X

x<n≤x+1/(2η)

a(n)   

dx.

P r o o f. See Lemma 1 of [1].

Lemma 4. Suppose T ≥ 1, χ is a character modq, and P

m∼M

|a(m)|

 M log

T . Then X

χ (mod q) T

\

−T

X

m∼M

a(m)χ(m) m

2

dt  (M + qT ) log

T.

P r o o f. See Theorem 3 on p. 38 of [15].

3. Mean value estimate (I)

Lemma 5. Assume that Y

 H  Y

, M H = Y , q ≤ Q, χ is a character modq, M (s, χ) is a Dirichlet polynomial and

H(s, χ) = X

h∼H

Λ(h)χ(h)

h

.

Let η = qQ/Y , b = 1 + 1/ log N and T

= log

Y . Then for T

≤ T ≤ Y , we have

min

 η, 1

T

 X

χ (good) 2T

\

T

|M (b + it, χ)H(b + it, χ)|

dt  η

log

N.

P r o o f. Let s = b + it and let χ be a good character modq. By (1) and the zero free region of the L-function, we know that for |t| ≤ 2Y ,

(4) X

c1H<h≤c2H

Λ(h)χ(h)

h

= E

(c

H)

− (c

H)

1 − s + O(log

Y ).

So, for T

≤ |t| ≤ 2Y ,

(5) H(s, χ)  log

Y.

According to the discussion in [2], there are O(log

Y ) sets S(V, W ), where S(V, W ) = {t

(χ) : χ runs through all good characters mod q, j = 1, . . . , J(χ), |t

(χ) − t

(χ)| ≥ 1 (r 6= s)}. For t

(χ) ∈ S(V, W ),

V ≤ M

|M (b + it

(χ), χ)| < 2V, W ≤ H

|H(b + it

(χ), χ)| < 2W,

where Y

≤ M

V , Y

≤ H

W and V  M

, W  H

log

Y . Thus

(6) X

χ (good) 2T

\

T

|M (b + it, χ)H(b + it, χ)|

dt  V

W

Y

log

Y |S(V, W )|, where S(V, W ) is one of the sets with the above properties.

Assume Y

≤ H ≤ Y

, where k is a positive integer, k ≥ 8 and kδ  1. Applying the mean value estimate (see Theorem 3 on p. 632 of [16]) and Lemma 1 to M (s, χ) and H

(s, χ), we have

|S(V, W )|  V

(M + qT ) log

Y,

|S(V, W )|  W

(H

+ qT ) log

Y,

where d = c/δ

. Applying the Hal´asz method (see Theorem 6 on p. 650 of [16]) to M (s, χ) and H

(s, χ), we have

|S(V, W )|  (V

M + V

qT M ) log

Y,

|S(V, W )|  (W

H

+ W

qT H

) log

Y.

Thus,

V

W

|S(V, W )|  V

W

F log

Y,

where

F = min{V

(M + qT ), V

M + V

qT M, W

(H

+ qT ),

W

H

+ W

qT H

}.

It will be proved that

(7) min

 η, 1

T



V

W

F  η

Y log

N.

We consider four cases.

(a) F ≤ 2V

M , 2W

H

. Then

V

W

F  V

W

min{V

M, W

H

}

≤ V

W

(V

M )

(W

H

)

= V

W M

H

 Y log

N.

(b) F > 2V

M, 2W

H

. Then

V

W

F  V

W

min{V

qT, V

qT M, W

qT, W

qT H

}

≤ V

W

(V

)

(V

M )

(W

)

qT = qT M

. Since k ≥ 8, we have H ≥ Y

≥ Y

, M

 Y

and so

min

 η, 1

T



V

W

F  η

T Y

qT  η

Y

. (c) F ≤ 2V

M, F > 2W

H

. Then

V

W

F  V

W

min{V

M, W

qT, W

qT H

}

≤ V

W

(V

M )

(W

qT H

)

 M H

(qT )

,

since V  M

. As H ≥ Y

≥ (

)

Y

, we have

min

 η, 1

T



V

W

F  η

T

Y

 Y Q



3k

(qT )

Y

 η

Y

. (d) F > 2V

M , F ≤ 2W

H

. Then

V

W

F  V

W

min{V

qT, V

qT M, W

H

}

≤ V

W

(V

qT )

(V

qT M )

(W

H

)

= (qT )

HM

.

