SMALL PRIME SOLUTIONS OF A PAIR OF LINEAR
EQUATIONS IN FIVE PRIME VARIABLES
Zhen Cui
Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, P.R.CHINA
e-mail: [email protected]
Hongze Li
Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, P.R.CHINA
e-mail: [email protected]
1
Abstract
Assume an additional congruent condition on the coefficients, we prove that the pair of
P
linear equations 5j=1 aλj pj = bλ (λ = 1, 2) has solutions in primes pj satisfying pj ¿ (|b1 | + |b2 | +
1) maxi,j |aij |2318+ε . This improves the exponent 79680 without assuming the additional condition of
the second author’s.
Keywords Diophantine equations, Deuring-Heilbronn phenomenon, Large sieve inequality, Zero-free
region.
MR2000 Subject Classification. 11P32, 11P05, 11N36, 11P55.
1. Introduction
For a pair of integers (b1 , b2 ), we consider the solubility of the diophantine equations
5
X
aλj pj = bλ ;
λ = 1, 2,
(1.1)
j=1
where pj are prime variables and the coefficients aλj are non-zero integers satisfying some necessary
conditions(see Theorem 0 below). For 1 ≤ i 6= j ≤ 5, write
¯
¯
¯
¯
¯
¯
¯
¯
¯ a1i a1j ¯
¯ a1i b1 ¯
∆ij = ¯
¯, ∆ib = ¯
¯.
¯ a2i a2j ¯
¯ a2i b2 ¯
We also write B = maxi,j {2, |bij |}. The solubility of equations (1.1) was discussed by Hua in [1, §31]
and [2, Chapter 12], in connection with generalization of the Vinogradov theorem and conjectures
concerning primes represented by polynomials with integral coefficients. In [3], by refining the techniques they developed in solving the problem of Baker’s constant, Liu and Tsang solved this problem
qualitatively. Precisely speaking, they proved the following
Theorem 0. Suppose that
∆ij 6= 0
for
all
1≤i<j≤5
and
gcd(∆ij )1≤i<j≤5 = 1.
(1.2)
Suppose also the corresponding congruences
5
X
aλj lj ≡ bλ
(modp),
λ = 1, 2
j=1
1
Project supported partially by the National Natural Science Foundation of China(Grant No. 10771135).
1
(1.3)
is soluble with 1 ≤ l1 , · · · , l5 ≤ p − 1 for each prime p, and the following pair of linear equations
5
X
aλj yj = 0,
λ = 1, 2
(1.4)
j=1
is soluble in positive real yj . Then the pair of equations in (1.1) is soluble in primes p1 , · · · , p5
satisfying
max{p1 , · · · , p5 } ¿ (|b1 | + |b2 | + 1)B A ,
where A > 0 is an effective absolute constant.
Recently, Li [4] showed that the constant A in Theorem 0 can be take as 79680. In this paper, we
improve his result under the following extra condition,
(∆ij , ∆kl ) = 1, for all 1 ≤ i < j ≤ 5 and 1 ≤ k < l ≤ 5 such that (i, j) 6= (k, l).
(1.5)
Theorem 1. Suppose (1.2)—(1.5). Then the pair of equations in (1.1) is soluble in primes
p1 , · · · , p5 satisfying
max{p1 , · · · , p5 } ¿ B 2318+ε ,
where the implied constant depends only on ε.
We prove our Theorem 1 by the circle method, and the idea of the proof will be explained in §2. At
this stage, we only point out that in contrast to the earlier works [3, 4] which treat the enlarged major
arcs by the Deuring-Heilbronn phenomenon, we appeal to a new approach which was first developed by
J.Y. Liu and Zhan [5]. This approach had been successfully applied to a number of additive problems
concerning primes, see for example [6, 7, 8, 9, 10]. Whenever this approach is applicable, it is not only
technically simpler, but also gives better result than the Deuring-Heilbronn phenomenon.
