article14_3. 255KB Jun 04 2011 12:06:56 AM
Journal de Th´eorie des Nombres
de Bordeaux 17 (2005), 925–948
On the exceptional set of Lagrange’s equation
with three prime and one almost–prime variables
par Doychin TOLEV
´sume
´. Nous consid´erons une version affaiblie de la conjecture
Re
sur la repr´esentation des entiers comme somme de quatre carr´es
de nombres premiers.
Abstract. We consider an approximation to the popular conjecture about representations of integers as sums of four squares
of prime numbers.
1. Introduction and statement of the result
The famous theorem of Lagrange states that every non–negative integer
n can be represented as
(1.1)
x21 + x22 + x23 + x24 = n,
where x1 , . . . , x4 are integers. There is a conjecture, which asserts that
every sufficiently large integer n, such that n ≡ 4 (mod 24), can be represented in the form (1.1) with prime variables x1 , . . . , x4 . This conjecture
has not been proved so far, but there are various approximations to it
established.
We have to mention first that in 1938 Hua [9] proved the solvability of
the corresponding equation with five prime variables. In 1976 Greaves [3]
and later Shields [19] and Plaksin [18] proved the solvability of (1.1) with
two prime and two integer variables (in [18] and [19] an asymptotic formula
for the number of solutions was found).
In 1994 Br¨
udern and Fouvry [2] considered (1.1) with almost–prime variables and proved that if n is large enough and satisfies n ≡ 4 (mod 24) then
(1.1) has solutions in integers of type P34 . Here and later we denote by
Pr any integer with no more than r prime factors, counted according to
multiplicity. Recently Heath–Brown and the author [8] proved that, under
the same conditions on n, the equation (1.1) has solutions in one prime and
three P101 – almost–prime variables and also in four P25 – almost–prime
variables. These results were sharpened slightly by the author [21], who
Manuscrit re¸cu le 7janvier 2004.
Doychin Tolev
926
established the solvability of (1.1) in one prime and three P80 – almost–
primes and, respectively, in four P21 – almost–primes.
There are several papers, published during the last years, devoted to
the study of the exceptional set of the equation (1.1) with prime variables.
Suppose that Y is a large real number and denote by E1 (Y ) the number
of positive integers n ≤ Y satisfying n ≡ 4 (mod 24) and which cannot
be represented in the form (1.1) with prime variables x1 , . . . , x4 . In 2000
J.Liu and M.-C. Liu [16] proved that E1 (Y ) ≪ Y 13/15+ε , where ε > 0 is
arbitrarily small. This result was improved considerably by Wooley [22],
who established that E1 (Y ) ≪ Y 13/30+ε . Recently L. Liu [15] established
that E1 (Y ) ≪ Y 2/5+ε .
In the paper [22] Wooley obtained other interesting results, concerning
the equation (1.1). We shall state one of them. Denote by R(n) the number
of solutions of (1.1) in three prime and one integer variables. It is expected
that R(n) can be approximated by the expression 21 π 2 S(n) n (log n)−3 ,
where S(n) is the corresponding singular series (see [22] for the definition).
Wooley proved that the set of integers n, for which R(n) fails to be close
to the expected value, is remarkably thin. More precisely, let ψ(t) be any
monotonically increasing and tending to infinity function of the positive
variable t, such that ψ(t) ≪ (log t)B for some constant B > 0. Let Y be a
large real number and denote by E ∗ (Y, ψ) the number of positive integers
n ≤ Y such that
1
R(n) − π 2 S(n) n (log n)−3 > n (log n)−3 ψ(n)−1 .
2
Theorem 1.2 of [22] asserts that
E ∗ (Y, ψ) ≪ ψ(Y )4 (log Y )6 .
We note that if the integer n satisfies
(1.2)
n ≡ 3, 4, 7, 12, 15 or 19 (mod 24) ,
then
1 ≪ S(n) ≪ log log n.
Therefore, if E2 (Y ) denotes the number of positive integers n ≤ Y , satisfying (1.2) and which cannot be represented in the form (1.1) with three
prime and one integer variables, then
(1.3)
E2 (Y ) ≪ (log Y )6+ε
for any ε > 0.
The purpose of the present paper is to obtain an estimate of almost the
same strength as (1.3) for the exceptional set of the equation (1.1) with
three prime and one almost–prime variables. We shall prove the following
On the exceptional set of Lagrange’s equation
927
Theorem. Let Y be a large real number and denote by E(Y ) the number
of positive integers n ≤ Y satisfying n ≡ 4 (mod 24) and which cannot be
represented in the form
(1.4)
p21 + p22 + p23 + x2 = n ,
where p1 , p2 , p3 are primes and x = P11 . Then we have
(1.5)
E(Y ) ≪ (log Y )1053 .
