J Wu

Chen's double sieve, Goldbach's conjecture and the twin prime problem

For every even integer N , denote by D(N ) and D 1,2 (N ) the number of representations of N as a sum of two primes and as a sum of a prime and an integer having at most two prime factors, respectively. In this paper, we give a new upper bound for D(N ) and a new lower bound for D 1,2 (N ), which improve the corresponding results of Chen. We also obtain similar results for the twin prime problem

Let Ω(n) be the number of all prime factors of the integer n with the convention Ω(1) = 0. For an even integer N 4, we define D(N ) as the number of representations of N as a sum of two primes:

D(N ) := |{p N : Ω(N -p) = 1}|,
where and in what follows, the letter p, with or without subscript, denotes a prime number. The well known Goldbach conjecture can be stated as D(N ) 1 for every even integer N 4. A more precise version of this conjecture was proposed by Hardy & Littlewood [15]:

(1.1) D(N ) ∼ 2Θ(N ) (N → ∞),
2000 Mathematics Subject Classification: 11P32, 11N35, 11N05.

where Θ(N ) := C N N (log N ) 2 and

C N := p|N, p>2 p -1 p -2 p>2 1 - 1 (p -1) 2 .
Certainly, the asymptotic formula (1.1) is extremely difficult. Although the lower bound problem remains open, the upper bound problem has a rich history. In 1949 Selberg [START_REF] Selberg | On elementary methods in prime number theory and their limitations[END_REF] proved

(1.2) D(N ) {16 + o(1)}Θ(N )
with the help of his well known λ 2 -upper bound sieve. By applying Linnik's large sieve method, C.D. Pan [START_REF] Pan | A new application of the Yu.V. Linnik large sieve method[END_REF] in 1964 improved 16 to 12. In 1966, Bombieri & Davenport [START_REF] Bombieri | Small differences between prime numbers[END_REF] obtained 8 instead.

Their proof is based on the linear sieve formulas and the mean value theorem of Bombieri-Vinogradov. It seems very difficult to prove (1.2) with a constant strictly less than 8 by the method in [START_REF] Bombieri | Small differences between prime numbers[END_REF]. Firstly the linear sieve formulas (see Lemma 2.2 below)

(1.3) XV (z)f log Q log z + error S(A; P, z) XV (z)F log Q log z + error are the best possible in the sense that taking

A = B ν := {n : 1 n x, Ω(n) ≡ ν (mod2)} (ν = 1, 2),
the upper and lower bounds in (1.3) are respectively attained by ν = 1 and ν = 2 (see [START_REF] Halberstam | Sieve Methods[END_REF], page 239). Secondly it is hopeless to try to improve the level of distribution 1 2 in Bombieri-Vinogradov's theorem.

In 1978, Chen [START_REF] Chen | On the Goldbach's problem and the sieve methods[END_REF] introduced a new idea in Selberg's sieve and proved (1.4) D(N ) 7.8342 Θ(N ) (N N 0 ).

His sieve machine involves two variables and is quite complicated. Roughly speaking, for the sequence A = {N -p : p N } he introduced two new functions h(s) and H(s) such that (1.3) holds with f (s) + h(s) and F (s) -H(s) in place of f (s) and F (s), respectively. The key innovation is to prove h(s) > 0 and H(s) > 0 via three weighted inequalities (see [START_REF] Chen | On the Goldbach's problem and the sieve methods[END_REF], [START_REF] Ross | A short intervals result in additive prime number theory[END_REF], (47), (64), (90), and (91)). It is worth pointing out that he did not give complete proofs for these three inequalities. Among the three inequalities, the third one is the most complicated (with 43 terms) and it seems quite difficult to reconstruct a proof. Indeed, combining any one of these three inequalities with the Chen-Iwaniec switching principle (see [START_REF] Chen | On the representation of a large even integer as the sum of a prime and the product of at most two primes[END_REF] and [START_REF] Iwaniec | Primes of the type φ(x, y) + A where φ is a quadratic form[END_REF]) leads to a constant less than 8. In order to derive a better result, Chen further introduced a very complicated iterative method. In 1980, C.B. Pan [START_REF] Pan | On the upper bound of the number of ways to represent an even integer as a sum of two primes[END_REF] applied essentially the first weighted inequality of Chen to get 7.988. According to [START_REF] Pan | Goldbach Conjecture[END_REF], Chen's proof is very long and somewhat difficult to follow, but his idea is clear.

In this paper, inspired by the ideas in [START_REF] Wu | Sur la suite des nombres premiers jumeaux[END_REF] we shall first try to give a more comprehensive treatment on Chen's double sieve and prove an upper bound sharper than (1.4).

Theorem 1. For sufficiently large N , we have D(N ) 7.8209 Θ(N ).

The improvement comes from a new weighted inequality (see Lemma 4.2 below), which is still quite complicated with 21 terms, but much simpler than Chen's third and more powerful than his second and third inequality. Recently Cai & Lu [START_REF] Cai | On the upper bound for π 2 (x)[END_REF] give another weighted inequality (with 31 terms), which is simpler but slightly weaker than Chen's third.

One way of approaching the lower bound problem in (1.1) is to give a non trivial lower bound for the quantity D 1,2 (N ) := |{p N : Ω(N -p) 2}|.

In this direction, Chen [START_REF] Chen | On the representation of a large even integer as the sum of a prime and the product of at most two primes[END_REF] proved, by his system of weights and switching principle, the following famous theorem: Every sufficiently large even integer can be written as sum of a prime and an integer having at most two prime factors. More precisely he established

(1.5) D 1,2 (N ) 0.67 Θ(N ) (N N 0 ).
Then Halberstam & Richert [START_REF] Halberstam | Sieve Methods[END_REF] obtained a better constant 0.689 in place of 0.67 by a careful numerical calculation. As they indicated in [START_REF] Halberstam | Sieve Methods[END_REF], it would be interesting to know whether a more elaborate weighting procedure could be adapted to the purpose of (1.5). This might lead to numerical improvements and could be important. In 1978 Chen improved the constant 0.689 of Halberstam & Richert to 0.7544 and to 0.81 by two more elaborate systems of weights ( [START_REF] Chen | On the representation of a large even integer as the sum of a prime and the product of at most two primes (II)[END_REF], [START_REF] Chen | Further improvement on the constant in the proposition '1+2': On the representation of a large even integer as the sum of a prime and the product of at most two primes (II) (in Chinese)[END_REF]). Very recently by improving Chen's weighting device Cai and Lu [START_REF] Cai | On Chen's theorem[END_REF] obtained 0.8285, which they described as being near to the limit of what could be obtained by the method employed.

The second aim of this paper is to propose a larger constant.

Theorem 2. For sufficiently large N , we have

D 1,2 (N ) 0.836 Θ(N ).
The proof of Theorem 2 is based on a modified version of Chen's weights (see Lemma 9.2 below), the linear sieve and the mean value theorems of Pan & Ding [START_REF] Pan | A new mean value theorem[END_REF] and of Fouvry [START_REF] Fouvry | Autour du théorème de Bombieri-Vinogradov[END_REF].

A conjecture of the same nature is the twin prime problem, which can be stated as π 2 (x) := |{p x : Ω(p + 2) = 1}| → ∞ (x → ∞).

Similar to (1.1), Hardy & Littlewood [START_REF] Hardy | Some problems of 'partitio numerorum' III : On the expression of a number as a sum of primes[END_REF] conjectured

(1.6) π 2 (x) ∼ Π(x) (x → ∞),
where Π(x) := Cx (log x) 2 and C := 2 where the constant a is half of the corresponding constant in the Goldbach problem. Due to the sieve of Rosser-Iwaniec and mean value theorems of Bombieri, Fouvry, Friedlander and Iwaniec, the history of (1.7) is much richer than that of (1.2). We refer the reader to [START_REF] Wu | Sur la suite des nombres premiers jumeaux[END_REF] and [START_REF] Cai | On the upper bound for π 2 (x)[END_REF] for a detailed historical description of this problem. In particular Wu [START_REF] Wu | Sur la suite des nombres premiers jumeaux[END_REF] obtained 3.418 in place of a + o(1) by placing these new mean value theorems in Chen's method. The main difficulty for applying these mean value theorems in [START_REF] Wu | Sur la suite des nombres premiers jumeaux[END_REF] is to not destroy the fact that the error terms are affected by well factorisable coefficients. Recently Cai & Lu [START_REF] Cai | On the upper bound for π 2 (x)[END_REF] improved the constant 3.418 to 3.406. Our argument in proving Theorem 1 allows us to give a better result.

Theorem 3. For sufficiently large x, we have

π 2 (x) 3.3996 Π(x).
As an analogue of Theorem 2, Chen [START_REF] Chen | On the representation of a large even integer as the sum of a prime and the product of at most two primes[END_REF] proved that

(1.8) π 1,2 (x) 0.335 Π(x) (x x 0 ), where π 1,2 (x) := |{p x : Ω(p + 2) 2}|.
The constant 0.335 was improved by many mathematicians. Like (1.7), the history of (1.8) is much richer than that of (1.5). A detailed historical description on this problem can be found in the recent paper of Cai [START_REF] Cai | A remark on Chen's theorem[END_REF]. In particular he obtained 1.0974 in place of 0.335, which is an improvement of Wu's constant 1.05 [START_REF] Wu | Sur la suite des nombres premiers jumeaux[END_REF]. Here we can propose a slightly better result.

Theorem 4. For sufficiently large x, we have

π 1,2 (x) 1.104 Π(x).
Remark 1. (i) Theorems 1 and 3 show that the principal terms in the linear sieve formulas can be improved in the special cases A = {N -p : p N } or A = {p + 2 : p x} (see the end of Section 3). This seems to be interesting and important. Our argument is quite general, which works for all sequences satisfying the Chen-Iwaniec switching principle.

(ii) Certainly we could obtain a better constant than 3.3996 in Theorem 3 if we used mean value theorems of ( [START_REF] Fouvry | Autour du théorème de Bombieri-Vinogradov[END_REF], Corollary 2), ( [START_REF] Fouvry | On the switching principle in sieve theory[END_REF], Lemma 6) and ( [START_REF] Iwaniec | Sums and difference of quartic norms[END_REF], Proposition) as in the proof of Theorem 4. But the numerical computation involved would be quite complicated.

The Chen theorem in short intervals was first studied by Ross [START_REF] Ross | A short intervals result in additive prime number theory[END_REF]. Let α ∈ (0, 1) be a fixed constant and define, for θ ∈ (0, 1), x 2 and even integer N 4,

D 1,2 (N, θ) := |{αN p αN + N θ : Ω(N -p) 2}|, π 1,2 (x, θ) := |{x p x + x θ : Ω(p + 2) 2}|.
He proved (see [START_REF] Wu | Sur l'équation p + 2 = P 2 dans les petits intervalles[END_REF]) that for θ 0.98, N N 0 (θ) and x x 0 (θ),

D 1,2 (N, θ) ≫ Ξ(N, θ), π 1,2 (x, θ) ≫ Π(x, θ),
where

Ξ(N, θ) := N θ (log N ) 2 p|N, p>2 p -1 p -2 p>2 1 - 1 (p -1) 2 and Π(x, θ) := 2x θ (log x) 2 p>2 1 - 1 (p -1) 2 .
The constant 0.98 was further improved to 0.973 by Wu [START_REF] Wu | Sur l'équation p + 2 = P 2 dans les petits intervalles[END_REF], to 0.9729 by Salerno & Vitolo [START_REF] Salerno | p + 2 = P 2 in short intervals[END_REF] and to 0.972 by Cai & Lu [START_REF] Cai | Chen's theorem in short intervals[END_REF].

Our method allows us to take a smaller exponent.

Theorem 5. For every θ 0.971, N N 0 (θ) and x x 0 (θ), we have

D 1,2 (N, θ) 0.012 Ξ(N, θ), π 1,2 (x, θ) 0.006 Π(x, θ).
Acknowledgement. The author would like to thank E. Fouvry for his generous help in writing this article, and the referee for his very careful reading of the manuscript. § 2. Preliminary lemmas

This section is devoted to present the formula of the Rosser-Iwaniec linear sieve and some mean value theorem on the distribution of primes in arithmetic progressions, which will be needed later. Before stating these results, it is necessary to recall some definitions.

Let k be a positive integer and τ k (n) the number of ways of writing n as the product of k positive integers. An arithmetical function λ(q) is of level Q and of order k if λ(q) = 0 for q > Q and |λ(q)| τ k (q) for q 1.

We say that λ is well factorable if for every decomposition

Q = Q 1 Q 2 (Q 1 , Q 2 1) there exist two arithmetical functions λ 1 and λ 2 of level Q 1 , Q 2 and of order k such that λ = λ 1 * λ 2 . Lemma 2.1. If λ ′ is an arithmetical function of level Q ′ ( Q) and of order k ′ , then λ * λ ′ is well factorable of level QQ ′ and of order k + k ′ .
Let A be a finite sequence of integers and P a set of prime numbers. For z 2, we put P (z) := p<z, p∈P p and define the sieve function S(A; P, z) := |{a ∈ A : (a, P (z)) = 1}|.

If d is a square-free integer with all its prime factors belonging to P, we denote by A d the set of elements of A divisible by d and we write the following approximate formula (2.1)

|A d | = w(d) d X + r(A, d),
where X > 1 is independent of d, and w(d) is a multiplicative function satisfying (2.2) 0 w(p) < p for p ∈ P.

We also define

V (z) := p<z 1 - w(p) p
and suppose that there exists an absolute constant K > 1 such that

(2.3) V (z 1 ) V (z 2 ) log z 2 log z 1 1 + K log z 1 (z 2 z 1 2).
The formula of the Rosser-Iwaniec linear sieve [START_REF] Iwaniec | A new form of the error term in the linear sieve[END_REF] is stated as follows.

Lemma 2.2. Let 0 < ε < λ + l (q)r(A, q) and (2.5)

S(A; P, z) XV (z) f log Q log z + E - l<L q|P (z) λ - l (q)r(A, q).
In these formulas, L depends only on ε, the λ ± l are well factorable functions of order 1 and of level Q, and E ≪ ε + ε -8 e K /(log Q) 1/3 . The functions F, f are defined by

(2.6) F (u) = 2e γ /u, f (u) = 0 (0 < u 2), (uF (u)) ′ = f (u -1), (uf (u)) ′ = F (u -1) (u > 2),
where γ is Euler's constant.

As usual, we denote by µ(q) Möbius' function, ϕ(q) Euler's function and ν(q) the number of distinct prime factors of q. Define π(y; q, a, m) := mp y mp≡a(mod q) 1, li(y) := y 2 dt log t and E 0 (y; q, a, m) := π(y; q, a, m) -li(y/m) ϕ(q) .

The next lemma is due to Pan & Ding [START_REF] Pan | A new mean value theorem[END_REF], which implies Bombieri-Vinogradov's theorem. Here we state it in the form of ( [START_REF] Pan | Goldbach Conjecture[END_REF], Corollary 8.12). Lemma 2.3. Let f (m) ≪ 1 and α ∈ (0, 1]. Let r 1 (y) be a positive function depending on x and satisfying r 1 (y) ≪ x α , y x.