Notation. As usual, ϕ(n), µ(n) and Λ(n) stand for the function of Euler, Möbius, and von Mangoldt
respectively. We use χ mod q and χ0 mod q to denote a Dirichlet character and the principal character
modulo q, and L(s, χ) is the Dirichlet L-function. For integers a, b, ... we denote by [a, b, ...] their least
common multiple. For a large integer N ≥ 1000B 2318+ε , let L = log N . While r ∼ R means
R < r ≤ 2R. And if there is no ambiguity, we express ab + θ as a/b + θ. The same convention will be
applied for quotients. The letter ε denotes a positive constant which is arbitrarily small and whose
value may vary at different occurrences. The letters c and cj stand for absolute positive constants,
but the value of c without subscript may vary at different occurrences.
In mathematical formulae, we will write “s.t.” for “similar terms”. For example, “A1 B2 C3 D4 E5 +
s.t.” means the sum of all possible terms Aα Bβ Cγ Dδ Eι with (α, β, γ, δ, ι) being any permutation of
(1, 2, 3, 4, 5).
2. Outline of the method
Let b = (b1 , b2 ). Denote by I(b) the weighted number of solutions of (1.1), i.e.,
I(b) =
X
(log p1 ) · · · (log p5 ),
where the sum is over all M < p1 , · · · , p5 ≤ N and
P5
j=1 aλj pj
= bλ (λ = 1, 2) with M = N/(1000B 3 ).
We will investigate I(b) by the circle method. To this end, we set
P = B 38 N ε ,
Q = B 4 N P −18 .
2
(2.1)
For any integers h1 , h2 , q such that
1 ≤ h1 , h2 ≤ q ≤ P,
(h1 , h2 , q) = 1,
(2.2)
let
M(h1 , h2 , q) := {(x1 , x2 ) ∈ R : |xλ − hλ /q| ≤ 1/qQ,
λ = 1, 2},
and define the major arcs M and the minor arcs m as follows:
·
¸
[
1
1 2
M=
M(h1 , h2 , q),
m=
,1 +
\M,
(2.3)
Q
Q
S
where the union
is over all h1 , h2 and q in (2.2). It follows from 2P ≤ Q that the major arcs
M(h1 , h2 , q) are mutually disjoint. For any real number x1 , x2 , define
xb := b1 x1 + b2 x2 ,
xj = a1j x1 + a2j x2 ,
Let
X
S(α) =
1 ≤ j ≤ 5.
(2.4)
(log p)e(pα).
M <p≤N
Then we have
Z
I(b) =
5
1+1/Q Z 1+1/Q Y
1/Q
1/Q
Z
S(xj )e(−xb )dx1 x2 =
j=1
Z
+
M
.
(2.5)
m
The integral on the minor arcs m was settled by the following
Lemma 2.1. We have
Z Y
5
S(xj )e(−xb )dx1 x2 ¿ N 3 B −9−ε .
m j=1
Proof: It is proved in [3, Lemma 3.1] that for any (x1 , x2 ) ∈ m, we have
min{|S(x1 )|, |S(x2 )|} ¿ N B 1/2 P −1/4 L4 .
Consequently,
Z Y
5
m j=1
S(xj )e(−xb )dx1 x2 ¿ N 3 B 1/2 P −1/4 L6 .
Now the choice of P in (2.1) gives the conclusion of the lemma.
¤
The integral on the major arcs M causes the main difficulty, which was settled by the following
Theorem 2. Let conditions be as in Theorem 1 and M be as in (2.3) with P, Q determined by
(2.1). Then we have
Z
5
Y
M j=1
¡
¢
Sj (xj )e(−xb )dx1 x2 = S(b, P )I(b) + O I(b)L−1 ,
(2.6)
where S(b, P ) and I(b) are defined in (2.9) and (2.10) respectively.
The proof of this theorem forms the bulks of this paper, Section 3 − 4. One easily sees from (2.1)
that our major arcs are quite large. Historically, the enlarged major arcs was treated by the DeuringHeilbronn phenomenon (see for example [3, 11, 4]). But here we observe that under the assumption
(1.5), we can save the factor r1/2 in Lemma 3.4 below. With this saving, (2.6) can be derived from
the large sieve inequality, Gallagher’s lemma and classical results on the distribution of zeros of L−
functions. This approach has also been used by Bauer, J.Y. Liu, and Zhan [6], J.Y. Liu and M.C. Liu
[7], and by Choi and J.Y. Liu [12].