As one may expect, the proof of this result is technically more complicated than the proof of Theorem 1.2 of [22]. We use a combination of the
circle method and the sieve methods.
In the circle method part we apply the approach of Wooley [22], adapted
for our needs. On the set of minor arcs we apply the method of Kloosterman, introduced in the classical paper [14]. This technique was, actually,
applied also by Wooley in the estimation of the sums T1 and T2 in section 3
of [22]. In his analysis, however, only Ramanujan’s sums appear, whiles in
our situation we have to deal with much more complicated sums, defined
by (8.19). Fortunately, these sums differ very slightly from sums considered
by Br¨
udern and Fouvry [2], so we can, in fact, borrow their result for our
needs.
The sieve method part is rather standard. We apply a weighted sieve
with weights of Richert’s type and proceed as in chapter 9 of Halberstam
and Richert’s book [4].
In many places we omit the calculations because they are similar to those
in other books or papers, or because they are standard and straightforward.
We note that one can obtain slightly stronger result (with smaller power
in (1.5) and with variable x having fewer prime factors) by means of more
elaborate computational work.
Acknowledgement. The main part of this paper was written during the
visit of the author to the Institute of Mathematics of the University of
Tsukuba. The author would like to thank the Japan Society of Promotion
of Science for the financial support and to the staff of the Institute for the
excellent working conditions. The author is especially grateful to Professor
Hiroshi Mikawa for the interesting discussions and valuable comments.
The author would like to thank also Professor Trevor Wooley for informing about his paper [22] and providing with the manuscript.
2. Notations and some definitions
Throughout the paper we use standard number–theoretic notations. As
usual, µ(n) denotes the M¨
obius function, ϕ(n) is the Euler function, Ω(n)
is the number of prime divisors of n, counted according to multiplicity,
Doychin Tolev
928
τ (n) is the number of positive divisors of n. The greatest common divisor
and, respectively, the least common multiple of the integers m1 , m2 are
denoted by (m1 , m2 ) and [m1 , m2 ]. However, if u and v are real numbers
then (u, v) means the interval with endpoints u and v. The meaning is
always clear from the context. We use bold style letters to denote four–
dimensional
vectors. The letter p is reserved for prime numbers. If p > 2
then p· stands for the Legendre symbol. To denote summation over the
P
P
positive integers n ≤ Z we write n≤Z . Furthermore, x(q) , respectively,
P
x(q)∗ means that x runs over a complete, respectively, reduced system of
residues modulo q. By [α] we denote the integer part of the real number α,
e(α) = e2πiα and eq (α) = e(α/q).
We assume that ε > 0 is an arbitrarily small positive number and A
is an arbitrarily large number; they can take different values in different
formulas. Unless it is not specified explicitly, the constants in the O –
terms and ≪ – symbols depend on ε and A. For positive U and V we write
U ≍ V as an abbreviation of U ≪ V ≪ U .
Let N be a sufficiently large real number. We define
(2.1)
(2.2)
X = N 1/2 ,
P = Xδ
Q = N P −1 (log N )−E ,
for some constant
δ ∈ (0, 9/40) ,
M = X(log N )−4E−4 ,
where E > 1 is a large constant, which we shall specify later.
To apply the sieve method we need information about the number of
solutions of (1.4) in integers x lying in arithmetical progressions and in
primes p1 , p2 , p3 . For technical reasons we attach logarithmic weights to
the primes and a smooth weight to the variable x. More precisely, we
consider the function
(
1
if t ∈ (9/20, 11/20) ,
exp (20t−10)
2 −1
(2.3)
ω0 (t) =
0
otherwise
and let
(2.4)
ω(x) = ω0 (xX −1 ) .
For any integer n ∈ (N/2, N ] and for any squarefree integer k, such that
(k, 6) = 1, we define
X
(2.5)
I(n, k) =
(log p1 ) (log p2 ) (log p3 ) ω(x) .
p21 +p22 +p23 +x2 =n
M 3.