Let r 2 (m) be a positive function depending on x and y, and satisfying

mr 2 (m) ≪ x, m x α , y x.
Then for every A > 0, there exists a constant B = B(A) > 0 such that

q √ x/(log x) B µ(q) 2 3 ν(q) max y x max (a,q)=1 m x 1-α (m,q)=1 f (m)E 0 (y; q, a, m) ≪ x (log x) A , q √ x/(log x) B µ(q) 2 3 ν(q) max y x max (a,q)=1 m x 1-α (m,q)=1 f (m)E 0 (mr 1 (y); q, a, m) ≪ x (log x) A , q √ x/(log x) B µ(q) 2 3 ν(q) max y x max (a,q)=1 m x 1-α (m,q)=1 f (m)E 0 (mr 2 (m); q, a, m) ≪ x (log x) A .
In order to prove Theorem 5, it is necessary to generalize the mean value theorem of Pan & Ding in short intervals. Such a result was established by Wu ([27], theorem 2). Lemma 2.4. Let f (m) ≪ 1, ε be an arbitrarily small positive number and define H(y, h, q, a, m) := π(y + h; q, a, m) -π(y; q, a, m) -li((y + h)/m) -li(y/m) ϕ(q) .

Then for any A > 0, there exists a constant B = B(A) > 0 such that

q Q µ(q) 2 3 ν(q) max (a,q)=1 max h x θ max x/2<y x m M, (m,q)=1
f (m)H(y, h, q, a, m) ≪ x θ (log x) A for x 10, 3 5 + ε θ 1, Q = x θ-1/2 /(log x) B and M = x (5θ-3)/2-ε . In the proofs of Theorem 3 and 4, we shall need some mean value theorems with well factorable or almost well factorable coefficients.

Let M 1, N 1 and X := M N . Let {α m } and {β n } be two sequences of order k supported in [M, 2M ] and [N, 2N ] respectively. We also suppose the conditions below:

(i) For all B, the equality n≡n0(mod k), (n,d)=1

β n = 1 ϕ(k) (n,dk)=1 β n + O B,k N τ k (d)/(log 2N ) B holds for d 1, k 1 and (k, n 0 ) = 1.
(ii) If n has a prime factor p with p < exp{(log log n) 2 }, then β n = 0.

The following result is an immediate consequence of Corollary 2 of [START_REF] Fouvry | Autour du théorème de Bombieri-Vinogradov[END_REF], Lemma 6 of [START_REF] Fouvry | On the switching principle in sieve theory[END_REF] and the proposition of [START_REF] Iwaniec | Sums and difference of quartic norms[END_REF]. Lemma 2.5. Under the conditions (i) and (ii) above, for any A and for any ε > 0 we have

(q,a)=1 λ(q) mn≡a(mod q) α m β n - 1 ϕ(q) (mn,q)=1 α m β n ≪ ε,A X (log X) A
uniformly for |a| (log X) A and ν := log N/ log X (ε ν 1 -ε). Here λ(q) is a well factorisable function of order 1 and of level Q := X θ(ν)-ε , where θ(ν) is given by

θ(ν) =                                      6-5ν 10
for ε ν Proof. The value (6 -5ν)/10 in [ε, 1/15] comes from Corollary 2 (ii) of [START_REF] Fouvry | Autour du théorème de Bombieri-Vinogradov[END_REF]. The intervals [1/15, 1/10] and [1/10, 3/14] follow from the proposition of [START_REF] Iwaniec | Sums and difference of quartic norms[END_REF] by decomposing λ = λ 1 * λ 2 with

Q 1 = Q = x 1/2-ε , Q 2 = R = N x -ε and Q 1 = Q = x 5/8-ε N -5/4 , Q 2 = R = N x -ε ,
respectively. The remaining case is Lemma 6 of [START_REF] Fouvry | On the switching principle in sieve theory[END_REF].

The next lemma is Corollary 2 (i) of [START_REF] Fouvry | Autour du théorème de Bombieri-Vinogradov[END_REF]. This is the first result, which is valid uniformly for |a| X and has the level of distribution > 1 2 . Lemma 2.6. Under the conditions (i) and (ii) above, for any A and for any ε > 0 we have

(q,a)=1 λ(q) mn≡a(mod q) α m β n - 1 ϕ(q) (mn,q)=1 α m β n ≪ ε,A X (log X) A
uniformly for |a| X and ε ν := log N/ log X 1 10 . Here λ(q) is a well factorisable function of order 1 and of level Q := X 5(1-ν)9-ε .

As usual define π(y; q, a) := p y, p≡a(mod q)

1.

The following result is due to Bombieri, Friedlander & Iwaniec ([2], theorem 10).

Lemma 2.7. Let λ be a well factorable function of order k and of level Q = x 4/7-ε . For any ε > 0 and any A, we have uniformly for x 3 and |a| (log x) A , (q,a)=1 λ(q) π(x; q, a) -li(x)

ϕ(q) ≪ ε,k,A x (log x) A .
When we use the weighted inequality, some coefficients are merely "almost well factorable". So we need the following results, due to Fouvry & Grupp ([13], theorem 2 and the corollary).

Lemma 2.8. Let λ be a well factorable function of level Q 1 and of order k, ξ an arithmetical function satisfying the conditions |ξ(q 2 )| log x and ξ(q 2 ) = 0 (q 2 > Q 2 ) and let Λ be the von Mangoldt function. Then we have for any integer a, any ε > 0 and any A > 0, (q1q2,a)=1 λ(q 1 )ξ(q 2 ) π(x; q 1 q 2 , a) -li(x) ϕ(q 1 q 2 ) ≪ a,ε,k,A x (log x) A , so long as one of the following three conditions is true:

Q 2 Q 1 , Q 1 Q 2 x 4/7-ε , (C.1) Q 2 Q 1 , Q 1 Q 6 2 x 2-ε , (C.2) ξ(q) = Λ(q), Q 1 Q 2 x 11/20-ε , Q 2 x 1/3-ε . (C.3)
The next two lemmas also are useful when we apply the switching principle. Lemma 2.9 ( [START_REF] Wu | Sur la suite des nombres premiers jumeaux[END_REF], Lemma 7). Let λ be a well factorable function of level Q := x 4/7-ε and of order k. Let η > 0 and {ε i } 1 i r be real numbers such that

ε i η, ε 1 + ε 2 + • • • + ε r = 1.
Then for any integer a, any ε > 0 and any A > 0, we have

(q,a)=1 λ(q) p1•••pr ≡a(mod q) x ε i <pi 2x ε i (1 i r) 1 - 1 ϕ(q) (p1•••pr ,q)=1 (x ε i <pi 2x ε i (1 i r) 1 ≪ a,ε,k,A x (log x) A .
Lemma 2.10 ( [START_REF] Wu | Sur la suite des nombres premiers jumeaux[END_REF], Lemma 12). Let x 2 and y = x 1/u . Then

n x p|n⇒p y 1 = x log y ω(u) + O x (log y) 2 ,
where ω(u) is Buchstab's function defined by ω(u) = 1/u (1 u 2) and uω(u) ′ = ω(u -1) (u 2).

Moreover we have ω(u) 0.561522 (u 3.5) and ω(u) 0.567144 (u 2). § 3. Chen's double sieve

We shall sieve the sequence

A := {N -p : p N }.
Let δ > 0 be a sufficiently small number and k ∈ Z. Put

Q := N 1/2-δ , d := Q/d, L := log N, W k := N δ 1+k .
Let ∆ be a real number with 1 + L -4 ∆ < 1 + 2L -4 . We put P(N ) := {p : (p, N ) = 1} and denote by π [Y,Z) the characteristic function of the set P(N ) ∩ [Y, Z). For k ∈ Z + and N 2, let U k (N ) be the set of all arithmetical functions σ which can be written as the form

σ = π [V1/∆,V1) * • • • * π [Vi/∆,Vi) ,
where i is an integer with 0 i k, and V 1 , . . . , V i are real numbers satisfying

(3.1)              V 2 1 Q, V 1 V 2 2 Q, • • • • • • • • • • • • • V 1 • • • V i-1 V 2 i Q, V 1 V 2 • • • V i W k .
By convention, σ is the characteristic function of the set {1} if i = 0. From this definition and Lemma 2.1, we see immediately the following result.

Lemma 3.1. (i) We have U k (N ) ⊂ U k+1 (N ) for k ∈ Z + . (ii) Let σ = π [V1/∆,V1) * • • • * π [Vi/∆,Vi) ∈ U k (N ). Then σ is well factorable of level V := V 1 • • • V i
and of order i. If λ is well factorable of level Q/V and of order 1, then σ * λ is well factorable of level Q and of order k + 1.

Let F and f be defined as in (2.6) and let For k ∈ Z + , N 0 2 and s ∈ [1, 10], we define H k,N0 (s) and h k,N0 (s) as the supremum of h -∞ such that for all N N 0 and σ ∈ U k (N ) one has the following inequalities

Φ(N, σ, s) {A(s) -h} Θ(N, σ) and Φ(N, σ, s) {a(s) + h} Θ(N, σ)
respectively.

From this definition, we deduce immediately the following result.

Lemma 3.2. For k ∈ Z + , N N 0 , s ∈ [1, 10] and σ ∈ U k (N ), we have

(3.3) Φ(N, σ, s) {A(s) -H k,N0 (s)}Θ(N, σ)
and

(3.4) Φ(N, σ, s) {a(s) + h k,N0 (s)}Θ(N, σ).
Obviously H k,N0 (s), h k,N0 (s) are decreasing on N 0 , and they are also decreasing on k by Lemma 3.1. Hence we can write

H k (s) := lim N0→∞ H k,N0 (s), H(s) := lim k→∞ H k (s), h k (s) := lim N0→∞ h k,N0 (s), h(s) := lim k→∞ h k (s). Lemma 3.3. For N 2 and σ = π [V1/∆,V1) * • • • * π [Vi/∆,Vi) ∈ U k (N ), we have L -5k ≪ δ,k d σ(d)/d ≪ δ,k 1, (3.5) d σ(d) ≪ δ,k V 1 • • • V i , (3.6) Θ(N, σ) ≫ δ,k N/L 5k+2 . (3.7) Proof. Let σ = π [V1/∆,V1) * • • • * π [Vi/∆,Vi) ∈ U k (N ). We have (3.8) d σ(d) d = 1 j i pj ∈P(N )∩[Vj/∆,Vj ) 1 p j .
The prime number theorem of the form p x 1 = li(x) + O(x e -2(log x) 1/2 ) implies

pj ∈P(N )∩[Vj/∆,Vj ) 1 p j = Vj /∆ pj <Vj 1 p j - Vj /∆ pj <Vj , pj |N 1 p j = log log V j log(V j /∆) + O e -log 1/2 (Vj /∆) + L V j log L .
Therefore our choice of ∆ and (3.1) give us (3.9)

pj ∈P(N )∩[Vj /∆,Vj) 1/p j ≍ δ,k L -5 . Now (3.5) follows from (3.8) and (3.9). Since σ(d) = 0 implies d V 1 • • • V i , the second inequality in (3.5) implies (3.6). Noticing Θ(N, σ) ≫ N L -2 d σ(d)/d, we obtain (3.7) by the first inequality in (3.5). Proposition 1. For k ∈ Z + and s ∈ [1, 10], we have H k (s) 0 and h k (s) 0.
Proof. We shall prove only the first inequality. The second one can be treated similarly. Let

σ = π [V1/∆,V1) * • • • * π [Vi/∆,Vi) ∈ U k (N ). We use Lemma 2.2 with X = li(N ) ϕ(d) , w(p) = p/(p -1) if p ∈ P(N ), 0 otherwise
to estimate σ(d)S(A d ; P(dN ), d 1/s ). By Merten's formula and (3.1), we can infer that for any ε > 0

(3.10) V (d 1/s ) = {1 + O δ,k (ε)} 2sC dN e γ log d .
By using Lemma 2.2 and (3.10), we deduce (3.11) and summing over d, we obtain

σ(d)S(A d ; P(dN ), d 1/s ) 4li(N ) σ(d)C dN ϕ(d) log d A log(Q/V ) log d 1/s + O δ,k (ε) (3.11) + l<L σ(d) q|P (d 1/s ) λ + l (q)r(A d , q), where λ + l (q) is well factorable of level Q/V with V := V 1 • • • V i and of order 1. If σ(d) = 0, we have d ∈ [V /∆ i , V ], which implies 0 log V -log d i log ∆ 2kL -4 . From this we deduce that A log(Q/V )/ log d 1/s = A(s)+ O δ,k (ε). Inserting
(3.12) Φ(N, σ, s) {A(s) + O δ,k (ε)}Θ(N, σ) + R, where R := l<L d σ(d) q|P (d 1/s ) λ + l (q)r(A d , q). Let q | P (d 1/s ). It is clear that µ(q) 2 = 1 and (N d, q) = 1. Thus we have r(A d , q) = |A dq | -li(N )/ϕ(dq) = π(N ; dq, N ) -li(N )/ϕ(dq).
Hence we can see, by using Lemmas 3.1(ii) and 2.3, that

R ≪ ε q Q τ k+1 (q)|π(N ; dq, N ) -li(N )/ϕ(dq)| (3.13) ≪ δ,k,ε N/L 5k+3 .
From (3.7), (3.12) and (3.13), we deduce

Φ(N, σ, s) {A(s) + O δ,k (ε)}Θ(N, σ),
which implies, by the definition of H k,N0 (s), for any ε > 0 and sufficiently large N 0

H k,N0 (s) -O δ,k (ε). First making N 0 → ∞ and then ε → 0, we obtain H k (s) 0. Proposition 2. For 2 s s ′ 10, we have h(s) h(s ′ ) + s ′ -1 s-1 H(t) t dt and
H(s) H(s ′ ) + s ′ -1 s-1 h(t) t dt.
Proof. We shall only prove the first inequality as the second one can be established in the same way.