3
To derive Theorem 1 from Theorem 2, we need to bound S(b, P ) and I(b) from below. For χ mod q,
we define
C(χ, a) =
q
X
µ
χ̄(h)e
h=1
ah
q
¶
,
C(q, a) = C(χ0 , a).
For any (x1 , x2 ) ∈ M(h1 , h2 , q), denote by xλ = hλ q −1 + ηλ ,
|ηλ | < 1/qQ,
λ = 1, 2. Then
λ = 1, 2.
Let
½
hj := a1j h1 + a2j h2 ,
η j := a1j η1 + a2j η2 ,
hb := b1 h1 + b2 h2 ,
η b := b1 η1 + b2 η2 ,
(2.7)
j = 1, · · · , 5.
If χ1 , · · · , χ5 are characters modq, then we write
X0 µ hb ¶
B(b, q, χ1 , · · · , χ5 ) =
e −
C(χ1 , h1 ) · · · C(χ5 , h5 ),
q
(2.8)
h
where here and throughout the sum
P0
h
B(b, q) = B(b, q, χ0 , χ0 , χ0 ),
is over all h1 , h2 in (2.2). We also write
A(b, q) =
B(b, q)
,
ϕ5 (q)
S(b, x) =
X
A(b, q).
(2.9)
q≤x
It is known that ([3], Lemma 4.2),
Lemma 2.2. Assuming (1.2), S(b, ∞) converges absolutely and S(b, P ) > c2 for some absolute
constant c2 > 0.
Same as Lemma 7.2 in [13], it is also easily checked that
Lemma 2.3. Assuming (1.2), and N ≥ B 2318+ε . Then we have
Z
3
−1
I(b) := N |∆45 |
D
dx1 dx2 dx3 À N 3 B −9 ,
(2.10)
where
D := {(x1 , x2 , x3 ) : M/N ≤ x1 , x2 , x3 , f4 (x1 , x2 , x3 ), f5 (x1 , x2 , x3 ) ≤ 1},
(2.11)
with
−1 ),
f4 (x1 , x2 , x3 ) := ∆−1
45 (∆51 x1 + ∆52 x2 + ∆53 x3 + ∆b5 N
−1
f5 (x1 , x2 , x3 ) := ∆54 (∆41 x1 + ∆42 x2 + ∆43 x3 + ∆b4 N −1 ).
Proof: This lemma can be proved in precisely the same way as [3, Lemma 4.4] with the parameters
taken into consideration.
¤
Now we give
Proof of Theorem 1: It follows from Theorem 2, Lemma 2.1, Lemma 2.2 and Lemma 2.3 that
¡
¢
¡
¢
I(b) = S(b, P )I(b) + O I(b)L−1 + O N 3 B −9−ε À N 3 B −9 .
¤
4
3. An explicit expression
In this section, we establish an explicit expression for the integral in Theorem 2 (see Lemma 3.6
below), from which and the estimates in the following section, we will derive Theorem 2.
It is known that ([14, p.24, Lemma 1.5]),
Lemma 3.1. Let χ(modq) is a character induced by the primitive character χ∗ (modq ∗ ). Write
q = q 0 q 00 such that (q ∗ , q 00 ) = 1 and every prime factor of q 0 divides q ∗ , Then for (a, q) > 1 we have

µ
¶ µ
¶ µ
¶
a
q
q

∗
∗

χ̄
χ
µ ∗


∗ (a, q)

(a, q) µ
q¶
q (a, q)


q0
q
C(χ, a) =
C(χ∗ , 1),
q∗ =
;
×ϕ(q)ϕ−1

(a, q)
(a, q 0 )



q0


q ∗ 6=
.
 0,
(a, q 0 )
Consequently, we always have
|C(χ, a)| ≤ (a, q)1/2 q 1/2 .