Furthermore, if p > 2 is a prime, then
(3.3)
f (pl , n, k) = 0
for
l>1
Doychin Tolev
930
and
(3.4)
h1 (p, n)
h (p)
2
f (p, n, k) =
h3 (p, n)
h4 (p)
if
if
if
if
p ∤ kn ,
p ∤ k, p | n,
p | k, p ∤ n,
p | k, p | n,
where the quantities hj are defined by
−1 n
−n
−1
h1 (p, n) =
(3.5)
+
3
,
p
1
+
3
+
(p − 1)3
p
p
p
−1
1
h2 (p) =
(3.6)
p+3
,
(p − 1)2
p
1
n
−1
2 −n
(3.7)
h3 (p, n) =
+1 ,
p
+ 3p
+
(p − 1)3
p
p
p
−1
−1
3p
h4 (p) =
(3.8)
+1 .
2
(p − 1)
p
The proof of formulas (3.2) – (3.8) is standard and uses only the basic
properties of the Gauss sums (see Hua [10], chapter 7, for example). We
leave the verification to the reader.
From (3.2) – (3.8) we easily get
(3.9)
f (q, n, k) ≪ τ 4 (q) q −2 (q, kn) .
This estimate implies that the series (2.9) is absolutely convergent. We
apply Euler’s identity and we use (3.1) – (3.5) to obtain
Y
S(n, k) = 8
1 + f (p, n, k) .
p>3
From this formula and (3.4), after some rearrangements, we get
(3.10)
S(n, k) = 8ξ(n) ψn (k) ,
where
(3.11)
ξ(n) =
Y
p>3
(3.12)
Y 1 + h2 (p)
,
1 + h1 (p, n)
1 + h1 (p, 0)
p|n
ψn (k) =
Y
p|k
ψn (p)
On the exceptional set of Lagrange’s equation
931
and where
(3.13)
ψn (p) =
1 + h3 (p, n)
1
+ h1 (p, n)
if p ∤ n ,
1 + h4 (p)
1 + h2 (p)
if p | n .
From (3.1), (3.5) – (3.8), (3.11) and (3.13) we obtain the estimates
1 ≪ ξ(n) ≪ log log n ,
(3.14)
(3.15)
0 < ψn (p) < 5
if
p≥7
or if
p=5
and
5 ∤ n,
ψn (5) = 0 if 5 | n
(3.16)
and
ψn (p) = 1 + O
(3.17)
1
p
,
where the constant in the O –term in the last formula is absolute. We leave
the easy verification of formulas (3.14) – (3.17) to the reader.
Let us note that from (3.10), (3.12) and (3.16) it follows
S(n, k) = 0 if 5 | (n, k) ,
(3.18)
which we, of course, expect, having in mind the definition (2.5) of I(n, k)
and the conditions (3.1).
4. Proof of the Theorem
A central rˆ
ole in the proof of the Theorem plays the following Proposition, which asserts that the difference between the quantities I(n, k) and
Σ(n, k), defined by (2.5) and (2.6), is small on average with respect to n
and k.
Proposition. Suppose that the set F consists of integers n ∈ (N/2, N ],
satisfying the congruence n ≡ 4 (mod 24), and denote by F the cardinality
of F. Let γ(k) be a real valued function, defined on the set of positive
integers and such that
(4.1)
γ(k) = 0
if
(6, k) > 1
or
µ(k) = 0
and
|γ(k)| ≤ τ (k) .
(4.2)
Suppose also that
(4.3)
D = Xη
for some constant
η ∈ (0, 1/8)
Doychin Tolev
932
and consider the sum
E=
(4.4)
X
γ(k)
X
n∈F
k≤D
Then we have
I(n, k) − Σ(n, k) .
E ≪ F 3/4 X 2 (log N )262 + F X 2 (log N )−E .
(4.5)
The constant in Vinogradov’s symbol depends only on the constants δ,
E, and η, included, respectively, in (2.1), (2.2) and (4.3).
We shall prove the Proposition in sections 5 – 8. In this section we shall
use it to establish the Theorem.
Let F be the set of integers n ∈ (N/2, N ] satisfying n ≡ 4 (mod 24) and
which cannot be represented in the form (1.4) with primes p1 , p2 , p3 and
with x = P11 . Let F be the cardinality of F. We shall establish that
F ≪ (log N )1052 .
(4.6)
Obviously, this implies the estimate (1.5).
To study the equation (1.4) with an almost–prime variable x we apply a
weighted sieve of Richert’s type.
Let η, ν, ν1 , θ be constants such that
(4.7)
0 < θ,
0 < η < 1/8 ,
0 < ν < ν1 ,
ν + ν1 < η .
Denote
z = Xν ,
(4.8)
z1 = X ν1 ,
D = Xη
and
P=
(4.9)
Y
p.