If k ≥ 9, then H ≤ Y

≤ Y

, M  Y

, and so min

 η, 1

T



V

W

F  η

Y

 Y Q



k

Y

 η

Y

. If k = 8, then Y

≤ H  Y

, M  Y

, and so

min

 η, 1

T



V

W

F  η

Y

 Y Q



8

Y

 η

Y

. Combining the above, we obtain (7). Hence, Lemma 5 follows.

Lemma 6. Assume that M  Y

, M L = Y , q ≤ Q, χ is a character modq, M (s, χ) is a Dirichlet polynomial, and

F (s, χ) = M (s, χ) X

l∼L

χ(l) l

. Let η = qQ/Y , b = 1 + 1/ log N and T

= p

L/q. Then for T

≤ T ≤ Y , we have

min

 η, 1

T

 X

χ (mod q) 2T

\

T

|F (b + it, χ)|

dt  η

Y

.

P r o o f. Perron’s formula yields X

l∼L

χ(l) l

= 1

2πi

4iT

\

−4iT

L(b + it + s, χ) (c

L)

− (c

L)

s ds

+ O

 Y

T

 + O

 1 L



= 1 2πi

12−b+4iT

\

1 2−b−4iT

L(b + it + s, χ) (c

L)

− (c

L)

s ds

+ O

 Y

T

 + O

 1 L

 + O

 L

(qT )

T

 ,  

  X

l∼L

χ(l) l

2

 L

log T

6T

\

−6T

   L

 1

2 + iu, χ

   

2

du

1 + |t − u|

+ O

 Y

T

 + O

 1 L

 + O

 L

(qT )

T



.

By Lemma 4 and the mean value estimate for the fourth power of the L-function, we have

X

χ (mod q) 2T

\

T

|F (b + it, χ)|

dt

 L

log N X

χ (mod q) 2T

\

T

|M (b + it, χ)|

dt



\

−6T

   L

 1

2 + iu, χ

   

2

du

1 + |t − u|



+ O

 Y

T

+ 1

L

+ L

(qT )

T

 X

χ (mod q) 2T

\

T

|M (b + it, χ)|

dt

 L

log

N

 X

χ (mod q) 2T

\

T

|M (b + it, χ)|

dt



2

×

 X

χ (mod q) 2T

\

T

dt

6T

\

−6T

|L(

+ iu, χ)|

1 + |t − u| du



2

+ O

 Y

T

+ 1

L

+ L

(qT )

T



1 + qT M



 L

 1 + qT

M



2

(qT )

log

N + Y

. Hence,

min

 η, 1

T

 X

χ (mod q) 2T

\

T

|F (b + it, χ)|

dt

 min

 η, 1

T



Y

(Y

+ qT )

(qT )

log

N + η

Y

 η

Y



Y

+ Y Q



2

 Y Q



2

log

N + η

Y

 η

Y

. Thus Lemma 6 follows.

4. Mean value estimate (II)

Lemma 7. Assume that M HK = Y , q ≤ Q, χ is a character mod q,

M (s, χ), H(s, χ) and K(s, χ) are Dirichlet polynomials and G(s, χ) =

M (s, χ)H(s, χ)K(s, χ). Let η = qQ/Y , b = 1 + 1/ log N , T

= log

Y .

Assume further that for T

≤ |t| ≤ 2Y , M (b + it, χ)  log

Y and

H(b + it, χ)  log

Y . Moreover , suppose that M and H satisfy one of the following four conditions:

1) M H  Y

, Y

 H, M

/H  Y

, Y

 M , H

/M  Y

, Y

 M

H;

2) M H  Y

, Y

 H, M

H  Y

, Y

 M

H, H

/M  Y

, Y

 M

H;

3) M H  Y

, Y

 H, M

H  Y

, Y

 M , M

H  Y

, Y

 M

H;

4) M H  Y

, Y

 H, M

H  Y

, Y

 M

H, H

/M  Y

, Y

 M

H, Y

 M H.

Then for T

≤ T ≤ Y , we have (8) min

 η, 1

T

 X

χ (good) 2T

\

T

|G(b + it, χ)|

dt  η

log

N.

P r o o f. We only show that for T = 1/η = Y /(qQ),

(9) I = X

χ (good) 2T

\

T

|G(b + it, χ)|

dt  log

N.