Let k 0 and

σ = π [V1/∆,V1) * • • • * π [Vi/∆,Vi) ∈ U k (N )
. By Buchstab's identity, we write

(3.14) Φ(N, σ, s) = Φ(N, σ, s ′ ) - d σ(d) d 1/s ′ p<d 1/s S(A dp ; P(dN ), p).
Next we shall give an upper bound for the last double sums S. The idea is to prove that the characteristic function of dp belongs to U k+1 (N ). Thus S can be estimated by a function H k+1,N0 . We put

V := V 1 • • • V i , V := Q/V and α j := V 1/s ′ ∆ j . Let r be the integer satisfying α r V 1/s < α r+1 . Noticing that σ(d) = 0 ⇒ V 1/s ′ d 1/s ′ and V 1/s d 1/s , we deduce S d σ(d) α0 p<αr S(A dp ; P(dN ), p) + R 1 (3.15) = 1 j r d, p σ(d)π [αj-1 ,αj) (p)S(A dp ; P(dpN ), (dp) 1/s * ) + R 1 ,
where s * := log d/ log p -1 and

R 1 := d σ(d) αr p<d 1/s S(A dp ; P(dN ), p). We would prove that σ * π [αj-1,αj ) ∈ U k+1 (N ). It suffices to verify that V 1 , V 2 , . . . , V i , α j satisfy (3.1) for j r. If V i α j , then V 1 V 2 • • • V i α 2 j V V 2/s = Q 2/s V 1-2/s Q and α j V 1/s ′ V 1/s ′ i W 1/s ′ k W k+1 . If V 1 • • • V l α j V l+1 • • • V i , we have V 1 • • • V l α j V l+1 • • • V 2 n V α 2 j V V 2/s Q for l < n i. Thus σ * π [αj-1,αj ) ∈ U k+1 (N ).
Since s * depends on d and p, we replace it by a suitable quantity independent of d and p such that we can use (3.3) with H k+1,N0 . For this we introduce s 1 := log(V /α j )/ log α j ,

s 2 := log(V /α j-i-1 )/ log α j-1 . Noticing that σ(d)π [αj-1,αj ) (p) = 0 ⇒ s 1 s * s 2 , we deduce from (3.15) that S 1 j r d, p σ(d)π [αj-1,αj ) (p)S(A dp ; P(dpN ), (dp) 1/s1 ) + R 1 + R 2 where R 2 := 1 j r d, p σ(d)π [αj-1,αj ) (p) S(A dp ; P(dpN ), (dp) 1/s * ) -S(A dp ; P(dpN ), (dp) 1/s1 ) . Now we can use (3.3) in Lemma 3.2 to write S 1 j r {A(s 1 ) -H k+1,N0 (s 1 )}Θ(N, σ * π [αj-1,αj ) ) + R 1 + R 2 4li(N ) d σ(d)C dN ϕ(d) log d α0 p<αr A(s * ) -H k+1,N0 (s * ) (p -2)(1 -log p/ log d) + R 1 + R 2 4li(N ) d σ(d)C dN ϕ(d) log d d 1/s ′ p<d 1/s A(s * ) -H k+1,N0 (s * ) (p -2)(1 -log p/ log d) + R 1 + R 2 + R 3 ,
where we have used the fact that A(s) -H k+1,N0 (s) is increasing on s, and the notation

R 3 := 4li(N ) d σ(d)C dN ϕ(d) log d V 1/s ′ p<d 1/s ′ A(s * ) -H k+1,N0 (s * ) ϕ(p)(1 -log p/ log d) .
Applying the prime number theorem, an integration by parts shows that

d 1/s ′ p<d 1/s A(s * ) -H k+1,N0 (s * ) (p -2)(1 -log p/ log d) = s ′ -1 s-1 A(t) -H k+1,N0 (t) t dt + O δ,k (ε). Hence (3.16) S s ′ -1 s-1 A(t) -H k+1,N0 (t) t dt + O δ,k (ε) Θ(N, σ) + R 1 + R 2 + R 3 . It remains to estimate R 1 , R 2 , R 3 . Observing that σ(d) = 0 ⇒ V /∆ i d < V , we have d 1/s V 1/s ∆ i/s . Thus log(log d 1/s / log α r ) log(1 + log ∆ 1+i/s / log(V 1/s /∆)) ≪ δ,k L -5
. By using the prime number theorem and the previous estimate, we have

R 1 ≪ d σ(d) αr p<d 1/s N/dp (3.17) ≪ N L -5 d σ(d)/ϕ(d) ≪ δ,k Θ(N, σ)/L 3 .
Similarly we can show that

(3.18) R 3 ≪ δ,k Θ(N, σ)/L 3 .
By the definition of R 2 , we easily see that

R 2 ≪ d σ(d) α0 p<αr (dp) 1/s * p ′ <(dp) 1/s 1 N/dpp ′ .
Using a similar preceding argument, we can show that

R 2 ≪ δ,k N L 4 d σ(d) ϕ(d) log d V 1/s ′ p<V 1/s 1 p (3.19) ≪ δ,k Θ(N, σ) L 3 .
Combining (3.16)-(3.19), we obtain the desired upper bound, for N N 0 ,

(3.20) S s ′ -1 s-1 A(t) -H k+1,N0 (t) t dt + O δ,k (ε) Θ(N, σ).
Inserting it in (3.14), estimating the first sum on the right-hand side of (3.14) by (3.9) and noticing the relation

a(s ′ ) -a(s) = s ′ s A(t -1) dt, we find that, for N N 0 (ε, δ, k), Φ(N, σ, s) a(s) + h k,N0 (s ′ ) + s ′ -1 s-1 H k+1,N0 (t) t dt + O δ,k (ε) Θ(N, σ).
From the definition of h k,N0 (s), we deduce that, for any ε > 0 and for sufficiently large N 0 ,

h k,N0 (s) h k,N0 (s ′ ) + s ′ -1 s-1 H k+1,N0 (t) t dt + O δ,k (ε).
Taking N 0 → ∞ and then ε → 0, we obtain

h k (s) h k (s ′ ) + s ′ -1 s-1 H k+1 (t) t dt
which implies the required inequality. This completes the proof. Similarly the definition of h k,N0 (s) and the fact that a(s) = 0 for 1 s 2 show that h(s) is increasing on [START_REF] Bombieri | Small differences between prime numbers[END_REF][START_REF] Bombieri | Primes in arithmetic progressions to large moduli[END_REF]. Propositions 2 and 1 imply that h(s) is decreasing on [START_REF] Bombieri | Primes in arithmetic progressions to large moduli[END_REF][START_REF] Chen | On the Goldbach's problem and the sieve methods[END_REF]. This concludes the proof.

The central results in this section are Propositions 3 and 4 below. Before stating it, it is necessary to introduce some notation.

Let 1 s 3 s ′ 5 and s κ 3 κ 2 κ 1 s ′ . Define

α 1 := κ 1 -2, α 4 := s ′ -s ′ /κ 2 -1, α 7 := s ′ -s ′ /κ 1 -s ′ /κ 3 , α 2 := s ′ -2, α 5 := s ′ -s ′ /κ 3 -1, α 8 := s ′ -s ′ /κ 1 -s ′ /κ 2 , α 3 := s ′ -s ′ /s -1, α 6 := s ′ -2s ′ /κ 2 , α 9 := κ 1 -κ 1 /κ 2 -1. Let 1 [a,b] (t) be the characteristic function of the interval [a, b]. We put σ(a, b, c) := b a log c t -1 dt t , σ 0 (t) := σ(3, t + 2, t + 1) 1 -σ(3, 5, 4) .
We can prove that H(s) satisfies some functional inequalities.

Proposition 3. For 5 s ′ 3 s 2 and s ′ -s ′ /s 2, we have

(3.21) H(s) Ψ 1 (s) + 3 1 H(t)Ξ 1 (t; s) dt,
where Ψ 1 (s) is defined as in Lemma 5.1 below and Ξ 1 (t; s) = Ξ 1 (t; s, s ′ ) is given by

Ξ 1 (t; s) := σ 0 (t) 2t log 16 (s -1)(s ′ -1) + 1 [α2,3] (t) 2t log (t + 1) 2 (s -1)(s ′ -1) + 1 [α3,α2] (t) 2t log t + 1 (s -1)(s ′ -1 -t) . Proposition 4. Let 5 s ′ 3 s 2 and s κ 3 < κ 2 < κ 1 s ′ satisfy s ′ -s ′ /s 2, 1 α i 3 (1 i 9), α 1 < α 4 , α 5 < α 8 .
Then we have

(3.22) H(s) Ψ 2 (s) + 3 1 H(t)Ξ 2 (t; s) dt,
where Ψ 2 (s) is defined as in Lemma 5.2 below, and

Ξ 2 (t; s) = Ξ 2 (t; s, s ′ , κ 1 , κ 2 , κ 3 ) is given by Ξ 2 (t; s) := σ 0 (t) 5t log 1024 (s -1)(s ′ -1)(κ 1 -1)(κ 2 -1)(κ 3 -1) + 1 [α2,3] (t) 5t log (t + 1) 5 (s -1)(s ′ -1)(κ 1 -1)(κ 2 -1)(κ 3 -1) + 1 [α9,α1] (t) 5t log t + 1 (κ 2 -1)(κ 1 -1 -t) + 1 [α5,α2] (t) 5t log t + 1 (κ 3 -1)(s ′ -1 -t) + 1 [α3,α2] (t) 5t log t + 1 (s -1)(s ′ -1 -t) + 1 [α1,α2] (t) 5t log (t + 1) 2 (κ 1 -1)(κ 2 -1) + 1 [α7,α5] (t) 5t(1 -t/s ′ ) log s ′2 (κ 1 s ′ -s ′ -κ 1 t)(κ 3 s ′ -s ′ -κ 3 t) + 1 [α5,α8] (t) 5t(1 -t/s ′ ) log s ′ (s ′ -1 -t) κ 1 s ′ -s ′ -κ 1 t + 1 [α6,α8] (t) 5t(1 -t/s ′ ) log s ′ κ 2 s ′ -s ′ -κ 2 t + 1 [α8,α2] (t) 5t(1 -t/s ′ ) log(s ′ -1 -t).
We shall prove these two propositions in Section 6. It is easy to see that Ξ i (t; s) is positive and that for s ∈ [1, 3) there exist parameters s ′ , κ i such that Ψ i (s) > 0. Therefore H(s) > 0 for s ∈ [1, 3) and then Proposition 2 implies that h(s) > 0 for s ∈ [1, 3). In Sections 7 and 8, we shall give numeric solution of (3.20) and (3.21), and prove Theorems 1 and 3. § 4. Weighted inequalities for sieve function The aim of this section is to present two weighted inequalities for sieve function. The first is essentially due to Chen ([10], [START_REF] Ross | A short intervals result in additive prime number theory[END_REF]). The second is new, which is not only much simpler than the third weighted inequality of Chen ([10], (64), (90) and (91)) but also more powerful.

Lemma 4.1. Let 1 s < s ′ 10. For N 2, k 0 and σ ∈ U k (N ), we have 2Φ(N, σ, s) d σ(d)(Ω 1 -Ω 2 + Ω 3 ) + O δ,k (N 1-η ),
where η = η(δ, k) > 0 and Ω i = Ω i (d) is given by

Ω 1 := 2S(A d ; P(dN ), d 1/s ′ ), Ω 2 := d 1/s ′ p<d 1/s (p,N )=1 S(A dp ; P(dN ), d 1/s ′ ), Ω 3 := d 1/s ′ p1<p2<p3<d 1/s (p1p2p3,N )=1 S(A dp1p2p3 ; P(dp 1 N ), p 2 ).
Proof. By the Buchstab identity, we have

2S(A d ; P(dN ), d 1/s ) = Ω 1 -2 d 1/s ′ p<d 1/s (p,N )=1
S(A dp ; P(dN ), p), (4.1)

d 1/s ′ p1<d 1/s (p1,N )=1 S(A dp1 ; P(dN ), p 1 ) = Ω 0 + d 1/s ′ p 1 p 3 <d 1/s (p1p3,N )=1 S(A dp1p3 ; P(dp 1 N ), p 3 ), (4.2) d 1/s ′ p3<d 1/s (p3,N )=1 S(A dp3 ; P(dN ), p 3 ) = Ω 2 - d 1/s ′ p 1 <p 3 <d 1/s (p1p3,N )=1 S(A dp1p3 ; P(dN ), p 1 ), (4.3)
where Ω 0 :=

d 1/s ′ p1<d 1/s (p1,N )=1 S(A dp1 ; P(dN ), d 1/s ).
Inserting (4.2)-(4.3) into (4.1), dropping the term Ω 0 (which is non-negative) and replacing p 1 p 3 by p 1 < p 3 , we find that

2S(A d ; P(dN ), d 1/s ) Ω 1 -Ω 2 + ∆ 1 ,
where

∆ 1 := d 1/s ′ p1<p3<d 1/s (p1p3,N )=1 S(A dp1p3 ; P(dN ), p 1 ) -S(A dp1p3 ; P(dp 1 N ), p 3 ) (4.4) = d 1/s ′ p 1 p 2 <p 3 <d 1/s (p1p2p3,N )=1 S(A dp1p2p3 ; P(dp 1 N ), p 2 ).
By the inequality S(A dp 2 1 p3 ; P(dN ), p 1 ) ≪ N/dp 2 1 p 3 and the fact that d W k , we easily see

d 1/s ′ p1<p3<d 1/s S(A dp 2 1 p3 ; P(dN ), p 1 ) ≪ d 1/s ′ p1<p3<d 1/s N/dp 2 1 p 3 ≪ δ,k N 1-η /d
for some η = η(δ, k) > 0. Inserting it in (4.4), we obtain that

(4.5) ∆ 1 = Ω 3 + O δ,k (N 1-η /d).
Finally we complete the proof with (3.5).

Lemma 4.2. Let 1 s κ 3 < κ 2 < κ 1 s ′ 10. For N 2, k 0 and σ ∈ U k (N ), we have

5Φ(N, σ, s) d σ(d)(Γ 1 -Γ 2 -Γ 3 -Γ 4 + Γ 5 + • • • + Γ 21 ) + O δ,k (N 1-η ),
where η = η(δ, k) > 0 and Γ i = Γ i (d) is given by

Γ 1 := 4S(A d ; P(dN ), d 1/s ′ ) + S(A d ; P(dN ), d 1/κ1 ), Γ 2 := d 1/s ′ p<d 1/s (p,N )=1 S(A dp ; P(dN ), d 1/s ′ ), Γ 3 := d 1/s ′ p<d 1/κ 2 (p,N )=1 S(A dp ; P(dN ), d 1/s ′ ), Γ 4 := d 1/s ′ p<d 1/κ 3 (p,N )=1 S(A dp ; P(dN ), d 1/s ′ ), Γ 5 := d 1/s ′ p1<p2<d 1/κ 2 (p1p2,N )=1 S(A dp1p2 ; P(dN ), d 1/s ′ ), Γ 6 := d 1/s ′ p1<d 1/κ 1 , d 1/κ 2 p2<d 1/κ 3 (p1p2,N )=1 S(A dp1p2 ; P(dN ), d 1/s ′ ), Γ 7 := d 1/s ′ p1<p2<d 1/κ 1 (p1p2,N )=1
S(A dp1p2 ; P(dN ), p 1 ),

Γ 8 := d 1/s ′ p1<d 1/κ 1 p2<d 1/κ 2 (p1p2,N )=1
S(A dp1p2 ; P(dN ), p 1 ),

Γ 9 := d 1/κ 1 p1<p2<p3<d 1/κ 3 (p1p2p3,N )=1 S(A dp1p2p3 ; P(dN ), p 2 ), Γ 10 := d 1/κ 1 p1<p2<d 1/κ 2 p3<d 1/s (p1p2p3,N )=1 S(A dp1p2p3 ; P(dN ), p 2 ), Γ 11 := d 1/κ 1 p1<d 1/κ 2 p2<p3<d 1/κ 3 (p1p2p3,N )=1 S(A dp1p2p3 ; P(dN ), p 2 ), Γ 12 := d 1/s ′ p1<p2<d 1/κ 1 , d 1/κ 3 p3<d 1/s (p1p2p3,N )=1 S(A dp1p2p3 ; P(dN ), p 2 ), Γ 13 := d 1/s ′ p1<d 1/κ 1 p2<d 1/κ 2 p3<d 1/s (p1p2p3,N )=1 S(A dp1p2p3 ; P(dN ), p 2 ), Γ 14 := Γ 15 := d 1/κ 1 p1<d 1/κ 2 p2<d 1/κ 3 p3<d 1/s (p1p2p3,N )=1 S(A dp1p2p3 ; P(dN ), p 2 ), Γ 16 := d 1/κ 2 p1<p2<p3<p4<d 1/κ 3 (p1p2p3p4,N )=1
S(A dp1p2p3p4 ; P(dN ), p 3 ),

Γ 17 := d 1/κ 2 p1<p2<p3<d 1/κ 3 p4<d 1/s (p1p2p3p4,N )=1
S(A dp1p2p3p4 ; P(dN ), p 3 ),

Γ 18 := d 1/κ 2 p1<p2<d 1/κ 3 p3<p4<d 1/s (p1p2p3p4,N )=1
S(A dp1p2p3p4 ; P(dN ), p 3 ),

Γ 19 := d 1/κ 1 p1<d 1/κ 2 , d 1/κ 3 p2<p3<p4<d 1/s (p1p2p3p4,N )=1
S(A dp1p2p3p4 ; P(dN ), p 3 ),

Γ 20 := d 1/κ 2 p1<d 1/κ 3 p2<p3<p4<p5<d 1/s (p1p2p3p4p5,N )=1
S(A dp1p2p3p4p5 ; P(dN ), p 4 ),

Γ 21 := d 1/κ 3 p1<p2<p3<p4<p5<p6<d 1/s (p1p2p3p4p5p6,N )=1
S(A dp1p2p3p4p5p6 ; P(dN ), p 5 ).