3
de Bordeaux 17 (2005), 925–948
On the exceptional set of Lagrange’s equation
with three prime and one almost–prime variables
par Doychin TOLEV
´sume
´. Nous consid´erons une version affaiblie de la conjecture
Re
sur la repr´esentation des entiers comme somme de quatre carr´es
de nombres premiers.
Abstract. We consider an approximation to the popular conjecture about representations of integers as sums of four squares
of prime numbers.
1. Introduction and statement of the result
The famous theorem of Lagrange states that every non–negative integer
n can be represented as
(1.1)
x21 + x22 + x23 + x24 = n,
where x1 , . . . , x4 are integers. There is a conjecture, which asserts that
every sufficiently large integer n, such that n ≡ 4 (mod 24), can be represented in the form (1.1) with prime variables x1 , . . . , x4 . This conjecture
has not been proved so far, but there are various approximations to it
established.
We have to mention first that in 1938 Hua [9] proved the solvability of
the corresponding equation with five prime variables. In 1976 Greaves [3]
and later Shields [19] and Plaksin [18] proved the solvability of (1.1) with
two prime and two integer variables (in [18] and [19] an asymptotic formula
for the number of solutions was found).
In 1994 Br¨
udern and Fouvry [2] considered (1.1) with almost–prime variables and proved that if n is large enough and satisfies n ≡ 4 (mod 24) then
(1.1) has solutions in integers of type P34 . Here and later we denote by
Pr any integer with no more than r prime factors, counted according to
multiplicity. Recently Heath–Brown and the author [8] proved that, under
the same conditions on n, the equation (1.1) has solutions in one prime and
three P101 – almost–prime variables and also in four P25 – almost–prime
variables. These results were sharpened slightly by the author [21], who
Manuscrit re¸cu le 7janvier 2004.
Doychin Tolev
926
established the solvability of (1.1) in one prime and three P80 – almost–
primes and, respectively, in four P21 – almost–primes.
There are several papers, published during the last years, devoted to
the study of the exceptional set of the equation (1.1) with prime variables.
Suppose that Y is a large real number and denote by E1 (Y ) the number
of positive integers n ≤ Y satisfying n ≡ 4 (mod 24) and which cannot
be represented in the form (1.1) with prime variables x1 , . . . , x4 . In 2000
J.Liu and M.-C. Liu [16] proved that E1 (Y ) ≪ Y 13/15+ε , where ε > 0 is
arbitrarily small. This result was improved considerably by Wooley [22],
who established that E1 (Y ) ≪ Y 13/30+ε . Recently L. Liu [15] established
that E1 (Y ) ≪ Y 2/5+ε .
In the paper [22] Wooley obtained other interesting results, concerning
the equation (1.1). We shall state one of them. Denote by R(n) the number
of solutions of (1.1) in three prime and one integer variables. It is expected
that R(n) can be approximated by the expression 21 π 2 S(n) n (log n)−3 ,
where S(n) is the corresponding singular series (see [22] for the definition).
Wooley proved that the set of integers n, for which R(n) fails to be close
to the expected value, is remarkably thin. More precisely, let ψ(t) be any
monotonically increasing and tending to infinity function of the positive
variable t, such that ψ(t) ≪ (log t)B for some constant B > 0. Let Y be a
large real number and denote by E ∗ (Y, ψ) the number of positive integers
n ≤ Y such that
1
R(n) − π 2 S(n) n (log n)−3 > n (log n)−3 ψ(n)−1 .
2
Theorem 1.2 of [22] asserts that
E ∗ (Y, ψ) ≪ ψ(Y )4 (log Y )6 .
We note that if the integer n satisfies
(1.2)
n ≡ 3, 4, 7, 12, 15 or 19 (mod 24) ,
then
1 ≪ S(n) ≪ log log n.
Therefore, if E2 (Y ) denotes the number of positive integers n ≤ Y , satisfying (1.2) and which cannot be represented in the form (1.1) with three
prime and one integer variables, then
(1.3)
E2 (Y ) ≪ (log Y )6+ε
for any ε > 0.
The purpose of the present paper is to obtain an estimate of almost the
same strength as (1.3) for the exceptional set of the equation (1.1) with
three prime and one almost–prime variables. We shall prove the following
On the exceptional set of Lagrange’s equation
927
Theorem. Let Y be a large real number and denote by E(Y ) the number
of positive integers n ≤ Y satisfying n ≡ 4 (mod 24) and which cannot be
represented in the form
(1.4)
p21 + p22 + p23 + x2 = n ,
where p1 , p2 , p3 are primes and x = P11 . Then we have
(1.5)
E(Y ) ≪ (log Y )1053 .
