An additive problem with primes and almost-primes
by
T. P. Peneva and D. I. Tolev (Plovdiv)
1. Introdu tion. In 1937 I. M. Vinogradov [10℄ proved that forevery
suÆ ientlylargeoddintegerN theequation
p
1 +p
2 +p
3
=N
hasa solutioninprimenumbersp
1 ,p
2 ,p
3 .
Twoyears latervan derCorput [9℄used themethodof Vinogradovand
establishedthatthere existin nitelymanyarithmeti progressions onsist-
ingofthree di erentprimes. A orrespondingresultforprogressionsoffour
ormore primeshasnotbeenproved sofar. In 1981,however, D. R.Heath-
Brown[5℄proved thatthereexistin nitelymanyarithmeti progressionsof
fourdi erentterms,threeofwhi hareprimesandthefourthisP
2
(asusual,
P
r
denotesanintegerwithnomore thanrprimefa tors, ounteda ording
to multipli ity). One of the main points in [5℄ is a result of Bombieri{
Vinogradov'stype forthesum
X
x<p
2
;p
3
2x
p
1 +p
3
=2p
2
p2 2p30(modd) u(p
1 )u(p
2 )u(p
3 );
where d is squarefree, (d;6) = 1; u(n) = (logn)=log3x for n 5 and
u(n)=0otherwise.
Re entlyTolev [8℄foundan analogousresult forthequantity
J
k;l (N)=
X
p1+p2+p3=N
p1l(modk) logp
1 logp
2 logp
3
;
whereN isa suÆ ientlylargeoddinteger and(l;k)=1. In [8℄the Hardy{
Littlewood ir le method and the Bombieri{Vinogradov theorem were ap-
plied,as wellassome arguments belonging to H.Mikawa.
1991 Mathemati sSubje tClassi ation: Primary11N36.
It would be interesting to prove that there exist in nitely many arith-
meti progressions of three di erent primes su h that for two of them, p
1
and p
2
, say, both the numbers p
1
+2, p
2
+2 are almost-primes. In the
presentpaperwestudythisproblem. OurmaintoolisaresultofBombieri{
Vinogradov'stype whi hweestablish usingthemethoddevelopedin[8℄.
Let x be a suÆ iently large real number and k
1 , k
2
be odd integers.
Denote byD
k1;k2
(x)thenumberof solutionsof theequation
(1) p
1 +p
2
=2p
3
inprimesp
1 , p
2 ,p
3
su h that
(2) x<p
1
;p
2
;p
3
3x
and
p
1
+20 (mod k
1 ); p
2
+20 (mod k
2 ):
Let usalso de ne
(x)=
X
x<m
1
;m
2
;m
3
3x
m1+m2=2m3
1
logm
1 logm
2 logm
3
;
0
=2 Y
p>2
1
1
(p 1) 2
:
We prove thefollowing
Theorem. For ea h A>0 thereexists B =B(A)>0 su h that
XX
k1;k2
p
x=(logx) B
(k
1 k
2
;2)=1
D
k1;k2 (x)
0 (x)
'(k
1 )'(k
2 )
Y
pj(k1;k2) p 1
p 2
x
2
(logx) A
:
Forsquarefree odd k we de ne J
k
(x) asthe numberof solutions of the
equation (1)inprimessatisfying(2)and su hthat
(p
1
+2)(p
2
+2)0(modk):
The Theoremstated above implies
Corollary 1.For ea h A>0 thereexists B =B(A)>0 su h that
X
k
p
x =(logx) B
(k;2)=1
2
(k)
J
k
(x)
0 (x)
Y
pjk
2p 5
(p 1)(p 2)
x
2
(logx) A
:
Remark. We shall prove the Theorem and Corollary 1 with B =
16A+100.
Corollary 2. There exist in nitely many triples p
1 , p
2 , p
3
of distin t
primes su h that p
1 +p
2
=2p
3
and (p
1 +2)(p
2
+2)=P
9 .
2. Notations. LetxbeasuÆ ientlylargerealnumberandAapositive
onstant. The onstantsinO-termsand -symbolsareabsoluteordepend
only on A. We shall denote by m, n, d, d
1 , d
2
, a, q, k, k
1 , k
2
, l, r, h, f
integers, byp,p
1 ,p
2 ,p
3
prime numbersand byy,z,t, real numbers. As
usual (n), '(n) denote Mobius's fun tion and Euler's fun tion;
k (n) is
thenumberof integer solutionsof theequation d
1 :::d
k
=n; (n)=
2 (n).
We denote by (m;n) and [m;n℄ the greatest ommon divisor and the least
ommon multiple of m and n, respe tively. For real y, z, however, (y;z)
denotes the open interval on the real line with endpoints y and z. The
meaning is always lear from the ontext. Instead of m n (mod k) we
shall writefor simpli itym n(k). We shall also use the notation e(t) =
exp(2it). The letter denotessome positive realnumber, notthe same in
all appearan es. This onvention allowsusto write
(logt)e
p
logt
e
p
logt
;
forexample.