Proof. Let S := S(A d ; P(dN ), d 1/s ). By using the Buchstab identity, we have

2S = 2S(A d ; P(dN ), d 1/s ′ ) - d 1/s ′ p<d 1/s (p,N )=1 S(A dp ; P(dN ), p) (4.6) -Γ 3 + d 1/s ′ p1<p2<d 1/κ 2 (p1p2,N )=1
S(A dp1p2 ; P(dN ), p 1 ) -

d 1/κ 2 p<d 1/s (p,N )=1
S(A dp ; P(dN ), p)

=: 2S(A d ; P(dN ), d 1/s ′ ) -E 1 -Γ 3 + D ′ 1 -E 2 .
We can also write, always by Buchstab's identity,

S = S(A d ; P(dN ), d 1/s ′ ) - d 1/s ′ p<d 1/κ 3 (p,N )=1 S(A dp ; P(dN ), p) (4.7) - d 1/κ 3 p<d 1/s (p,N )=1
S(A dp ; P(dN ), p).

But we have

d 1/s ′ p<d 1/κ 3 (p,N )=1 S(A dp ; P(dN ), p) = Γ 4 -Γ 7 -Γ 8 - d 1/κ 1 p1<p2<d 1/κ 3 (p1p2,N )=1 S(A dp1p2 ; P(dN ), p 1 ) -Γ 6 + d 1/s ′ p1<p2<d 1/κ 1 , d 1/κ 2 p3<d 1/κ 3 (p1p2p3,N )=1
S(A dp1p2p3 ; P(dN ), p 1 ).

Inserting these relations into (4.7), it yields that

S = S(A d ; P(dN ), d 1/s ′ ) -Γ 4 + Γ 6 + Γ 7 + Γ 8 (4.8) - d 1/κ 3 p<d 1/s (p,N )=1
S(A dp ; P(dN ), p) +

d 1/κ 1 p1<p2<d 1/κ 3 (p1p2,N )=1 S(A dp1p2 ; P(dN ), p 1 )
-

d 1/s ′ p1<p2<d 1/κ 1 , d 1/κ 2 p3<d 1/κ 3 (p1p2p3,N )=1 S(A dp1p2p3 ; P(dN ), p 1 ) =: S(A d ; P(dN ), d 1/s ′ ) -Γ 4 + Γ 6 + Γ 7 + Γ 8 -E 3 + D 2 -E 4 .
Similar to (4.8), we can prove that

S = S(A d ; P(dN ), d 1/s ′ ) -Γ 2 + Γ 5 - d 1/s ′ p1<p2<p3<d 1/κ 2 (p1p2p3,N )=1
S(A dp1p2p3 ; P(dN ), p 1 ) (4.9)

+ d 1/s ′ p1<d 1/κ 2 p2<d 1/s (p1p2,N )=1 + d 1/κ 2 p1<p2<d 1/s (p1p2,N )=1 S(A dp1p2 ; P(dN ), p 1 ) =: S(A d ; P(dN ), d 1/s ′ ) -Γ 2 + Γ 5 -E 5 + D ′′ 1 + D ′′′ 1 .
Finally we write (4.10) S = S(A d ; P(dN ), d 1/κ1 ) -

d 1/κ 1 p<d 1/κ 3 (p,N )=1 + d 1/κ 3 p<d 1/s (p,N )=1
S(A dp ; P(dN ), p). This implies

d 1/κ 1 p<d 1/κ 3 (p,N )=1
S(A dp ; P(dN ), p)

d 1/κ 1 p1<p2<d 1/κ 3 (p1p2,N )=1 S(A dp1p2 ; P(dN ), p 2 ) + d 1/κ 1 p1<d 1/κ 3 p2<d 1/s (p1p2,N )=1 S(A dp1p2 ; P(dN ), p 2 ).
Inserting it into (4.10), we obtain S S(A d ; P(dN ), d 1/κ1 ) -

d 1/κ 1 p1<p2<d 1/κ 3 (p1p2,N )=1 S(A dp1p2 ; P(dN ), p 2 ) (4.11) - d 1/κ 1 p1<d 1/κ 3 p2<d 1/s (p1p2,N )=1 S(A dp1p2 ; P(dN ), p 2 ) - d 1/κ 3 p<d 1/s (p,N )=1 S(A dp ; P(dN ), p) =: S(A d ; P(dN ), d 1/κ1 ) -E 6 -E 7 -E 3 .
Now by adding up the inequalities (4.6), (4.8), (4.9) and (4.11) and by noticing the estimate

D 2 -E 6 Γ 9 + O δ,k (N 1-η /d), we get (4.12) 5S Γ 1 -Γ 2 -Γ 3 -Γ 4 + Γ 5 + • • • + Γ 9 + ∆ 2 + O δ,k (N 1-η /d) where ∆ 2 := D 1 -E 1 -E 2 -2E 3 -E 4 -E 5 -E 7
and

D 1 := D ′ 1 + D ′′ 1 + D ′′′ 1 = d 1/s ′ p1<p2<d 1/s (p1p2,N )=1
S(A dp1p2 ; P(dN ), p 1 ).

Clearly we have E 1

d 1/s ′ p1<p2<d 1/s (p1p2,N )=1 S(A dp1p2 ; P(dN ), p 2 ).
Thus an application of Bechstab's identity gives us

D 1 -E 1 d 1/s ′ p1<p2<p3<d 1/s (p1p2p3,N )=1 S(A dp1p2p3 ; P(dN ), p 2 ) + O δ,k (N 1-η /d).
From this, we can deduce

D 1 -E 1 -E 5 d 1/s ′ p1<p2<p3<d 1/s (p1p2p3,N )=1 S(A dp1p2p3 ; P(dN ), p 2 ) - d 1/s ′ p1<p2<p3<d 1/κ 2 (p1p2p3,N )=1 S(A dp1p2p3 ; P(dN ), p 2 ) + O δ,k (N 1-η /d) = d 1/s ′ p1<p2<d 1/κ 2 p3<d 1/s (p1p2p3,N )=1 S(A dp1p2p3 ; P(dN ), p 2 ) + d 1/s ′ p1<d 1/κ 2 p2<p3<d 1/s (p1p2p3,N )=1 S(A dp1p2p3 ; P(dN ), p 2 ) + d 1/κ 2 p1<p2<p3<d 1/s (p1p2p3,N )=1 S(A dp1p2p3 ; P(dN ), p 2 ) + O δ,k (N 1-η /d) =: D 3 + D 4 + D 5 + O δ,k (N 1-η /d).
We have

D 3 -E 4 Γ 10 + Γ 12 + Γ 13 .
By splitting D 4 into 4 subsums, we have

D 4 = Γ 11 + Γ 14 + Γ 15 + d 1/κ 1 p1<d 1/κ 2 , d 1/κ 3 p2<p3<d 1/s (p1p2p3,N )=1 S(A dp1p2p3 ; P(dN ), p 2 ).
Similarly by splitting E 7 into 2 subsums, we have

E 7 = E 8 + d 1/κ 1 p1<d 1/κ 2 , d 1/κ 3 p2<d 1/s (p1p2,N )=1
S(A dp1p2 ; P(dN ), p 2 )

E 8 + d 1/κ 1 p1<d 1/κ 2 , d 1/κ 3 p2<p3<d 1/s (p1p2p3,N )=1 S(A dp1p2p3 ; P(dN ), p 3 ),
where

E 8 := d 1/κ 2 p1<d 1/κ 3 p2<d 1/s (p1p2,N )=1 S(A dp1p2 ; P(dN ), p 2 ).
By noticing that

d 1/κ 1 p1<d 1/κ 2 , d 1/κ 3 p2<p3<d 1/s (p1p2p3,N )=1 S(A dp1p2p3 ; P(dN ), p 2 ) -S(A dp1p2p3 ; P(dN ), p 3 ) = Γ 19 + O δ,k (N 1-η /d),
we can deduce

D 4 -E 7 Γ 11 + Γ 14 + Γ 15 + Γ 19 -E 8 + O δ,k (N 1-η /d). Since E 2 d 1/κ 2 p1<p2<p3<d 1/s (p1p2p3,N )=1 S(A dp1p2p3 ; P(dN ), p 3 ),
we have

D 5 -E 2 d 1/κ 2 p1<p2<p3<p4<d 1/s (p1p2p3p4,N )=1 S(A dp1p2p3p4 ; P(dN ), p 3 ) + O δ,k (N 1-η /d) =: D 6 + O δ,k (N 1-η /d).
Similarly

E 3 d 1/κ 3 p1 p2<p3<p4<d 1/s (p1p2p3p4,N )=1 S(A dp1p2p3p4 ; P(dN ), p 4 ) =: E ′ 3 , E 3
d 1/κ 3 p1 p2<p3<p4<p5<d 1/s (p1p2p3p4p5,N )=1 S(A dp1p2p3p4p5 ; P(dN ), p 5 ) =: E ′′ 3 , E 8
d 1/κ 2 p1<d 1/κ 3 p2 p3<p4<d 1/s (p1p2p3p4,N )=1 S(A dp1p2p3p4 ; P(dN ), p 4 ) =: E ′ 8 , D 6 = Γ 16 + Γ 17 + Γ 18 + d 1/κ 2 p1<d 1/κ 3 p2<p3<p4<d 1/s (p1p2p3p4,N )=1 S(A dp1p2p3p4 ; P(dN ), p 3 ) + d 1/κ 3 p1<p2<p3<p4<d 1/s (p1p2p3p4,N )=1 S(A dp1p2p3p4 ; P(dN ), p 3 ) =: Γ 16 + Γ 17 + Γ 18 + D ′ 6 + D ′′ 6 . Since D ′ 6 -E ′ 8 = Γ 20 + O δ,k (N 1-η /d)
and

D ′′ 6 -E ′ 3 -E ′′ 3 = d 1/κ 3 p1<p2<p3<p4<p5<d 1/s (p1p2p3p4p5,N )=1 S(A dp1p2p3p4p5 ; P(dN ), p 4 ) - d 1/κ 3 p1<p2<p3<p4<p5<d 1/s (p1p2p3p4p5,N )=1 S(A dp1p2p3p4p5 ; P(dN ), p 5 ) + O δ,k (N 1-η /d) = Γ 21 + O δ,k (N 1-η /d),
we have

D 6 -2E 3 -E 8 Γ 16 + Γ 17 + Γ 18 + D ′ 6 + D ′′ 6 -E ′ 3 -E ′′ 3 -E ′ 8 = Γ 16 + Γ 17 + Γ 18 + Γ 20 + Γ 21 + O δ,k (N 1-η /d).
Combining these estimations leads to the following inequalities (4.13)

∆ 2 D 3 + D 4 + D 5 -E 2 -2E 3 -E 4 -E 7 Γ 10 + • • • + Γ 15 + Γ 19 + D 6 -2E 3 -E 8 + O δ,k (N 1-η /d) Γ 10 + • • • + Γ 21 + O δ,k (N 1-η /d).
Now the desired result follows from (4.12) and (4.13). § 5. Functional inequalities between H(s) and h(s)

In this section, we start from two weighted inequalities for the sieve function to deduce two functional inequalities between H(s) and h(s). They will be used to prove Propositions 3 and 4 in the next section. Lemma 5.1. For 5 s ′ 3 s 2 and s ′ -s ′ /s 2, we have

H(s) Ψ 1 (s) + 1 2 1-1/s ′ 1-1/s h(s ′ t) t(1 -t) dt + H(s ′ ),
where Ψ 1 (s) = Ψ 1 (s, s ′ ) is given by

Ψ 1 (s) := - s ′ -1 2 log(t -1) t dt + 1 2 1-1/s ′ 1-1/s log(s ′ t -1) t(1 -t) dt -I 1 (s)
and I 1 (s) = I 1 (s, s ′ ) is given by

I 1 (s) := max φ 2 1/s ′ t u v 1/s ω φ -t -u -v u dt du dv tu 2 v .
Proof. Our starting point is the inequality in Lemma 4.1. We need to estimate all terms in the right-hand side of this inequality. Firstly, (3.3) of Lemma 3.2 gives us (5.1)

d σ(d)Ω 1 2 A(s ′ ) -H k,N0 (s ′ ) Θ(N, σ).
Secondly, by an argument similar to the proof of (3.20), we can prove, for any ε > 0 and N N 0 (ε, δ, k),

(5.2) d σ(d)Ω 2 1-1/s ′ 1-1/s a(s ′ t) + h k+1,N0 (s ′ t) t(1 -t) dt -ε Θ(N, σ).
(N ), Q 1/2 ) + O(Q 1/2 ).
In the set E, d is not determined uniquely by e. This causes technical difficulty. In order to avoid it, we define E ′ and B ′ , similar to E and B, with the condition (n, N P (p 2 )/p 1 ) = 1 replaced by (n, dN P (p 2 )/p 1 ) = 1 and E by E ′ respectively. Obviously the difference S(B;

P(N ), Q 1/2 ) - S(B ′ ; P(N ), Q 1/2 ) is d σ(d) d 1/s ′ p1<p2<p3<d 1/s n N/dp1p2p3 (n,d)=1 1 ≪ δ,k N 1-η
where η = η(δ, k) > 0. Hence (5.4)

d σ(d)Ω 3 S(B ′ ; P(N ), Q 1/2 ) + O δ,k (N 1-η ).
In order to estimate S(B ′ ; P(N ), Q 1/2 ), we use Theorem 5.2 in [START_REF] Halberstam | Sieve Methods[END_REF] with

X := e∈E ′ p2<p3 κ(d,e)
1, w(q) = q/ϕ(q) if µ(q) 2 = (q, N ) = 1 0 otherwise to write (5.5) S(B;

P(N ), Q 1/2 ) 8(1 + ε) C N X log N + O(R 1 + R 2 ),
where

R 1 := q<Q q|P (Q 1/2 ) 3 ν1(q) e∈E ′ (e,q)=1 p2<p3 κ(d,e) ep3≡N (mod q) 1 - 1 ϕ(q) p2<p3 κ(d,e) 1 , R 2 := q<Q q|P (Q 1/2 ) 3 ν1(q) /ϕ(q) e∈E ′ (e,q)>1 p2<p3 κ(d,e) 1.
We first estimate R 2 . Noticing that for e ∈ E ′ we have e N 1-η and the smallest prime factor of e is min{p 1 , V k /∆} W 1/s ′ k , we can deduce (5.6)

R 2 N q Q 3 ν1(q) ϕ(q) e N 1-η (e,q) W 1/s ′ k 1 e ≪ N L m W 1/s ′ k 1 m q Q q≡0(mod m) 3 ν1(q) ϕ(q) ≪ N W 1/3s ′ k m W 1/s ′ k 1 m q Q q≡0(mod m) 1 q ≪ N W 1/2s ′ k m W 1/s ′ k 1 m 2 ≪ δ,k Θ(N, σ) L 3 .
Next we estimate R 1 . Let g(a) := e∈E ′ ,e=a 1. Obviously for each e = dnp 1 p 2 ∈ E ′ , the integers d, n, p 1 , p 2 are pairwisely coprime. Therefore they are uniquely determined by e. Thus g(a) 1 and there are some injections r 0 (e) = d and r(e) = p 2 . Then we have

R 1 ≪ q Q (q,N )=1 µ(q) 2 3 ν1(q) a∈I(a) (a,q)=1 g(a) p2<p3 κ(d,a) ap3≡N (mod q) 1 - 1 ϕ(q) p2<p3 κ(d,a) 1 ,
where I(a) := r 0 (a)r 0 (a) 2/s ′ , N/r 0 (a) 1/s ′ .