For j = 1, · · · , 5, let χj (modrj ) be primitive characters and denote by r = [r1 , · · · , r5 ]. The
following is Lemma 4.3 in [3].
Lemma 3.2. For r|q, write q = q 0 q 00 such that (r, q 00 ) = 1 and every prime factor of q 0 divides r.
We have
(1) B(b, q, χ1 χ0 , · · · , χ5 χ0 ) = B(b, q 0 , χ1 χ0 , · · · , χ5 χ0 )ϕ5 (q 00 )A(b, q 00 ),
(2) B(b, q 0 , χ1 χ0 , · · · , χ5 χ0 ) = 0 unless q 0 = r.
We also copy the following lemma, which is Lemma 4.2 in [3],
Lemma 3.3. Let A(b, q) be defined as in (2.9). Then A(b, q) is a multiplicative function of q and
∞
X
q=1
χ0
|A(b, q)| =
Y
(1 + |A(p)|) ¿ log6 L.
p
Lemma 3.4. Let χj mod rj with j = 1, · · · , 5 be primitive characters. Set r = [r1 , · · · , r5 ], and let
be the principal character modq. Then
X 1
|B(b, q, χ1 χ0 , · · · , χ5 χ0 )| ¿ r−1/2+ε Lc .
5 (q)
ϕ
q≤P
r|q
Proof: By Lemma 3.2, the above sum is equal to
ϕ−5 (r)|B(b, r, χ1 χ0 , · · · , χ5 χ0 )|
X
|A(b, q)|.
(3.1)
q≤P/r
(r,q)=1
By Lemma 3.3, the last sum in (3.1) is ¿ L. To bound the first sum, we first note that, it is
easily checked that B(b, q, χ1 χ0 , · · · , χ5 χ0 ) is a multiplicative function of q. For given h1 , h2 , suppose
p|hi , p|hj , we have
p|(a1i h1 + a2i h2 , a1j h1 + a2j h2 ) = (∆ij h1 , ∆ji h2 ) = ∆ij .
Now assume (1.5), this means that there are at most two of 1 ≤ i ≤ 5 such that p|hi , and when this
happens, we have most pt choices of the pair (h1 , h2 ), this together with Lemma 3.1 gives
ϕ−5 (pt )|B(b, pt , χ1 χ0 , · · · , χ5 χ0 )| ¿ p−t/2 .
5
(3.2)
This completes the proof of the lemma.
Denote by
Z N
Φ(η) =
e(uη)du,
¤
Z
N
Ψ(η, ρ) =
M
uρ−1 e(uη)du.
(3.3)
M
The next lemma gives an asymptotic formula of S(xj ) for xj ∈ M.
Lemma 3.5. Let T = P 253/10 . Then for any (x1 , x2 ) ∈ M(h1 , h2 , q), we have
S(xj ) = S1 (η j ) + S2 (η j ) + S3 (η j )
with
S1 (η j ) :=
C(q, hj )
Φ(η j ),
ϕ(q)
and
S2 (η j ) := −
X
X
1
C(χ, hj )
Ψ(η j , ρ)
ϕ(q)
χ mod q
|γ|≤T
¾
½
1/2+ε N
2
S3 (η j ) = O q
(1 + |η j |N )L ,
T
where ρ = β + iγ denotes a non-trivial zero (possibly the Siegel zero) of the Dirichlet L−function
L(s, χ).
Proof: The lemma can be proved in precisely the same way as [8, Lemma 3.1].
¤
For j = 0, 1, · · · , 5 we define
X X µ hb ¶ Z ∞ Z ∞ ©
ª
Ij :=
e −
S2 (η 1 ) · · · S2 (η j )S1 (η j+1 ) · · · S1 (η 5 ) + s.t. e(−η b )dη1 dη2 , (3.4)
q
−∞ −∞
q≤P
h
i.e., the contribution from those products with 5 − j pieces of S1 and j pieces of S2 to the integral.
Now we state the main result of this section.
Lemma 3.6. Let P be as in (2.1) and T = P 253/10 as in Lemma 3.5. Then we have
Z
5
Y
M j=1
S(xj )e(−xb )dx1 x2 =
5
X
Ij + O(N 3 B −9−ε ).