As one may expect, the proof of this result is technically more complicated than the proof of Theorem 1.2 of [22]. We use a combination of the
circle method and the sieve methods.
In the circle method part we apply the approach of Wooley [22], adapted
for our needs. On the set of minor arcs we apply the method of Kloosterman, introduced in the classical paper [14]. This technique was, actually,
applied also by Wooley in the estimation of the sums T1 and T2 in section 3
of [22]. In his analysis, however, only Ramanujan’s sums appear, whiles in
our situation we have to deal with much more complicated sums, defined
by (8.19). Fortunately, these sums differ very slightly from sums considered
by Br¨
udern and Fouvry [2], so we can, in fact, borrow their result for our
needs.
The sieve method part is rather standard. We apply a weighted sieve
with weights of Richert’s type and proceed as in chapter 9 of Halberstam
and Richert’s book [4].
In many places we omit the calculations because they are similar to those
in other books or papers, or because they are standard and straightforward.
We note that one can obtain slightly stronger result (with smaller power
in (1.5) and with variable x having fewer prime factors) by means of more
elaborate computational work.
Acknowledgement. The main part of this paper was written during the
visit of the author to the Institute of Mathematics of the University of
Tsukuba. The author would like to thank the Japan Society of Promotion
of Science for the financial support and to the staff of the Institute for the
excellent working conditions. The author is especially grateful to Professor
Hiroshi Mikawa for the interesting discussions and valuable comments.
The author would like to thank also Professor Trevor Wooley for informing about his paper [22] and providing with the manuscript.
2. Notations and some definitions
Throughout the paper we use standard number–theoretic notations. As
usual, µ(n) denotes the M¨
obius function, ϕ(n) is the Euler function, Ω(n)
is the number of prime divisors of n, counted according to multiplicity,
Doychin Tolev
928
τ (n) is the number of positive divisors of n. The greatest common divisor
and, respectively, the least common multiple of the integers m1 , m2 are
denoted by (m1 , m2 ) and [m1 , m2 ]. However, if u and v are real numbers
then (u, v) means the interval with endpoints u and v. The meaning is
always clear from the context. We use bold style letters to denote four–
dimensional
vectors. The letter p is reserved for prime numbers. If p > 2
then p· stands for the Legendre symbol. To denote summation over the
P
P
positive integers n ≤ Z we write n≤Z . Furthermore, x(q) , respectively,
P
x(q)∗ means that x runs over a complete, respectively, reduced system of
residues modulo q. By [α] we denote the integer part of the real number α,
e(α) = e2πiα and eq (α) = e(α/q).
We assume that ε > 0 is an arbitrarily small positive number and A
is an arbitrarily large number; they can take different values in different
formulas. Unless it is not specified explicitly, the constants in the O –
terms and ≪ – symbols depend on ε and A. For positive U and V we write
U ≍ V as an abbreviation of U ≪ V ≪ U .
Let N be a sufficiently large real number. We define
(2.1)
(2.2)
X = N 1/2 ,
P = Xδ
Q = N P −1 (log N )−E ,
for some constant
δ ∈ (0, 9/40) ,
M = X(log N )−4E−4 ,
where E > 1 is a large constant, which we shall specify later.
To apply the sieve method we need information about the number of
solutions of (1.4) in integers x lying in arithmetical progressions and in
primes p1 , p2 , p3 . For technical reasons we attach logarithmic weights to
the primes and a smooth weight to the variable x. More precisely, we
consider the function
(
1
if t ∈ (9/20, 11/20) ,
exp (20t−10)
2 −1
(2.3)
ω0 (t) =
0
otherwise
and let
(2.4)
ω(x) = ω0 (xX −1 ) .
For any integer n ∈ (N/2, N ] and for any squarefree integer k, such that
(k, 6) = 1, we define
X
(2.5)
I(n, k) =
(log p1 ) (log p2 ) (log p3 ) ω(x) .
p21 +p22 +p23 +x2 =n
M 3.