Since d 1/s ′ < r(e) < d 1/s and er(e) N , we can write

R 1 ≪ R (1) 1 + R (2) 1 + R (3) 1 ,
where R Replacing (n, N P (p 2 )/p 1 ) = 1 by (n, N P (p 2 )) = 1 in the definition of X, the difference is ≪ δ,k N L 2 /dd 1/s ′ ≪ δ,k N/L 2 d. Thus we can obtain, by Lemma 2.10, that

= q Q (q,N )=1 µ(q) 2 3 ν1(q) a∈I1(a) (a,q)=1 g(a) p3 r0(a) 1/s ap3≡N (mod q) 1 - 1 ϕ(q) p3 r0(a) 1/s 1 , R (2) 1 := q Q (q,N )=1 µ(q) 2 3 ν1(q) a∈I2(a) (a,q)=1 g(a) p3 N/a ap3≡N (mod q) 1 - 1 ϕ(q) p3 N/a 1 , R (3) 1 := q Q (q,N )=1 µ(q) 2 3 ν1(q) a∈I(a) (a,q)=1 g(a) p3 r(a) ap3≡N (mod q) 1 - 1 ϕ(q) (1) 1 :
X = d σ(d) d 1/s ′ p1<p2<p3<d 1/s (p1p2,dN )=1 n N/dp1p2p3 (n, N P (p2))=1 1 + O δ,k (N/L 2 d) (1 + ε) d σ(d) d 1/s ′ p1<p2<p3<d 1/s N ω(log(N/dp 1 p 2 p 3 )/ log p 2 ) dp 1 p 2 p 3 log p 2 + O δ,k N L 2 d .
By applying the prime number theorem, we can deduce

X (1 + ε)N d σ(d) d log d 1/s ′ t u v 1/s ω φ d,N -t -u -v u dt du dv tu 2 v , where φ d,N := log(N/d)/ log d. Obviously σ(d) = 0 implies φ d,N 2. Thus (5.8) X (1 + ε)I 1 (s)N d σ(d) d log d (1 + ε)I 1 (s)N d σ(d) ϕ(d) log d .
Combining (5.4)-(5.8) and noticing C N C dN , we obtain, for any ε > 0 and N N 0 (ε, δ, k),

(5.9)

d σ(d)Ω 3 {2I 1 (s) + ε}Θ(N, σ).
Inserting (5.1), (5.2) and (5.9) into the inequality of Lemma 4.1 and noticing that (5.10)

A(s ′ ) = A(s) + s ′ -1 2 log(t -1) t dt, a(s ′ t) = log(s ′ t -1), we find that Φ(N, σ, s) A(s) -Ψ 1 (s) - 1 2 1-1/s ′ 1-1/s h k+1,N0 (s ′ t) t(1 -t) dt -H k,N0 (s ′ ) + ε Θ(N, σ).
By the definition of H k,N0 (s), we must have

H k,N0 (s) Ψ 1 (s) + 1 2 1-1/s ′ 1-1/s h k+1,N0 (s ′ t) t(1 -t) dt + H k,N0 (s ′ ) -ε.
Making N 0 → ∞ and then ε → 0 yields

H k (s) Ψ 1 (s) + 1 2 1-1/s ′ 1-1/s h k+1 (s ′ t) t(1 -t) dt + H k (s ′ ).
Now it remains to take k → ∞ to get the desired result.

Lemma 5.2. For 5 s ′ 3 s 2, s ′ -s ′ /s 2 and s κ 3 < κ 2 < κ 1 s ′ , we have

H(s) Ψ 2 (s) + 4 5 H(s ′ ) + 1 5 H(κ 1 ) + 1 5 1-1/s ′ 1-1/s h(s ′ t) t(1 -t) dt + 1 5 1-1/s ′ 1-1/κ2 h(s ′ t) t(1 -t) dt + 1 5 1-1/s ′ 1-1/κ3 h(s ′ t) t(1 -t) dt + 1 5 1/κ2 1/s ′ dt t 1/κ2 t H(s ′ -s ′ t -s ′ u) u(1 -t -u) du + 1 5 1/κ1 1/s ′ dt t 1/κ3 1/κ2 H(s ′ -s ′ t -s ′ u) u(1 -t -u) du + 1 5 1/κ1 1/s ′ dt t 1/κ2 t H((1 -t -u)/t) u(1 -t -u) du,
where

Ψ 2 (s) = Ψ 2 (s, s ′ , κ 1 , κ 2 , κ 3 ) is given by Ψ 2 (s) := - 2 5 s ′ -1 2 log(t -1) t dt - 2 5 κ1-1 2 log(t -1) t dt - 1 5 κ2-1 2 log(t -1) t dt + 1 5 1-1/s ′ 1-1/s log(s ′ t -1) t(1 -t) dt + 1 5 1-1/κ1 1-1/κ3 log(κ 1 t -1) t(1 -t) dt - 2 5
21 i=9 I 2,i (s)
and I 2,i (s) = I 2,i (s, s ′ , κ 1 , κ 2 , κ 3 ) is given by

I 2,i (s) := max φ 2 D2,i ω φ -t -u -v u dt du dv tu 2 v (i = 9, . . . , 15), I 2,i (s) := max φ 2 D2,i ω φ -t -u -v -w v
dt du dv dw tuv 2 w (i = 16, . . . , [START_REF] Pan | On the upper bound of the number of ways to represent an even integer as a sum of two primes[END_REF],

I 2,20 (s) := max φ 2 D2,20 ω φ -t -u -v -w -x w dt du dv dw dx tuvw 2 x , I 2,21 (s) := max φ 2 D2,21 ω φ -t -u -v -w -x -y x
dt du dv dw dx dy tuvwx 2 y .

The sets D 2,i (9 i 21) are defined as follows:

D 2,9 := {(t, u, v) : 1/κ 1 t u v 1/κ 3 }, D 2,10 := {(t, u, v) : 1/κ 1 t u 1/κ 2 v 1/s}, D 2,11 := {(t, u, v) : 1/κ 1 t 1/κ 2 u v 1/κ 3 }, D 2,12 := {(t, u, v) : 1/s ′ t u 1/κ 1 , 1/κ 3 v 1/s}, D 2,13 := {(t, u, v) : 1/s ′ t 1/κ 1 u 1/κ 2 v 1/s}, D 2,14 := {(t, u, v) : 1/s ′ t 1/κ 1 , 1/κ 2 u v 1/s}, D 2,15 := {(t, u, v) : 1/κ 1 t 1/κ 2 u 1/κ 3 v 1/
d σ(d)Γ 1 4A(s ′ ) + A(κ 1 ) -4H k,N0 (s ′ ) -H k,N0 (κ 1 ) Θ(N, σ).
Similar to (3.20), we can prove

d σ(d)Γ 2 1-1/s ′ 1-1/s a(s ′ t) + h k+1,N0 (s ′ t) t(1 -t) dt -ε Θ(N, σ), (5.12) d σ(d)Γ 3 1-1/s ′ 1-1/κ2 a(s ′ t) + h k+1,N0 (s ′ t) t(1 -t) dt -ε Θ(N, σ), (5.13) d σ(d)Γ 4 1-1/s ′ 1-1/κ3 a(s ′ t) + h k+1,N0 (s ′ t) t(1 -t) dt -ε Θ(N, σ). (5.14)
Similar to (3.20)

and in view of

A(s ′ -s ′ t -s ′ u) = A((1 -t -u)/t) = 1, we have d σ(d)Γ 5 1/κ2 1/s ′ dt t 1/κ2 t 1 -H k+2,N0 (s ′ -s ′ t -s ′ u) u(1 -t -u) du + ε Θ(N, σ), (5.15) d σ(d)Γ 6 1/κ1 1/s ′ dt t 1/κ3 1/κ2 1 -H k+2,N0 (s ′ -s ′ t -s ′ u) u(1 -t -u) du + ε Θ(N, σ), (5.16) d σ(d)Γ 7 1/κ1 1/s ′ dt t 1/κ1 t 1 -H k+2,N0 ((1 -t -u)/t) u(1 -t -u) du + ε Θ(N, σ), (5.17) d σ(d)Γ 8 1/κ1 1/s ′ dt t 1/κ2 1/κ1 1 -H k+2,N0 ((1 -t -u)/t) u(1 -t -u) du + ε Θ(N, σ). (5.18)
We have also, for i = 9, . . . , 21,

(5.19) d σ(d)Γ i {2I 2,i (s) + ε}Θ(N, σ).
As before, inserting (5.11)- (5.19) into the inequality of Lemma 4.2 and using the definition of H k,N0 (s), we can deduce

(5.20) 5H k,N0 (s) A(s, s ′ ) + B(s, s ′ ) -ε, where
A(s, s ′ ) := 5A(s) -4A(s ′ ) -A(κ 1 ) + 4H k,N0 (s ′ ) + H k,N0 (κ 1 ) + 1-1/s ′ 1-1/s a(s ′ t) t(1 -t) dt + 1-1/s ′ 1-1/κ2 a(s ′ t) t(1 -t) dt + 1-1/s ′ 1-1/κ3 a(s ′ t) t(1 -t) dt - 1/κ2 1/s ′ 1/κ2 t dt du tu(1 -t -u) - 1/κ1 1/s ′ 1/κ3 t dt du tu(1 -t -u) - 2
21 i=9 I 2,i (s) and B(s, s ′ ) := 1-1/s ′ 1-1/s + 1-1/s ′ 1-1/κ2 + 1-1/s ′ 1-1/κ3 h k+1,N0 (s ′ t) t(1 -t) dt + 1/κ2 1/s ′ dt t 1/κ2 t H k+2,N0 (s ′ -s ′ t -s ′ u) u(1 -t -u) du + 1/κ1 1/s ′ dt t 1/κ3 1/κ2 H k+2,N0 (s ′ -s ′ t -s ′ u) u(1 -t -u) du + 1/κ1 1/s ′ dt t 1/κ2 t H k+2,N0 ((1 -t -u)/t) u(1 -t -u) du.
For a b > 2, we have

1/b 1/a dt t 1/b t du u(1 -t -u) = 1/b 1/a du u u 1/a dt t(1 -t -u) = 1/b 1/a log(a -1 -au) -log(1/u -2) u(1 -u) du = 1-1/a 1-1/b log(at -1) t(1 -t) dt - a-1 b-1 log(t -1) t dt,
where we have used the change of variables t = 1 -u and t = 1/u -1 respectively. Similarly for a b c d > 2, we have

1/b 1/a dt t 1/d 1/c du u(1 -t -u) = 1-1/c 1-1/d log(at -1) t(1 -t) dt - 1-1/c 1-1/d log(bt -1) t(1 -t) dt.
By using these two relations and (5.10), a simple calculation shows

A(s, s ′ ) = 5Ψ 2 (s) + 4H k,N0 (s ′ ) + H k,N0 (κ 1 ).
Inserting this into (5.20) and making N → ∞, ε → 0 and k → ∞, we obtain the desired inequality. This completes the proof. § 6. Proofs of Propositions 3 and 4

We first prove a preliminary lemma. Let

1 [a,b] (t) be the characteristic function of the interval [a, b]. We put σ(a, b, c) := b a log c t -1 dt t , σ 0 (t) := σ(3, t + 2, t + 1) 1 -σ(3, 5 , 4) .
Lemma 6.1. Let 3 s ′ 5, 0 < a < b < 1 and 2 ac < bc 4. Then we have h( 4)

3 1 H(t) σ 0 (t) t dt. (6.1) H(s ′ ) 3 1 H(t) σ 0 (t) t log 4 s ′ -1 + 1 [s ′ -2,3] (t) t log t + 1 s ′ -1 dt, (6.2) b a h(ct) t(1 -t) dt log b -ab a -ab 3 1 H(t) σ 0 (t) + 1 [bc-1,3] (t) t dt (6.3) + 3 1 H(t) 1 [ac-1,bc-1] (t) t log (1 -a)(t + 1) a(c -1 -t) dt.
Proof. By Proposition 2, we have

H(s ′ ) 4 s ′ -1 h(u) u du 4 s ′ -1 h(4) + 3 u-1 H(t) dt t du u (6.4) = h(4) log 4 s ′ -1 + 3 1 H(t) 1 [s ′ -2,3] (t) t log t + 1 s ′ -1 dt.
From Proposition 2 and (6.4), we deduce h(4)

5 3 H(v) v dv h(4)σ(3, 5, 4) + 3 1 H(t) σ(3, t + 2, t + 1) t dt,
which implies the inequality (6.1). The inequality (6.2) follows immediately from (6.4) and (6.1). By using Proposition 2, we have

b a h(ct) t(1 -t) dt = c bc ac h(u) u(c -u) du c bc ac du u(c -u) h(4) + 3 u-1 H(t) dt t = h(4) + 3 1 H(t) 1 [bc-1,3] (t) t dt log b -ab a -ab + 3 1 H(t) 1 [ac-1,bc-1] (t) t log (1 -a)(t + 1) a(c -1 -t) dt,
which combines (6.1) to give (6.3). This completes the proof.

Proof of Proposition 3. By using Lemma 6.1, a simple calculation shows

1 2 1-1/s ′ 1-1/s h(s ′ t) t(1 -t) dt + H(s ′ ) 3 1 H(t)Ξ 1 (t; s) dt,
which, together with Lemma 4.1, implies the desired result.

Proof of Proposition 4. From (6.1)-( 6.3), we can deduce

1-1/s ′ 1-1/s h(s ′ t) t(1 -t) dt + 1-1/s ′ 1-1/κ2 h(s ′ t) t(1 -t) dt + 1-1/s ′ 1-1/κ3 h(s ′ t) t(1 -t) dt + 4H(s ′ ) + H(κ 1 ) (6.5) 3 1 H(t) 1 [α2,3] (t) t log (t + 1) 5 (s -1)(s ′ -1)(κ 1 -1)(κ 2 -1)(κ 3 -1) dt + 3 1 H(t) σ 0 (t) t log 1024 (s -1)(s ′ -1)(κ 1 -1)(κ 2 -1)(κ 3 -1) dt + 3 1 H(t) 1 [α5,α2] (t) t log t + 1 (κ 3 -1)(s ′ -1 -t) dt + 3 1 H(t) 1 [α4,α2] (t) t log t + 1 (κ 2 -1)(s ′ -1 -t) dt + 3 1 H(t) 1 [α3,α2] (t) t log t + 1 (s -1)(s ′ -1 -t) dt + 3 1 H(t) 1 [α1,α2] (t) t log t + 1 κ 1 -1 dt.
By the change of variable v = s ′ (1 -t -u), we have

1/κ2 1/s ′ dt t 1/κ2 t H(s ′ -s ′ t -s ′ u) u(1 -t -u) du = 1/κ2 1/s ′ dt t s ′ (1-2t) s ′ (1-1/κ2-t) s ′ H(v) v(s ′ -s ′ t -v) dv.
Interchanging the order of integration and a simple calculation show that

1/κ2 1/s ′ dt t 1/κ2 t H(s ′ -s ′ t -s ′ u) u(1 -t -u) du = 3 1 H(t) 1 [α6,α4] (t) t(1 -t/s ′ ) log s ′ κ 2 s ′ -s ′ -κ 2 t (6.6) + 1 [α4,α2] (t) t(1 -t/s ′ ) log(s ′ -1 -t) dt.
Similarly we can prove (6.7)

1/κ1 1/s ′ dt t 1/κ3 1/κ2 H(s ′ -s ′ t -s ′ u) u(1 -t -u) du = 3 1 H(t)L 1 (t) dt
where

L 1 (t) := 1 [α7,α5] (t) t(1 -t/s ′ ) log s ′2 (κ 1 s ′ -s ′ -κ 1 t)(κ 3 s ′ -s ′ -κ 3 t) + 1 [α5,α8] (t) t(1 -t/s ′ ) log s ′ (s ′ -1 -t) κ 1 s ′ -s ′ -κ 1 t + 1 [α8,α4] (t) t(1 -t/s ′ ) log (s ′ -1 -t)(κ 2 s ′ -s ′ -κ 2 t) s ′ , and (6.8
)

1/κ1 1/s ′ dt t 1/κ2 t H((1 -t -u)/t) u(1 -t -u) du = 3 1 H(t)L 2 (t) dt with L 2 (t) := 1 [α9,α1] (t) t log t + 1 (κ 2 -1)(κ 1 -1 -t) + 1 [α1,α4] (t) t log t + 1 κ 2 -1 + 1 [α4,α2] (t) t log(s ′ -1 -t).
Now by inserting (6.5)-(6.8) into Lemma 5.2, we easily deduce the required result. § 7. Proof of Theorem 1

We need to resolve the functional inequalities (3.21) and (3.22). It seems very difficult to give the exact solutions, because we only know that H(s) is decreasing. Next we shall give a numeric lower bound for solution by using discretion, which is sufficient to prove Theorem 1.