(3.5)
j=0
Proof: Firstly we show that on substituting S1 (η j ) + S2 (η j ) for S(xj ) in the integral in (3.5), the
resulting error is acceptable, i.e.,
Z
5
Y
M j=1
S(xj )e(−xb )dx1 x2
X X µ hb ¶
−
e −
q
q≤P
h
Z Z
5
Y
©
[−1/qQ,1/qQ]2
ª
S1 (η j ) + S2 (η j ) e(−η b )dη1 dη2
j=1
¿ N 3 B −9−ε .
(3.6)
The left hand side of (3.6) is
Z
¿
max
|η1 |,|η2 |≤1/qQ
+
XX
q≤P
|S3 (η 1 )|
Z Z
M
|S(x1 )S(x2 )S(x3 )S(x4 )|dx1 dx2 + s.t.
{|S3 (η 1 )| + · · · + |S3 (η 5 )|}5 dη1 dη2
h [−1/qQ,1/qQ]2
¿ N 3 B −9−ε .
6
Finally we extend the interval of integration in the second integral in (3.6) to (−∞, ∞). Denote by
Γ := R2 \[−1/qQ, 1/qQ]2 . Same as [13, Lemma 6.4], for any (η1 , η2 ) ∈ Γ, we have
max{|η 1 |, |η 2 |} ≥ 1/2B max{|η1 |, |η2 |} ≥ 1/2BqQ > N −1 .
Consequently, we have
min{|S1 (η 1 ) + S2 (η 1 )|, |S1 (η 2 ) + S2 (η 2 )|} ¿ qN (M/BqQ)−1/2 .
Same as (6.11) in [13], we have
Z Z Y
5
¯
¯
©
ª
¯S1 (η j ) + S2 (η j )¯ dη1 dη2 ¿ P 3/2 N 1/2 B 2 Q1/2 P 2 N 2 M −1 L 2 ¿ N 3 P −3 B −9−ε .
Γ
(3.7)
j=1
Now the conclusion of the lemma follows from (3.6) and (3.7).
¤
In the following section, we will also need the following
Lemma 3.7. Let N (σ, T, χ) denote the number of zeros of L(s, χ) in the region σ ≤ Res ≤
1, |Ims| ≤ T. Define
X
X X ∗
N (σ, T, χ),
N ∗ (σ, T, x) =
N (σ, T, q) =
N (σ, T, χ),
q≤x χ mod q
χ mod q
where * means that the summation is restricted to primitive characters χ mod q. Then we have
N (σ, T, q) ¿ (qT )(12/5+ε)(1−σ) ,
N ∗ (σ, T, x) ¿ (x2 T )(12/5+ε)(1−σ) .
It should be pointed out that the above log-free form of Lemma 3.7 is unnecessary for our purpose,
and the normal form of zero-density estimates works equally well.
Proof: The lemma follows from (1.1) of Huxley [15] and Theorem 1 of Jutila [16].
4. Estimation of Ij and the proof of Theorem 2
The purpose of this section is to establish the following Lemmas 4.1 and 4.2. At the end of this
section, we derive Theorem 2 from these two lemmas.
Lemma 4.1. Let Ij be defined in (3.4). Then for all Ij with j = 1, · · · , 5, we have
Ij ¿ I(b)L−1 .
(4.1)
Proof: We only treat the case j = 5 in (4.1) in detail. In other cases, the proof for (4.1) is similar.
By (3.4), Lemma 3.4 and Lemma 4.4 of [3], we have
5
X X µ hb ¶ Z ∞ Z ∞ Y
I5 =
e −
S2 (η j )e(−η b )dη1 dη2
q
−∞ −∞
q≤P
j=1
h
= N 3 |∆45 |−1
X
X
q≤P χ1 mod q
where J(b; ρ1 , ..., ρ5 ) :=
Z Y
5
D j=1
···
X
χ5 mod q
X
B(b, q, χ1 , ..., χ5 ) X
J(b; ρ1 , ..., ρ5 ), (4.2)
·
·
·
ϕ5 (q)
|γ1 |≤T
(N xj )ρj −1 dx1 dx2 dx3 with D defined in (2.11). Reducing the character
into primitive ones and write for r0 = [r1 , · · · , r5 ], we have
X
X −1/2+ε X ∗
X
I5 ¿ N 3 |∆45 |−1
···
r0
···
r1 ≤P
|γ5 |≤T
r5 ≤P
∗
X
χ5 mod r5 |γ1 |≤T
χ1 mod r1
7
···
X
|γ5 |≤T
|J(b, ρ1 , · · · , ρ5 )|.