Furthermore, if p > 2 is a prime, then
(3.3)
f (pl , n, k) = 0
for
l>1
Doychin Tolev
930
and
(3.4)
h1 (p, n)
h (p)
2
f (p, n, k) =
h3 (p, n)
h4 (p)
if
if
if
if
p ∤ kn ,
p ∤ k, p | n,
p | k, p ∤ n,
p | k, p | n,
where the quantities hj are defined by
−1 n
−n
−1
h1 (p, n) =
(3.5)
+
3
,
p
1
+
3
+
(p − 1)3
p
p
p
−1
1
h2 (p) =
(3.6)
p+3
,
(p − 1)2
p
1
n
−1
2 −n
(3.7)
h3 (p, n) =
+1 ,
p
+ 3p
+
(p − 1)3
p
p
p
−1
−1
3p
h4 (p) =
(3.8)
+1 .
2
(p − 1)
p
The proof of formulas (3.2) – (3.8) is standard and uses only the basic
properties of the Gauss sums (see Hua [10], chapter 7, for example). We
leave the verification to the reader.
From (3.2) – (3.8) we easily get
(3.9)
f (q, n, k) ≪ τ 4 (q) q −2 (q, kn) .
This estimate implies that the series (2.9) is absolutely convergent. We
apply Euler’s identity and we use (3.1) – (3.5) to obtain
Y
S(n, k) = 8
1 + f (p, n, k) .
p>3
From this formula and (3.4), after some rearrangements, we get
(3.10)
S(n, k) = 8ξ(n) ψn (k) ,
where
(3.11)
ξ(n) =
Y
p>3
(3.12)
Y 1 + h2 (p)
,
1 + h1 (p, n)
1 + h1 (p, 0)
p|n
ψn (k) =
Y
p|k
ψn (p)
On the exceptional set of Lagrange’s equation
931
and where
(3.13)
ψn (p) =
1 + h3 (p, n)
1
+ h1 (p, n)
if p ∤ n ,
1 + h4 (p)
1 + h2 (p)
if p | n .
From (3.1), (3.5) – (3.8), (3.11) and (3.13) we obtain the estimates
1 ≪ ξ(n) ≪ log log n ,
(3.14)
(3.15)
0 < ψn (p) < 5
if
p≥7
or if
p=5
and
5 ∤ n,
ψn (5) = 0 if 5 | n
(3.16)
and
ψn (p) = 1 + O
(3.17)
1
p
,
where the constant in the O –term in the last formula is absolute. We leave
the easy verification of formulas (3.14) – (3.17) to the reader.
Let us note that from (3.10), (3.12) and (3.16) it follows
S(n, k) = 0 if 5 | (n, k) ,
(3.18)
which we, of course, expect, having in mind the definition (2.5) of I(n, k)
and the conditions (3.1).
4. Proof of the Theorem
A central rˆ
ole in the proof of the Theorem plays the following Proposition, which asserts that the difference between the quantities I(n, k) and
Σ(n, k), defined by (2.5) and (2.6), is small on average with respect to n
and k.
Proposition. Suppose that the set F consists of integers n ∈ (N/2, N ],
satisfying the congruence n ≡ 4 (mod 24), and denote by F the cardinality
of F. Let γ(k) be a real valued function, defined on the set of positive
integers and such that
(4.1)
γ(k) = 0
if
(6, k) > 1
or
µ(k) = 0
and
|γ(k)| ≤ τ (k) .
(4.2)
Suppose also that
(4.3)
D = Xη
for some constant
η ∈ (0, 1/8)
Doychin Tolev
932
and consider the sum
E=
(4.4)
X
γ(k)
X
n∈F
k≤D
Then we have
I(n, k) − Σ(n, k) .
E ≪ F 3/4 X 2 (log N )262 + F X 2 (log N )−E .
(4.5)
The constant in Vinogradov’s symbol depends only on the constants δ,
E, and η, included, respectively, in (2.1), (2.2) and (4.3).
We shall prove the Proposition in sections 5 – 8. In this section we shall
use it to establish the Theorem.
Let F be the set of integers n ∈ (N/2, N ] satisfying n ≡ 4 (mod 24) and
which cannot be represented in the form (1.4) with primes p1 , p2 , p3 and
with x = P11 . Let F be the cardinality of F. We shall establish that
F ≪ (log N )1052 .
(4.6)
Obviously, this implies the estimate (1.5).
To study the equation (1.4) with an almost–prime variable x we apply a
weighted sieve of Richert’s type.
Let η, ν, ν1 , θ be constants such that
(4.7)
0 < θ,
0 < η < 1/8 ,
0 < ν < ν1 ,
ν + ν1 < η .
Denote
z = Xν ,
(4.8)
z1 = X ν1 ,
D = Xη
and
P=
(4.9)
Y
p.
3