Put s 0 := 1 and s i := 2 + 0.1 × (i + 1) for i = 1, . . . , 9. Since H(s) is decreasing on [START_REF] Bombieri | Small differences between prime numbers[END_REF][START_REF] Chen | On the Goldbach's problem and the sieve methods[END_REF], Proposition 4 allows us to deduce

(7.1) H(s i ) Ψ 2 (s i ) + 9 j=1 a i,j H(s j ),
where

a i,j := sj sj-1 Ξ 2 (t, s i ) dt (i = 1, . . . , 4; j = 1, . . . , 9).
Similarly Proposition 3 implies

(7.2) H(s i ) Ψ 1 (s i ) + 9 j=1 a i,j H(s j ),
where a i,j := sj sj-1 Ξ 1 (t, s i ) dt (i = 5, . . . , 9; j = 1, . . . , 9)

Table 1. Choice of parameters The parameters s ′ i , κ 1,i , κ 2,i and κ 3,i are chosen such that Ψ 1 (s i ) or Ψ 2 (s i ) is maximal.

i s i s ′ i κ 1,i κ 2,i κ 3,i Ψ 1 (s i ) Ψ 2 (s i ) 1 2
We put

A :=    a 1,1 • • • a 1,9 . . . . . . a 9,1 • • • a 9,9    , H :=    H(s 1 ) . . . H(s 9 )    , B :=           Ψ 2 (s 1 ) . . . Ψ 2 (s 4 ) Ψ 1 (s 5 ) . . . Ψ 1 (s 9 )          
. Then (7.1) and (7.2) can be written as

(7.3) (I -A)H B,
where I is the unit matrix.

In order to resolve (7.3), we first solve the system of linear equations

(7.4) (I -A)X = B,
by using Maple and obtain

X =                0.0223939 • • • 0.0217196 • • • 0.0202876 • • • 0.0181433 • • • 0.0158644 • • • 0.0129923 • • • 0.0100686 • • • 0.0078162 • • • 0.0072943 • • •               
. From (7.3) and (7.4), we deduce that

(I -A)(H -X) 0.
Since all elements of (I -A) -1 are positive, it follows that H X.

In particular we have H(2.2) 0.0223939. Now taking σ = {1} and s = 2.2 in (3.3) of Lemma 3.2, we find, for δ sufficiently small, N 0 sufficiently large and N N 0 ,

D(N ) S A; P(N ), N (1/2-δ)/2.2 = Φ(N, {1}, 2.2) A(2.2) -H k,N0 (2.2) 4C N li(N ) log(N 1/2-δ ) 8(1 -0.0223938)Θ(N ) 7.82085Θ(N ).
This completes the proof of Theorem 1.

Remark 2. (i)

The constant s 1 = 2.2 comes from the fact that Ψ 2 (s) attains the maximal value at s = s 1 (approximately). Since H(s) is decreasing on [START_REF] Bombieri | Small differences between prime numbers[END_REF][START_REF] Chen | On the Goldbach's problem and the sieve methods[END_REF], we have H(2.1) 0.0223939. In order to obtain a better lower (which leads to a smaller constant than 7.82085), we must look for a new weighted inequality (as in Lemma 4.1 and 4.2) such that the corresponding main term Ψ(2.1) has a lager lower bound than 0.015826357.

(ii) If we divide the interval [START_REF] Bombieri | Primes in arithmetic progressions to large moduli[END_REF][START_REF] Cai | A remark on Chen's theorem[END_REF] into more subintervals than 9, it is certain that we can obtain a better result. But the improvement is very minuscule. § 8. Proof of Theorem 3

In the case of the twin primes problem, we need to sieve the following sequence

B := {p + 2 : p x}.
Thinking to Lemmas 2.7 and 2.9, we have 4 7 for the level of distribution in place 1 2 in the Bombieri-Vinogradov theorem. Thus we can take Q := x 4/7-δ and d := Q/d in the definitions described in Section 3. As before, we can prove the corresponding Propositions 3 and 4 with the following modification: In the definition of Ψ 1 (s) we add a factor 7 8 before I 1 (s), and in the definition of Ψ 2 (s) we replace the factor 2 5 before the sum by 7 20 . When we use the switching principle to treat the terms Ω 3 and Γ i for 5 i 21, the related error terms can be estimated by using Lemma 2.9 which has 4 7 for the level of distribution (see [START_REF] Wu | Sur la suite des nombres premiers jumeaux[END_REF], page 380).

Table 2. Choice of parameters Thus for δ sufficiently small, x 0 sufficiently large and x x 0 , we have π 2 (x) S B; P(2), x (1/2-δ)/2.1

i s i s ′ i κ 1,i κ 2,i κ 3,i Ψ 1 (s i ) Ψ 2 (s i ) 1
3.5(1 -0.0287117) Π(x)

Π(x).

This completes the proof of Theorem 3. § 9. Chen's system of weights

Let

A := {N -p : p N }, P(q) := {p : (p, q) = 1}.

The inequality (9.1) below appeared in [START_REF] Chen | Further improvement on the constant in the proposition '1+2': On the representation of a large even integer as the sum of a prime and the product of at most two primes (II) (in Chinese)[END_REF] (page 479, [START_REF] Fouvry | Autour du théorème de Bombieri-Vinogradov[END_REF]) with (κ, σ) = ( 1 12 , 1 3.047 ), ( 1 9.2 , 1 3.41 ) without proof. Cai [START_REF] Cai | A remark on Chen's theorem[END_REF] gave a proof with an extra assumption 3σ + κ > 1. Here we present a proof without Cai's assumption. This removal is important in our argument. Lemma 9.1. Let 0 < κ < σ < 1 3 . Then we have

(9.1) D 1,2 (N ) S(A; P(N ), N κ ) -1 2 S 1 -S 2 -1 2 S 3 + 1 2 S 4 + O(N 1-κ ),
where S i = S i (κ, σ) (1 i 4) are defined by

S 1 := N κ p<N σ (p,N )=1 S(A p ; P(N ), N κ ), S 2 := N σ p1<p2<(N/p1) 1/2 (p1p2,N )=1
S(A p1p2 ; P(N p 1 ), p 2 ), The main difference between (9.1) and Chen's other weighted inequalities (see (34) of [START_REF] Chen | On the representation of a large even integer as the sum of a prime and the product of at most two primes[END_REF] and page 425 of [START_REF] Chen | On the representation of a large even integer as the sum of a prime and the product of at most two primes (II)[END_REF]) is the additional positive term S 4 . However a direct application of sieve to S 4 leads to zero contribution. In order to take advantage of S 4 , Chen used (9.1) with two different couples of parameters (κ, σ). Then an agreeable application of the Buchstab identity and switching principle leads to some compensation. This idea was also used by Cai & Lu [START_REF] Cai | Chen's theorem in short intervals[END_REF] and Cai [START_REF] Cai | A remark on Chen's theorem[END_REF]. Here we make some modifications of their argument such that this process is more powerful.

S 3 := N κ p1<N σ p2<(N/p1) 1/2 (p1p2,N )=1 S(A p1p2 ; P(N p 1 ), p 2 ),
Lemma 9.2. Let 0 < κ 1 < κ 2 < ρ < σ 2 < σ 1 < 1 3 such that 3κ 1 + ρ 1 2 . Then we have 4D 1,2 (N ) 4S(A; P(N ), N κ1 ) -Υ 1 -Υ 2 -Υ 3 + Υ 4 + Υ 5 + Υ 6 -2Υ 7 (9.4)
where

Υ 1 := N κ 1 p<N κ 2 (p,N )=1
S(A p ; P(N ), p),

Υ 2 := N κ 1 p<N σ 1 (p,N )=1 S(A p ; P(N ), N κ1 ), Υ 3 := N κ 1 p<N σ 2 (p,N )=1 S(A p ; P(N ), N κ1 ), Υ 4 := N κ 1 p1<p2<N κ 2 (p1p2,N )=1 S(A p1p2 ; P(N ), N κ1 ), Υ 5 := N κ 1 p1<N κ 2 p2<N 1/2-2κ 1 /p1 (p1p2,N )=1 S(A p1p2 ; P(N ), N κ1 ), Υ 6 := N κ 1 p1<N κ 2 ,N ρ p2<N σ 2 (p1p2,N )=1
S(A p1p2 ; P(N ), p 1 ),

Υ 7 := N σ 1 p1<p2<(N/p1) 1/2 (p1p2,N )=1 S(A p1p2 ; P(N p 1 ), p 2 ), Υ 8 := N σ 2 p1<p2<(N/p1) 1/2 (p1p2,N )=1 S(A p1p2 ; P(N p 1 ), p 2 ), Υ 9 := N κ 1 p1<N σ 1 p2<(N/p1) 1/2 (p1p2,N )=1
S(A p1p2 ; P(N p 1 ), p 2 ), Υ 10 := N κ 2 p1<N σ 2 p2<(N/p1) 1/2 (p1p2,N )=1 S(A p1p2 ; P(N p 1 ), N σ1 ),

Υ 11 := N κ 2 p1<p2<p3<N σ 2 (p1p2p3,N )=1 S(A p1p2p3 ; P(N ), p 2 ), Υ 12 := N κ 2 p1<N σ 2 , N σ 1 p2<p3<(N/p1) 1/2 (p1p2p3,N )=1 S(A p1p2p3 ; P(N p 1 ), p 2 ), Υ 13 := N κ 1 p1<p2<p3<p4<N κ 2 (p1p2p3p4,N )=1 S(A p1p2p3p4 ; P(N ), p 2 ), Υ 14 := N κ 1 p1<p2<p3<N κ 2 p4<N 1/2-2κ 1 /p3 (p1p2p3p4,N )=1 S(A p1p2p3p4 ; P(N ), p 2 ), Υ 15 := N κ 2 p1<N σ 2 p2<p3<p4<N σ 1 (p1p2p3p4,N )=1
S(A p1p2p3p4 ; P(N ), p 3 ).

Proof. The inequality (9.1) with (κ, σ) = (κ 2 , σ 2 ) implies

(9.5) 2D 1,2 (N ) 2S(A; P(N ), N κ2 ) -S 1 (κ 2 , σ 2 ) -2Υ 8 -S 3 (κ 2 , σ 2 ) + Υ 11 + O(N 1-κ2 ).
Thus we have

∆ Υ 5 + ∆ 1 , where ∆ 1 := - N κ 1 p1<p2<p3<N κ 2 (p1p2p3,N )=1 S(A p1p2p3 ; P(N ), p 1 ) - N κ 1 p1<p2<N κ 2 p3<N 1/2-2κ 1 /p2 (p1p2p3,N )=1 S(A p1p2p3 ; P(N ), p 1 ) - N κ 2 p1<N σ 2 p2<p3<N σ 1 (p1p2p3,N )=1
S(A p1p2p3 ; P(N p 1 ), p 3 ). Now the inequality (9.6) becomes

2D 1,2 (N ) 2S(A; P(N ), N κ1 ) -Υ 1 -Υ 3 + Υ 4 + Υ 5 + Υ 6 (9.7) -2Υ 8 -Υ 10 + Υ 11 + Υ 12 + ∆ 1 + O(N 1-κ2 ).
The inequality (9.1) with (κ, σ) = (κ 1 , σ 1 ) gives us

(9.8) 2D 1,2 (N ) 2S(A; P(N ), N κ1 ) -Υ 2 -2Υ 7 -Υ 9 + S 4 (κ 1 , σ 1 ) + O(N 1-κ1 ).
Adding (9.7) to (9.8) yields 4D 1,2 (N ) 4S(A;

P(N ), N κ1 ) -Υ 1 -Υ 2 -Υ 3 + Υ 4 + Υ 5 + Υ 6 (9.9) -2Υ 7 -2Υ 8 -Υ 9 -Υ 10 + Υ 11 + Υ 12 + ∆ 2 + O(N 1-κ1 ),
where

∆ 2 := ∆ 1 + N κ 1 p1<p2<p3<N σ 1 (p1p2p3,N )=1 S(A p1p2p3 ; P(N ), p 2 ).
Clearly all domains of summation in the three terms on the right-hand side of ∆ 1 are distinct and are contained in the domain of summation of the last triple sums on the right-hand side of ∆ 2 (since 3κ

1 + σ 1 > 3κ 1 + ρ 1 2 ). Therefore we have ∆ 2 - N κ 1 p1<p2<p3<N κ 2 (p1p2p3,N )=1 S(A p1p2p3 ; P(N ), p 1 ) -S(A p1p2p3 ; P(N ), p 2 ) - N κ 1 p1<p2<N κ 2 p3<N 1/2-2κ 1 /p2 (p1p2p3,N )=1 S(A p1p2p3 ; P(N ), p 1 ) -S(A p1p2p3 ; P(N ), p 2 ) + N κ 2 p1<N σ 2 p2<p3<N σ 1 (p1p2p3,N )=1 S(A p1p2p3 ; P(N ), p 2 ) -S(A p1p2p3 ; P(N ), p 3 ) = -Υ 13 -Υ 14 + Υ 15 + O(N 1-κ1 ).
Combining this with (9.9), we obtain the required result. This completes the proof.

to S(A p ; P(N ), p). The contribution of the error term in (2.4) is

≪ ε N κ 1 p<N κ 2 , (p,N )=1 q √ N /pL B , q|P (p) |r(A, pq)| ≪ ε d √ N /L B µ(d) 2 max y N max (a,d)=1 π(y; d, a) - li(y) ϕ(d) ≪ ε N L 3
by Lemma 2.3 with the same choice of f as above. Thus

Υ 1 {1 + O(ε)}N L N κ 1 p<N κ 2 V (p) ϕ(p) F log( √ N/p) log p + O N L 3 .
The standard procedure for replacing sums over primes by integrals yields

(10.3) Υ 1 {F 1 + O(ε)} Θ(N ),
where

F 1 := 2 e γ κ2 κ1 F (1/2t -1) t 2 dt = 4 e γ 1/2κ1-1 1/2κ2-1 F (t) dt.
Similarly we can prove

(10.4) Υ i {F i + O(ε)} Θ(N ) (i = 2, 3),
where

F 2 := 4 e γ (1/2-κ1)/κ1 (1/2-σ1)/κ1 F (t) 1 -2κ 1 t dt, F 3 := 4 e γ (1/2-κ1)/κ1 (1/2-σ2)/κ1 F (t) 1 -2κ 1 t dt.
3 • Lower bounds of Υ 4 and Υ 5