(4.3)
−1/2+ε
−1/10+ε
Applying the inequality r0
≤ r1
X −1/10+ε X
I5 ¿ I(b)
r1
r1 ≤P
¿ I(b)
to the above formula, we got
X −1/10+ε X ∗ X
M β1 −1 · · ·
r5
M β5 −1
r5 ≤P
χ1 mod r1 |γ1 |≤T

X

∗
−1/10+ε
· · · r5
X
X
r−1/10+ε
r≤P
∗
X
M β−1
χ mod r |γ|≤T
χ5 mod r5 |γ5 |≤T
5


=: I(b)(J5 + K5 )5 ,
(4.4)
where J5 and K5 denote the contribution from r ≤ LA and LA < r ≤ P respectively.
By Lemma 3.7, K5 can be easily estimated as
X
X ∗ X
X
X ∗ X
K5 =
r−1/10+ε
M β−1 ¿ L max
r−1/10+ε
M β−1
LA <R≤P
χ mod r |γ|≤T
LA <r≤P
r∼R
χ mod r |γ|≤T
max R−1/10+ε (R2 T )(12/5+ε)(1−σ) M σ−1 .
¿ L max
(4.5)
LA <R≤P 1/2≤σ≤1
The total exponent of R in the last line is
f (σ) := (24/5 + 2ε) (1 − σ) − 1/10 + ε,
say. Obviously f (σ) ≥ 0 for 1/2 ≤ σ ≤ σ0 , and f (σ) < 0 for σ0 < σ ≤ 1, where σ0 = (47 + 30ε)/(48 +
20ε). Thus, (4.5) becomes
K5 ¿ L
max
1/2≤σ≤σ0
P f (σ) T (12/5+ε)(1−σ) M σ−1 + L max LAf (σ) T (12/5+ε)(1−σ) M σ−1
σ0 <σ≤1
=: K51 + K52 ,
(4.6)
say. Clearly,
K51 ¿ L
max
1/2≤σ≤σ0
N {(24/5+3036/50+8ε)(1−σ)−1/10+ε}/61−(1−3/2318)(1−σ) ¿ L−A ,
if P < N 1/61 , where we have used B < N 1/(2318) . We remark that it is this estimate for K51 that
requires P < N 1/61 exactly. To estimate K52 , we note that
K52 ¿ L
max
σ0 <σ≤99/100
N {3036/3050+3/2318−1}(1−σ) + L
max
99/100≤σ≤1
LAf (σ) .
For σ0 < σ ≤ 99/100, the exponent of N above is < −1/50000; while for 99/100 ≤ σ ≤ 1, we have
f (σ) < −1/20. Hence, we have K52 ¿ L−A/20 , and consequently (4.12) becomes
K5 ¿ L−A/40 .
Q
(4.7)
Now we turn to J5 . By Satz VIII.6.2 of Prachar [17], there exists a positive constant c1 such that
χ mod q L(s, χ) is zero-free in the region
σ ≥ 1 − c1 / max{log q, log4/5 N },
|t| ≤ N,
(4.8)
except for the possible Siegel zero. But by Siegel’s theorem (see [18, §21]), the Siegel zero does not
exist in this situation, since q ≤ LA . Let η(N ) = c1 log−4/5 N. Then Lemma 3.3 gives
X X ∗ X
M β−1 ¿ L
max
(L2A T )(12/5+ε)(1−σ) N 99(σ−1)/100
J5 ¿
1/2≤σ≤1−η(N )
r≤LA χ mod r |γ|≤T
¿ L4A
max
1/2≤σ≤1−η(N )
N (12/5×26/69−99/100+8ε)(1−σ) ¿ L4A N −c3 η(N )
¿ L4A exp{−c4 L1/5 }.