I. First, we assume condition 1). On applying the mean value estimate and Hal´asz method to M

(s, χ), H

(s, χ) and K

(s, χ), we get

I  U

V

W

Y

F log

N, where

F = min{V

(M

+ qT ), V

M

+ V

qT M

, W

(H

+ qT ),

W

H

+ W

qT H

, U

(K

+ qT ), U

K

+ U

qT K

}.

We discuss the following cases:

(a) F ≤ 2V

M

, 2W

H

, 2U

K

. Then

U

V

W

F  U

V

W

min{V

M

, W

H

, U

K

}

≤ U

V

W

(V

M

)

(W

H

)

(U

K

)

= V

M

U

K

H  Y log

N.

(b) F ≤ 2V

M

, 2W

H

, F > 2U

K

. Then

U

V

W

F  U

V

W

min{V

M

, W

H

, U

qT, U

qT K

}

≤ U

V

W

(V

M

)

(W

H

)

(U

qT )

(U

qT K

)

= (qT )

M HK

 Y

.

(c) F ≤ 2V

M

, F > 2W

H

, F ≤ 2U

K

. Then

U

V

W

F  U

V

W

min{V

M

, W

qT, W

qT H

, U

K

}

≤ U

V

W

(V

M

)

(W

qT )

(W

qT H

)

(U

K

)

= (qT )

M KH

 Y

.

(d) F ≤ 2V

M

, F > 2W

H

, 2U

K

. Then U

V

W

F

 U

V

W

min{V

M

, W

qT, W

qT H

, U

qT, U

qT K

}

≤ U

V

W

(V

M

)

(W

qT )

(U

qT )

(U

qT K

)

= (qT )

M K

 Y

.

(e) F > 2V

M

, F ≤ 2W

H

, 2U

K

. Then

U

V

W

F  U

V

W

min{V

qT, V

qT M

, W

H

, U

K

}

≤ U

V

W

(V

qT )

(V

qT M

)

(W

H

)

(U

K

)

= (qT )

M

HK  Y

.

(f) F > 2V

M

, F ≤ 2W

H

, F > 2U

K

. Then U

V

W

F

 U

V

W

min{V

qT, V

qT M

, W

H

, U

qT, U

qT K

}

≤ U

V

W

(V

qT )

(W

H

)

(U

qT )

(U

qT K

)

= (qT )

HK

 Y

.

(g) F > 2V

M

, 2W

H

, F ≤ 2U

K

. Then U

V

W

F

 U

V

W

min{V

qT, V

qT M

, W

qT, W

qT H

, U

K

}

≤ U

V

W

(V

qT )

(W

qT )

(W

qT H

)

(U

K

)

= (qT )

H

K  Y

.

(h) F > 2V

M

, 2W

H

, 2U

K

. Then U

V

W

F

 U

V

W

min{V

, V

M

, W

, W

H

, U

, U

K

}qT

≤ U

V

W

(V

)

(V

M

)

(W

)

(U

)

qT

= qT M

 Y

,

since M  Y .

II. Next, we assume condition 2). On applying the mean value estimate and Hal´asz method to M

(s, χ)H(s, χ), H

(s, χ) and K

(s, χ), we get

I  U

V

W

Y

F log

N, where

F = min{V

W

(M

H + qT ), V

W

M

H + V

W

qT M

H, W

(H

+ qT ), W

H

+ W

qT H

, U

(K

+ qT ), U

K

+ U

qT K

}.

We consider several cases:

(a) F ≤ 2V

W

M

H, 2W

H

, 2U

K

. Then

U

V

W

F  U

V

W

min{V

W

M

H, W

H

, U

K

}

≤ U

V

W

(V

W

M

H)

(W

H

)

(U

K

)

= W

H

U

K

M  Y log

N.

(b) F ≤ 2V

W

M

H, 2W

H

, F > 2U

K

. Then U

V

W

F

 U

V

W

min{V

W

M

H, W

H

, U

qT, U

qT K

}

≤ U

V

W

(V

W

M

H)

(W

H

)

(U

qT )

(U

qT K

)

= (qT )

M HK

 Y

.

(c) F ≤ 2V

W

M

H, F > 2W

H

, F ≤ 2U

K

. Then U

V

W

F

 U

V

W

min{V

W

M

H, W

qT, W

qT H

, U

K

}

≤ U

V

W

(V

W

M

H)

(W

qT H

)

(U

K

)

= (qT )

U

K

M H

 (qT )

M H

K  Y

.