As before we can deduce, from Lemmas 2.2 and 2.3, that (10.5)

Υ i {F i + O(ε)} Θ(N ) (i = 4, 5),
where

F 4 := 2 κ 1 e γ κ2 κ1 dt t κ2 t f 1/2 -t -u κ 1 du u , F 5 := 2 κ 1 e γ κ2 κ1 dt t 1/2-2κ1-t κ2 f 1/2 -t -u κ 1 du u .
We have used the following fact to remove the condition (p 1 p 2 , N ) = 1:

p1|N p1 N κ N κ p2<N σ N p 1 p 2 + N κ p1<N σ p2|N p2 N κ N p 1 p 2 ≪ N 1-κ L.
4 • Upper bounds of Υ i for i = 7, 8, 9, 10, 13, 14

We shall only majorize Υ 7 and the others can be treated similarly. Since σ 1 > 1 4 , the quantity Υ 7 is equal to the number of primes p N such that N -p = p 1 p 2 p 3 with N σ1 p 1 < p 2 < (N/p 1 ) 1/2 , p 3 p 2 and (p 1 p 2 p 3 , N ) = 1. Define

M := {m : m = p 1 p 2 , N σ1 p 1 < p 2 < (N/p 1 ) 1/2 , (p 1 p 2 , N ) = 1}, B := {b : b = N -mp N, m ∈ M, p N/m}. It is clear that Υ 7 S(B; P(N ), N 1/2 ) + O(N 1/2 ).
By applying (2.4) of Lemma 2.2 with

X = m∈M li N m , w(p) = p/ϕ(p) if p ∈ P(N ), 0 otherwise, Q = √ N L B , we obtain (10.6) Υ 7 8 C N X log N {1 + O(ε)} + O ε √ N + R 3 + R 4 ,
where

R 3 := q √ N /L B (q,N )=1 µ(q) 2 m∈M (q,m)=1 mp N mp≡N (mod q) 1 - li(N/m) ϕ(q) , R 4 := q √ N /L B , (q,N )=1 µ(q) 2 ϕ(q) m∈M, (q,m)>1 li N m .
Let f (m) be the characteristic function of M. Since m N 3/4 for m ∈ M, Lemma 2.3 implies

(10.7) R 3 = q √ N /L B (q,N )=1 µ(q) 2 m N 5/6 (q,m)=1 f (m) mp N mp≡N (mod q) 1 - li(N/m) ϕ(q) ≪ N L 3 .
Noticing that (d, m) > 1 implies (d, m) N σ1 for m ∈ M, we have

R 4 ≪ N L q √ N µ(q) 2 ϕ(q) m N 3/4 , (q,m) N σ 1 1 m ≪ N L q N µ(q) 2 ϕ(q) d|q, d N σ 1 1 d n N 3/4 /d 1 n ≪ N q √ N µ(q) 2 ϕ(q) d|q, d N σ 1 1 d ≪ N N σ 1 <d N 1 d l √ N/d µ(dl) 2 ϕ(dl) ≪ N N σ 1 <d N µ(d) 2 dϕ(d) l √ N/d µ(l) 2 ϕ(l) .
Since the function µ(n) 2 /ϕ(n) is multiplicative and µ(p ν ) 2 /ϕ(p ν ) = 1/(p -1) for ν = 1 and = 0 for ν 2, it is plain to see that

l t µ(n) 2 ϕ(n) ≍ log t. Thus (10.8) R 4 ≪ N 1-σ1 L 2 .
By the prime number theorem, we obtain (10.9)

X = {1 + o(1)} N σ 1 p1<p2<(N/p1) 1/2 N p 1 p 2 log(N/p 1 p 2 ) = {1 + o(1)} N L 1/3 σ1 log(1/t -2) t(1 -t) dt = {1 + o(1)} N L 1/σ1-1 2 log(t -1) t dt.
Inserting (10.7)-(10.9) into (10.6) yields (10.10)

Υ 7 {F 7 + O(ε)} Θ(N ),
where

F 7 := 8 1/σ1-1 2 log(t -1) t dt.
Similarly we can prove that (10.11)

Υ i {F i + O(ε)} Θ(N ) (i = 8, 10),
where

F 8 := 8 1/σ2-1 2 log(t -1) t dt, F 10 := 8 σ2 κ2 log(1/σ 2 -1 -t/σ 2 ) t(1 -t) dt.
[We need to use the assumption 2σ [We need to use the assumption 3σ 1 + κ 1 > 1 in Υ 9 and Lemma 2.10 in Υ 13 and Υ 14 .] By inserting (10.2)-(10.5), (10.10)-(10.12) and by using the trivial lower bounds Υ i 0 (i = 6, 11, 12, 15) into (9.4), we get the following inequality [For the integrals F 13 and F 14 , we make use of ω(u) 0.561522 for u 3.5.] This completes the proof of Theorem 2.

Theorem 5 can be proved in the same way. The only difference is to replace Lemmas 2.3 and 2.6 by Lemma 2.4. Here, the choice of parameters is (θ, κ 1 , κ 2 , ρ, σ 2 , σ 1 ) = (0.971, (2θ -1)/12, 0.111, (2θ -1)/4, 0.271, 0.313). § 11. Proof of Theorem 4

The proof of Theorem 4 is very similar to that of Theorem 2. But we must use Lemmas 2.5, 2.7 and 2.8 in place of Lemmas 2.3 and 2.6. In order to take the advantage of these lemmas, we must carry out a more careful and delicate analysis. Thus the proof will be slightly complicated.

Suppose that the parameters satisfy the following conditions: λ + l (q)r(A ′ , pq)

where Q = x 4/7-ε and λ + l (q) is well factorable of level Q/P and of order 1. Denote by π P the characteristic function of the primes in the interval [P, 2P ). Noticing that P

x κ2 ⇒ P Q/P , Lemma 2.1 shows that π P * λ + l is well factorable of level Q and of order 2. Thus Lemma 2.7 allows us to deduce that P p<2P l<L q|P (p) λ + l (q)r(A ′ , pq) ≪ ε x/(log x) Our assumptions on κ 1 and ρ imply that p 2 1 p 2 x 4/7-ε and p 2 1

x 4/7-ε . As before we can apply (2.5) of Lemma 2. 4

• Upper bounds of Υ ′ i for i = 7, 8, 9, 10, 13, 14 We shall apply the technique of [START_REF] Fouvry | On the switching principle in sieve theory[END_REF]. Since σ 1 > 2 7 , the quantity Υ ′ 7 is equal to the number of primes p x such that p + 2 = p 1 p 2 p 3 with x σ1 p 1 < p 2 < (x/p 1 ) 1/2 and p 3 p 2 .

Introduce the set B := {b -2 : b = p 1 p 2 p 3 x, x σ1 p 1 < p 2 < p 3 }.

Then we have Υ ′ 7 = S(B; P(2), x 1/2 ) + O(x 1/2 ).

Let ∆ := 1 + L -4 . We cover the set B by cuboids B(t 1 , t 2 , t 3 ) := b -2 : b = p 1 p 2 p 3 x, p i ∈ [∆ ti , ∆ ti+1 ) for 1 i 3 where t i are integers satisfying x σ1 ∆ t1 ∆ t2 ∆ t3 and ∆ t1+t2+t3+3

x. In view of x 2/7 p 2 x (1-σ2)/2 x 2/5 , Lemma 2.5 with the choice

α m = 1 if m = p 1 p 3 0 otherwise , β n = 1 if n = p 2 0 otherwise
implies the inequality (q,2)=1

λ + l (q) |B(t 1 , t 2 , t 3 ) q | -|B(t 1 , t 2 , t 3 )| ϕ(q) ≪ x (log x) 18 , where λ + l (q) is well factorable of order 1 and of level Q = x θ(t2) with θ(t 2 ) = (2 + t 2 )/4. Thus we find by (2.4)

§ 10 .

 10 Proofs of Theorems 2 and 5 . . . . . . . . . . . . . . . . . . . . . . . 38 § 11. Proof of Theorem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 § 1. Introduction

2 .

 2 The methods of Selberg, Pan, Bombieri & Davenport and Chen work in a similar way and give upper bounds of this type (1.7) π 2 (x) {a + o(1)}Π(x),

(3. 2 )

 2 A(s) := sF (s)/2e γ and a(s) := sf (s)/2e γ , We introduce the notation Φ(N, σ, s) := d σ(d)S(A d ; P(dN ), d 1/s ), Θ(N, σ) := 4li(N ) d σ(d)C dN ϕ(d) log d .

For p 1

 1 < d 1/κ3 < d 1/s , we have S(A dp ; P(dN ), p) p p1<d 1/κ 3 (p1,N )=1 S(A dpp1 ; P(dN ), p 1 ) + d 1/κ 3 p1<d 1/s (p1,N )=1 S(A dpp1 ; P(dN ), p 1 ).

Finally we apply the

  switching principle to estimate d σ(d)Ω 3 . For this, we introduce E := {e : e = dnp 1 p 2 , σ(d) = 0, (n, p 1 , p 2 ) satisfies (5.3) below}, where(5.3) d 1/s ′ p 1 < p 2 < d 1/s , (p 1 p 2 , dN ) = 1, n N/dp 1 p 2 2 ,(n, N P (p 2 )/p 1 ) = 1; and B := {b : b = N -ep 3 , e ∈ E, p 2 < p 3 κ(d, e)}, where κ(d, e) := min{N/e, d 1/s }. The set B is a multiset and an element b may occur more than once. Clearly d σ(d)Ω 3 does not exceed the number of primes in the set B. Thus d σ(d)Ω 3 S(B; P

p3 r(a) 1 ,

 1 and I 1 (a) := r 0 (a)r 0 (a) 2/s ′ , N/r 0 (a) 1/s , I 2 (a) := N/r 0 (a) 1/s , N/r 0 (a) 1/s ′ . Applying Lemma 2.3 yields R (j) 1 ≪ δ,k N/L 5k+5 for j = 1, 2, 3. Hence (5.7) R 1 ≪ Θ(N, σ)/L 3 .

S 4 2 1 ≪ N 1 / 2 .

 4212 := N κ p1<p2<p3<N σ (p1p2p3,N )=1S(A p1p2p3 ; P(N p 1 ), p 2 ).Proof. Clearly the desired inequality (9.1) is equivalent to(9.2) D 1,2 (N )a∈A, (a,P (N κ ))=11-1 2 s 1 (a) -s 2 (a) -1 2 s 3 (a) + 1 2 s 4 (a) + O(N 1-κ ),where s 1 (a) :=N κ p<N σ p|a, (p,N )=1 1, s 2 (a) := N σ p1<p2<(N/p1) 1/2 p1p2|a, (p1p2,N )=1 p|(a/p1p2)⇒p p2 1, s 3 (a) := N κ p1<N σ p2<(N/p1) 1/2 p1p2|a, (p1p2,N )=1 p|(a/p1p2)⇒p p2 1, s 4 (a) := N κ p1<p2<p3<N σ p1p2p3|a, (p1p2p3,N )=1 p|(a/p1p2p3)⇒p p2 1. Let δ * (a) := 1 if Ω(a) 2, 0 otherwise.Then it is easy to seeD 1,2 (N ) a∈A, (a,P (N κ ))=1 δ * (a) = a∈A, (a,P (N κ ))=1 µ(a) 2 δ * (a) + O(N 1/2 ),where we have used the fact thata∈A (a,P (N κ ))=1 {1 -µ(a) 2 }δ * (a) N κ p N 1/Similarly if we write δ(a) := 1 -1 2 s 1 (a) -s 2 (a) -1 2 s 3 (a) + 1 2 s 4 (a),we can show that a∈A, (a,P (N κ ))=1δ(a) = a∈A, (a,P (N κ ))=1 µ(a) 2 δ(a) + O(N 1-κ ).Thus in order to prove (9.2) it suffices to verify that (9.3) δ * (a) δ(a) for a ∈ A, µ(a) 2 = 1 and (a, P (N κ )) = (a, N ) = 1. We first observe that (9.3) is trivial if Ω(a) 2, since δ * (a) = 1 and s 4 (a) = 0 in this case. It remains to show that δ(a) 0 in all other cases, which can be verified as follows: If Ω(a) = 3 and s 1 (a) = 0, then s 2 (a) = 1 and s 3 (a) = s 4 (a) = 0. Thus δ(a) = 0. If Ω(a) = 3 and s 1 (a) = 1, then s 3 (a) = 1 and s 2 (a) = s 4 (a) = 0. Thus δ(a) = 0. If Ω(a) = 3 and s 1 (a) = 2, then s 2 (a) = s 3 (a) = s 4 (a) = 0. Thus δ(a) = 0. If Ω(a) = 3 and s 1 (a) = 3, then s 2 (a) = s 3 (a) = 0 and s 4 (a) = 1. Thus δ(a) = 0. If Ω(a) 4 and s 1 (a) = 1, then s 3 (a) = 1 and s 2 (a) = s 4 (a) = 0. Thus δ(a) = 0. If Ω(a) 4 and s 1 (a) = 2, then s 2 (a) = s 3 (a) = s 4 (a) = 0. Thus δ(a) = 0. If Ω(a) 4 and s 1 (a) 3, then s 2 (a) = s 3 (a) = 0 and s 4 (a) = s 1 (a) -2. Thus δ(a) = 0. This completes the proof.

D 1 , 2

 12 (N ) {F (κ 1 , κ 2 , ρ, σ 2 , σ 1 ) + O(ε)} Θ(N ), where F (κ 1 , κ 2 , ρ, σ 2 , σ 1 ) := 1 4 (4F 0 -F 1 -F 2 -F 3 + F 4 + F 5 -2F 7 -2F 8 -F 9 -F 10 -F 13 -F 14 ).Taking κ 1 = 1 12 , κ 2 = 29 250 , ρ = 1 4 , σ 2 = 141 500 and σ 1 = 41 125 , a numerical computation gives us F (κ 1 , κ 2 , ρ, σ 2 , σ 1 ) 1 4 (4 × 13.473613 -3.891854 -20.432098 -17.327241 + 0.697375 + 2.118119 -2 × 0.004609 -2 × 0.434368 -5.161945 -5.468377 -0.023310 -0.182860) > 0.83607.

2 21 = κ 1 < κ 2 1 7 , 2 7 =

 2277 ρ < σ 2 < 29 100 < σ 1 < 1 3 , 3σ 1 + κ 1 > 1, 2σ 1 + σ 2 + κ 2 > 1.