(4.9)
8
Inserting (4.7) and (4.9) into (4.4), we get (4.1) for j = 5. This completes the proof of Lemma 4.1 for
j = 5. The estimate of other cases of Ij follows from the same arguments and larger P suffices for the
purpose. This completes the proof of the lemma.
¤
Lemma 4.2. With the notations of Theorem 2, we have
©
ª
I0 = S(b)I(b) + O I(b)P −1 .
Proof: This is Lemma 5.1 in [3].
¤
Proof of Theorem 2: Clearly, the assertion of Theorem 2 follows from the Lemmas 2.3, 4.1 and
4.2.
¤
Acknowledgements. The first named author would like to thank professor Liu Jianya for constant
discussions and valuable advices.
References
[1] L.K. Hua, Die Abschätzung von Expontial summen und ihre Angewendung in der Zahlentheorie, Enzykl. Maht.
Wiss. Band I, Teil 2, Heft 13, Leipzig(Chinese ed.), Beijing, 1963.
[2] L.K. Hua, Additive Theory of Prime Numbers, Translations of Mathematical Monographs, Vol. 13, Amer. Math.
Soc., Providence, R.I. 1965.
[3] M.C. Liu, and K.M. Tsang, Small prime solutions of a pair of linear equations in five variables, in: International
Symposium in Memory of Hua Loo Keng, Vol. I (Beijing,1988), 163-182, Springer, Berlin, 1991.
[4] H.Z. Li, A numerical bound for small prime solutions of a pair of linear equations in five variables(in Chinese),Chinese Ann. math, 24A(2003), 777-786.
[5] J.Y. Liu, and T. Zhan, Sums of five almost equal prime squares(II), Sci. China, 41(1998), 710-722.
[6] C. Bauer, M.C. Liu, and T. Zhan, On a sum of three prime squares, J. Number Theory, 85(2000), 336-359.
[7] J.Y. Liu, and M.C. Liu, The exceptional set in the four prime squares problem, Illinois Journal of Mathematics,
44(2000), 272-293.
[8] X.M. Ren, The Waring-Goldbach problem for cubes, Acta Arith., XCIV(2000), 287-301.
[9] J.Y. Liu, and T. Zhan, The Exceptional Set in Huas Theorem for Three Squares of Primes, Acta Mathematica
Sinica, English Series., 21(2005), 335-350.
[10] K.K. Stephen Choi, and A. Kumchev, Mean values of Dirichlet polynomials and applications to linear equations
with prime variables, Acta Arith., 123(2006), 125-142.
[11] M.C. Liu, and T.Z. Wang, A numerical bound for small prime solutions of some ternary linear equations, Acta
Arith., 86(1998), 343-383.
[12] K.K. Stephen Choi, and J.Y. Liu, Small prime solutions of quadratic equations, Canad. J. Math., 54(2002), 71-91.
[13] M.C. Liu, and K.M. Tsang, On pairs of linear equations in three prime variables and an application to Goldbach’s
problem, J.reine angew.Math. 399(1989),109-136.
[14] C.D. Pan, and C.B. Pan, Goldbach conjecture, Chinese Science Press, Beijing 1992.
[15] M.N. Huxley, Large values of Dirichlet polynomials (III), Acta Arith., 26(1975), 435-444.
[16] M. Jutila, On Linnik’s constant, Math. Scand. 41(1977), 45-62.
[17] K. Prachar, P rimzahlverteilung, Springer, Berlin 1957.
[18] H. Davenport, Multiplicative number theory, 2nd ed., Springer, Berlin 1980.
Zhen Cui
Hongze Li
Department of Mathematics
Department of Mathematics
Shanghai Jiao Tong University
Shanghai Jiao Tong University
Shanghai 200240
Shanghai 200240
P.R.CHINA
P.R.CHINA
e-mail:
e-mail:
[email protected]
9
[email protected]