(d) F ≤ 2V

W

M

H, F > 2W

H

, 2U

K

. Then U

V

W

F

 U

V

W

min{V

W

M

H, W

qT, W

qT H

, U

qT, U

qT K

}

≤ U

V

W

(V

W

M

H)

(W

qT H

)

(U

qT )

(U

qT K

)

= (qT )

M H

K

 Y

.

(e) F > 2V

W

M

H, F ≤ 2W

H

, 2U

K

. Then U

V

W

F

 U

V

W

min{V

W

qT, V

W

qT M

H, W

H

, U

K

}

≤ U

V

W

(V

W

qT )

(V

W

qT M

H)

(W

H

)

(U

K

)

= (qT )

M

H

K  Y

.

(f) F > 2V

W

M

H, F ≤ 2W

H

, F > 2U

K

. Then U

V

W

F

 U

V

W

min{V

W

qT, V

W

qT M

H, W

H

, U

qT, U

qT K

}

≤ U

V

W

(V

W

qT )

(W

H

)

(U

qT )

(U

qT K

)

= (qT )

H

K

 Y

.

(g) F > 2V

W

M

H, 2W

H

, F ≤ 2U

K

. Then U

V

W

F

 U

V

W

min{V

W

qT, V

W

qT M

H, W

qT, W

qT H

, U

K

}

≤ U

V

W

(V

W

qT )

(V

W

qT M

H)

(W

qT H

)

× (U

K

)

= (qT )

M

H

K  Y

.

(h) F > 2V

W

M

H, 2W

H

, 2U

K

. Then

U

V

W

F  U

V

W

min{V

W

, V

W

M

H, W

, W

H

, U

, U

K

}qT

≤ U

V

W

(V

W

)

(W

)

(U

)

(U

K

)

qT

= qT K

 Y

,

since Y

 M H (the latter follows from Y

 H and Y

 M

H).

III. Now, we assume condition 3). On applying the mean value estimate and Hal´asz method to M

(s, χ), H

(s, χ) and K

(s, χ), we get

I  U

V

W

Y

F log

N, where

F = min{V

(M

+ qT ), V

M

+ V

qT M

, W

(H

+ qT ), W

H

+ W

qT H

, U

(K

+ qT ), U

K

+ U

qT K

}.

Consider the following cases:

(a) F ≤ 2V

M

, 2W

H

, 2U

K

. Then

U

V

W

F  U

V

W

min{V

M

, W

H

, U

K

}

≤ U

V

W

(V

M

)

(W

H

)

(U

K

)

= V

M

U

K

H  Y log

N.

(b) F ≤ 2V

M

, 2W

H

, F > 2U

K

. Then

U

V

W

F  U

V

W

min{V

M

, W

H

, U

qT, U

qT K

}

≤ U

V

W

(V

M

)

(W

H

)

(U

qT )

(U

qT K

)

= (qT )

M HK

 Y

.

(c) F ≤ 2V

M

, F > 2W

H

, F ≤ 2U

K

. Then

U

V

W

F  U

V

W

min{V

M

, W

qT, W

qT H

, U

K

}

≤ U

V

W

(V

M

)

(W

qT )

(W

qT H

)

(U

K

)

= (qT )

M KH

 Y

.

(d) F ≤ 2V

M

, F > 2W

H

, 2U

K

. Then U

V

W

F

 U

V

W

min{V

M

, W

qT, W

qT H

, U

qT, U

qT K

}

≤ U

V

W

(V

M

)

(W

qT )

(W

qT H

)

(U

qT )

= (qT )

M H

 Y

.

(e) F > 2V

M

, F ≤ 2W

H

, 2U

K

. Then

U

V

W

F  U

V

W

min{V

qT, V

qT M

, W

H

, U

K

}

≤ U

V

W

(V

qT )

(V

qT M

)

(W

H

)

(U

K

)

= (qT )

M

HK  Y

.

(f) F > 2V

M

, F ≤ 2W

H

, F > 2U

K

. Then U

V

W

F

 U

V

W

min{V

qT, V

qT M

, W

H

, U

qT, U

qT K

}

≤ U

V

W

(V

qT )

(V

qT M

)

(W

H

)

(U

qT )

= (qT )

M

H  Y

.