  1 8 and 2 z Q 1/2 . Under the assumptions (2.1), (2.2) and (2.3),

	we have
	(2.4)	S(A; P, z) XV (z) F	log Q log z	+ E +	l<L q|P (z)

  1 + σ 2 + κ 2 > 1 in Υ 10 .]For the terms Υ 9 , Υ 13 and Υ 14 with p 1 N 1/10 , we can apply Lemma 2.6 instead of Lemma 2.3. A similar argument allows us to show that

	(10.12)	Υ i {F i + O(ε)} Θ(N )	(i = 9, 13, 14),
	where
	F 9 := F 13 := + 8 36 5 36 5	1/10 κ1 1/10 κ1 κ2 1/10 dt 1 log(1/σ 1 -1 -t/σ 1 ) t(1 -t) 2 dt 1 κ2 t1 dt 2 t 2 2 t 1 (1 -t 1 ) t 1 κ2 t1 dt 2 t 2 2 κ2 t2 dt 3 t 3	dt + 8 κ2 t2 dt 3 t 3 κ2 t3 ω	σ1 1/10 κ2 t3 1 -t 1 -t 2 -t 3 -t 4 log(1/σ 1 -1 -t/σ 1 ) t(1 -t) ω 1 -t 1 -t 2 -t 3 -t 4 dt, t 2 t 2 dt 4 t 4 ,	dt 4 t 4
	F 14 := + 8 36 5	1/10 κ1 κ2 1/10 dt 1 t 1 (1 -t 1 ) dt 1 t 1 κ2 t1 dt 2 t 2 2	κ2 t1 t2 κ2 dt 2 t 2 2 dt 3 t 3	κ2 t2 κ2 1/2-2κ1-t3 dt 3 t 3 1/2-2κ1-t3 κ2 ω 1 -t 1 -t 2 -t 3 -t 4 ω 1 -t 1 -t 2 -t 3 -t 4 t 2 t 2 dt 4 t 4 .	dt 4 t 4

  We divide the interval [x κ1 , x κ2 ] into O(L) subintervals of the form [P, 2P ) and apply (2.4) of Lemma 2.2 to S(A ′ p ; P(2), p) for p ∈ [P, 2P ). We have

	S(A ′ p ; P(2), p)	{1 + O(ε)}x L	V (p) ϕ(p)	F	log(Q/P ) log p	+	l<L q|P (p)

1 • Lower bounds of S(A ′ ; P(2), x κ1 ) By (2.5) of Lemma 2.2 and Lemma 2.7, we can easily prove (11.1) S(A ′ ; P(2), x κ1 ) {G 0 + O(ε)} Π(x) with G 0 := f (4/7κ 1 )/κ 1 e γ . 2 • Upper bounds of Υ ′ 1 , Υ ′ 2 and Υ ′ 3

  Since V (p) ∼ e -γ C/ log p, the prime number theorem implies that We divide the interval of summation [x κ1 , x σ1 ] of Υ ′ 2 into three parts:[x κ1 , x 2/7-ε ], [x 2/7-ε , x 29/100 ], [x 29/100 , x σ1 ], and use (2.4) of Lemma 2.2 to handle each sum. As before we apply Lemma 2.7, the condition (C.2) and (C.3) of Lemma 2.8, respectively, to control the corresponding error terms. We find

	and		Υ ′ 1	{1 + O(ε)}x log x	x κ 1 p<x κ 2	V (p) ϕ(p)	F	log(Q/p) log p	+ O	x (log x) 4 .
	(11.2)		Υ ′ 1	{G 1 + O(ε)} Π(x)	with	G 1 :=	7 4e γ	4/7κ1-1 4/7κ2-1	F (t) dt.
	(11.3)							Υ ′ 2	{G 2 + O(ε)} Π(x),
	where
	G 2 :=	1 κ 1 e γ	4/7κ1-1 2/7κ1	F (t) 4/7κ 1 -t	dt +	2/7κ1 13/50κ1	F (t) 2/κ 1 -t	dt +	13/50κ1 (11/20-σ1)/κ1	F (t) 11/20κ 1 -t	dt .
	Similarly
								Υ ′ 3	{G 3 + O(ε)} Π(x),
	where
	(11.4)		G 3 :=	1 κ 1 e γ	(4/7-κ1)/κ1 2/7κ1	F (t) 4/7κ 1 -t	dt +	2/7κ1 (2-6σ2)/κ1	F (t) 2/κ 1 -t	dt .
	3 • Lower bounds of Υ ′ 4 and Υ ′ 5
	In view of 3κ 2	3 7 < 4 7 , a similar argument proving (11.2) implies that
	(11.5)							Υ ′ 4	{G 4 + O(ε)} Π(x),
	where			G 4 :=	1 κ 1 e γ	κ2 κ1	dt t	t	κ2	f	4/7 -t -u κ 1	du u	.
												4

  of Lemma 2.2,S(B(t 1 , t 2 , t 3 ); P(2), x 1/2 ) 2C{1 + O(ε)} θ(t 2 )L |B(t 1 , t 2 , t 3 )| + O x (log x)18 .Since the number of cuboids B(t 1 , t 2 , t 3 ) is O (log x)15 , we have|B(t 1 , t 2 , t 3 )| θ(t 2 ) =x σ 1 p1<p2 (x/p1) 1/2 4x p 1 p 2 log(x/p 1 p 2 )(2 + log p 2 / log x)

	(t1,t2,t3)					+ O	x (log x) 2
	=	4x{1 + O(ε)} log x	σ1 t u (1-t)/2	dt du tu(1 -t -u)(2 + u)	.
	Combining these estimates, we obtain
	(11.7)		Υ ′ 7	{G 7 + O(ε)} Π(x),
	where	G 7 := 8	σ1 t u (1-t)/2	dt du tu(1 -t -u)(2 + u)	.
	Analogously we have
	(11.8)		Υ ′ 8	{G 8 + O(ε)} Π(x),
	where
		G 8 := 8

σ2 t u (1-t)/2 dt du tu(1 -t -u)(2 + u)

.

d 1/s ′ p1<d 1/κ 1 , d 1/κ 2 p2<p3<d 1/s (p1p2p3,N )=1 S(A dp1p2p3 ; P(dN ), p 2 ),

Buchstab's identity, when applied three times, gives the equality 2S(A; P(N ), N κ2 ) = 2S(A; P(N ), N κ1 ) -Υ 1 -

(p,N )=1 S(A p ; P(N ), N κ1 )

S(A p1p2p3 ; P(N ), p 1 ).

Similarly a simple application of Buchstab's identity yields

S(A p ; P(N ), N κ1 ) -

S(A p1p2 ; P(N ), p 1 ) -Υ 6 .

Clearly p 1 < N σ2 and σ 2 < σ 1 < 1 3 imply that N σ1 < (N/p 1 ) 1/2 . Thus by Buchstab's identity, we can write

Inserting these into (9.5), we find that

where ∆ := -

Next we shall further decompose ∆. In view of 3κ 1 + ρ 1 2 , we have N ρ N 1/2-2κ1 /p 1 provided p 1 N κ1 . Thus Buchstab's identity allows us to write

S(A p1p2p3 ; P(N ), p 1 ).

Remark 3. Lemmas 9.1 and 9.2 are also valid for

if we make some suitable modifications. For example, we have

where Υ ′ j is similarly defined as Υ j with the difference that A is replaced by A ′ , P(N ) by P(2), P(N p 1 ) by P(2p 1 ), (N/p 1 ) 1/2 by (x/p 1 ) 1/2 , N 1/2-2κ1 /p 2 by x 4/7-2κ1 /p 2 (in Υ 5 and Υ 14 ), N ρ by x ρ , N κi by x κi , N σi by x σi and that the conditions (p, N ) = 1, (p 1 p 2 , N ) = 1, (p 1 p 2 p 3 , N ) = 1 and (p 1 p 2 p 3 p 4 , N ) = 1 are eliminated. The assumption on the parameters is

The last condition is necessary in the proof of ∆ Υ ′ 5 + ∆ 1 . § 10. Proofs of Theorems 2 and 5

For simplicity, we write L := log N and use B to denote a suitable positive constant determined by Lemma 2.3. We shall estimate all terms Υ i in the inequality (9.4). For this we suppose that (10.1)

1 • Lower bound of S(A; P(N ), N κ1 )

We apply (2.5) of Lemma 2.2 with

In view of V (z) ∼ 2e -γ C N / log z (γ is the Euler constant) and C N ≫ 1, we can deduce (10.2)

We apply (2.4) of Lemma 2.2 with

For Υ ′ 9 , the assumption 3σ 1 + κ 1 > 1 allows us to write Υ ′ 9 = S(B ′ ; P(2), x 1/2 ) + O(x 1/2 ) with (11.9)

We decompose

, where B ′ 1 , . . ., B ′ 6 are defined as in (11.9) but we add respectively the extra conditions p 1 x 1/10 in B ′ 1 ;

p 1 > x 1/10 and p 1 p 2 x 1/2 in B ′ 2 ;

x (1-σ2)/2 < p 2 x 2/5 and p 1 p 2 > x 1/2 in B ′ 5 ;

p 2 x (1-σ2)/2 and p 1 p 2 > x 1/2 in B ′ 6 . Again we can use Lemma 2.5 with

Similarly in view of the assumption 2σ 1 + σ 2 + κ 2 > 1, we can prove

where

.

More easily we can prove that (11.12)

where

Inserting these estimations and the trivial lower bounds Υ ′ i 0 (i = 6, 11, 12, 15) into (9.10), we obtain [For the integrals F 13 and F 14 , we make use of ω(u) 0.561522 for u 3.5 and ω(u) 0.567144 for u 2.] This completes the proof of Theorem 4.


published or not. The documents may come

INIST and its position in the changing information market


Zahra Bouanani-Oukhaled

Christine Verdier

Sophie Dupuy- Chessa

Karan Fouladi

Laurent Breda

Ontological Model for EHR interoperability

Keywords: Electronic Health Record (EHR), Medical Ontologies, Interoperability

The main purpose of this paper is to design a data model for Electronic Health Records which main goal is to enable cooperation of various heterogeneous health information systems. We investigate the interest of the meta-ontologies proposed in [1] by instantiating it with real data. We tested the feasibility of our model on real anonymous medical data provided by the Médibase Systèmes company.

Introduction

Health information systems (HIS) are complex, heterogeneous, ambiguous (semantically) and rarely interoperable. As part of our research, we are particularly interested in one of the most common tools used in the field of health informatics: Electronic Health Records (EHRs). According to [START_REF] Schiza | Integrated Electronic Health Record Database Management System: A Proposal[END_REF], EHRs universally comprises medical history, clinical examinations, laboratory results and para-clinical examinations. We propose to design a data model for EHRs in order to enable cooperation of various heterogeneous health information systems and reduce the cognitive overload of physicians treating numerous and various medical data.

Despite the large number of works on medical data structuring and design, data exchange between healthcare professionals remain inefficient. Among the works dedicated to this problem, an interesting approach has been proposed by [START_REF] Sassi | Représentation, visualisation et communication de l'information au travers une représentation iconique des données[END_REF] whose main contribution is to make cooperating heterogeneous information systems together based on a semantic mediation architecture. We propose an instantiation and adaptation of this work.

In this article we firstly briefly describe Sassi's work [START_REF] Sassi | Représentation, visualisation et communication de l'information au travers une représentation iconique des données[END_REF] and secondly, our instantiation and adaptation of this model applied to the patient health record from a French company, Médibase Systèmes. We have illustrated features of our work on a "trajectory of care" representation in regard to a chronic disease (diabetes) on a real anonymous medical database.

Modeling and representation of medical data

In this work, we are interested in modeling, structuring and representing medical data as well as their interoperability. Among the works on modeling and representing medical data, some propose medical terminologies as UMLF [3] and SNOMED CT [START_REF] Wang | The SNOMED Clinical Terms Development Process: Refinement and Analysis of Content[END_REF]. These works propose a semantic representation of data but their use does not necessarily provide semantic interoperability. According to [START_REF] Schulz | Formal ontologies in biomedical knowledge representation[END_REF] the use of knowledge bases and formal ontologies for the structuring and the reorganization of data and medical knowledge are widely used. We can cite [START_REF] Martinez-Costa | Ontology patterns for clinical information modelling[END_REF] and [START_REF] Martinez-Costa | Ontology Content Patterns as Bridge for the Semantic Representation of Clinical Information[END_REF], which proposed an ontological framework to represent information and medical events.

"OntolUrgences" [START_REF] Charlet | Construire une ontologie médicale pour la recherche d'information : problématiques terminologiques et de modélisation[END_REF] is an ontology of emergency acts, and in [START_REF] Benmimoune | Ontology-based Medical Decision Support System to Enhance Chronic Patient's Lifestyle within E-care Telemonitoring Platform[END_REF], the authors designed an ontology based on a medical decision support system to enhance patient's lifestyle. Ontologies are widely used to model knowledge in the medical domain because of their ability to represent data semantics. But the fact is here: the increase of more and more complex, distributed medical data lead to a continuous difficulty to have a synoptic view of the patient health record. We propose in this work a standard model to represent any EHR independently of its own design.

Overview of the ICOP approach

The Iconic Project Management (ICOP) mediation systems transforms data coming from different medical databases into a synoptic health records on which medical data are chronologically represented, annotated, contextualized and "iconized". The system uses two meta-ontologies, the Meta-Ontology of Domain (MOD) and the Meta-Ontology of Tasks (MOT) in order to unify the different data from heterogeneous systems. The MOD describes and classifies medical objects and represents the concepts used in the different EHRs. The MOT is used to unify the medical vocabulary and the actions reflecting the medical professions.

Let us consider the following medical scenario: "A patient P1 (who has a NIDDM2 ) consults a diabetes specialist HP1. HP1 draws a report that can be communicated to all health professionals whatever their specialties and a biological laboratory test that must be read only by other diabetes specialists. Every legacy system, represents each concept with their own terminology. The ICOP system extracts these concepts, generates the meta data and adapts these concepts to the MOD and MOT. In our example, the report is considered as a simple object (text), communicated to all HP, a biological lab test is a complex object (<attribute, value, lower/upper bound>) only communicated to diabetes specialists. The concepts are then unified, annotated and contextualized. The figure 1 describes the meta model of the MOD.

Data model of patient health records

In our work, we propose an instantiation of the Icop mediation system adapted to the Médibase Systèmes company databases. We created an EHRs model (based on the company medical databases) containing anonymous real medical data. The model mainly contains data related to patients, doctors, medical prescription, drugs and biological examinations. (For more visibility of our model in figure 2, we retrieved all attributes of objects). The prescriptions are made according to Vidal database [10]. Diseases are coded according to the International Classification of Diseases (ICD10) [11].

Every medical data is then seen as a medical event, that represents the treatment of each clinical problem (e.g. allergies, diagnoses, complaints, labs, imaging, medications, immunizations and communication). We add a date (a time stamp) for each medical event. All medical events related to a same health problem are gathered in a care episode represented by a start-date, an end-date, a label (extracted from ICD-10 or any else classification like ICPC [START_REF]International Classification of Primary Care[END_REF] in primary care), an identification, and all medical events related to this care episode like prescription, medical exam or biological exam. In the previous example, the care episode is type 2 diabetes, (labelled E11 in ICD-10) that contains two medical events, the report, and the biological laboratory test.

To represent the transition between two consecutive care episodes, we defined a new concept, the trigger event that represents a dynamic point of view 3 which trigs the creation of a new care episode linked to the previous one. In the above example, the trigger event is the result of the laboratory examination. We pass a care episode (E11: NIDDM) to another (E10: IDDM 4 )

The graphical representation of medical events is supported by a concept vector generated from the EHR and representing all medical events related to a care episode. The concept vector is described below: VC CE : CareEpisode (Att: IdCareEpisode, Att: label, Att: label, Att : start-date, Att: end-date, P: Person, D: Document, MI : MedicalIcon, Pr : Prescription, ME : MedicalExam, BE : BiologicalExam, BM : BiometricMeasurement). For each significant table in the database, we instantiate the concrete objects and define the attributes of the concept vector. The graphical interface of the application (vieviewer) is provided with a chronological timeline. Care episodes are linked to the timeline, and contain icons representing each medical event.

Conclusion and perspectives

In this article, we have presented an original instantiation of a mediation system, which allows heterogeneous EHR to cooperate. We used some real anonymous medical patient and obtained very good feedbacks. Our perspectives are to design the patient's context, the physician's behaviors and improving the medical knowledge with machine learning methods. The final goal is to perform recommendations and diagnosis help.

Figure 1 .

 1 Figure 1. Meta-Ontology of Domain(MOD).

Figure 2 .

 2 Figure 2. Medical Data Model.

Corresponding Author. Zahra Bouanani-Oukhaled; E-mail: zahra.bouanani@osoft.fr.

NIDDM: Non-Insulin Dependent Diabetes Mellitus.