We de ne
(3)
H =
p
x
(logx)
16A+100
; Q=(logx) 4A+20
; =xQ 1
;
E
1
= [
qQ q 1
[
a=0
(a;q)=1
a
q 1
q
; a
q +
1
q
; E
2
=
1
;1 1
E
1
;
S
k ( )=
X
x<p3x
p 2(k)
e( p); S( )=S
1
( ); V( )= X
x<m3x e( m)
logm
;
E =
XX
k
1
;k
2
H
(k1k2;2)=1
D
k1;k2 (x)
0 (x)
'(k
1 )'(k
2 )
Y
pj(k1;k2) p 1
p 2
:
3. Proof of the orollaries. Supposethat 2
(k)=1and M
1 ,M
2 are
integers. The followingidentityholds:
(4) (k)
X
[k
1
;k
2
℄=k
k1jM1
k
2 jM
2
(k
1 )(k
2 )=
1 ifkjM
1 M
2 ,
0 ifk-M
1 M
2 .
A similar identity has been stated in [1, Lemma 8℄. For onvenien e we
Ifk-M
1 M
2
theequality(4)isobvious. SupposethatkjM
1 M
2
. Wehave
(k) X
[k1;k2℄=k
k
1 jM
1
k
2 jM
2
(k
1 )(k
2
)=(k) X
k1jk
k
1 jM
1 X
djk1
kd=k
1 jM
2
(k
1 )
kd
k
1
=
X
k
1 j(M
1
;k)
M
2 k
1
0(k)
X
dj(k
1
;M
2 k
1
=k)
(d)=
X
k
1 j(M
1
;k)
M
2 k
1
0(k)
(k1;M2k1=k)=1 1=1;
sin etheonlyintegerwhi hsatis esthe onditionsimposedinthelastsum
isk
1
=k=(k;M
2
): This ompletesthe proof of(4).
Using (4)we get
J
k (x)=
X
x<p
1
;p
2
;p
3
3x
p1+p2=2p3
(k) X
[k
1
;k
2
℄=k
k
1 j(p
1 +2)
k2j(p2+2)
(k
1 )(k
2 ) (5)
=(k) X
[k
1
;k
2
℄=k
(k
1 )(k
2 )D
k
1
;k
2 (x):
Supposethat 2
(k)= 2
(l)=1 and (k;2) =(l;2)=1. We de ne
t(l)= Y
pjl p 1
p 2
; %(l)=
(l)t(l)
'(l)
and
(6) L(k)=(k)
X
[k
1
;k
2
℄=k
(k
1 )(k
2 )
'(k
1 )'(k
2 )
t((k
1
;k
2 )):
It is learthat
L(k)=
(k)
t(k) X
[k1;k2℄=k
(k
1 )(k
2 )t(k
1 )t(k
2 )
'(k
1 )'(k
2 )
=
(k)
t(k) X
[k1;k2℄=k
%(k
1 )%(k
2 )
=
(k)
t(k) X
k
1 jk
X
djk
1
%(k
1 )%
kd
k
1
=
(k)%(k)
t(k) X
k
1 jk
X
djk
1
%(d)
=
(k)%(k)
t(k) X
djk
%(d)
k
d
=
(k)%(k) (k)
t(k)
X
djk
%(d)
(d)
=
(k)%(k) (k)
t(k)
Y
1+
%(p)
2
:
Usingthede nitionsof %(l) and t(l) we easily ompute
(7) L(k)=
Y
pjk
2p 5
(p 1)(p 2) :
Nowweapply(3), (5){(7) andthe Theoremto obtain
X
kH
(k;2)=1
2
(k)
J
k
(x)
0 (x)
Y
pjk
2p 5
(p 1)(p 2)
= X
kH
(k;2)=1
2
(k)
(k) X
[k
1
;k
2
℄=k
(k
1 )(k
2 )
D
k1;k2 (x)
0 (x)
'(k
1 )'(k
2 )
Y
pj(k
1
;k
2 )
p 1
p 2
XX
k
1
;k
2
H
(k
1 k
2
;2)=1
D
k
1
;k
2 (x)
0 (x)
'(k
1 )'(k
2 )
Y
pj(k
1
;k
2 )
p 1
p 2
x
2
(logx) A
:
Corollary1 isproved.
Consider thesequen e
A=f(p
1
+2)(p
2
+2)jx<p
1
;p
2
3x; (p
1 +p
2
)=2 primeg
and letB bethe setof odd primes. De ne
X =
0
(x); !(k)=k Y
pjk
2p 5
(p 1)(p 2) :
We applyTheorem10.3of [3℄ hoosing=2, =1=4, =4:1, =0:4. It
is lear thatwe may get ridthe extra fa tor 3
(d)
in the onditionR (; )
using,forexample, theCau hyinequality. We obtain
jfP
9 :P
9
2Agj x
2
log 5
x :
Sin e the ontributionof theterms forwhi h p
1
=p
2
is at most O(x),
thelastestimate provesCorollary 2.
4. Proof of the Theorem. Itis learthat
D
k
1
;k
2 (x)=
1 1=
\
S
k
1 ( )S
k
2
( )S( 2 )d =D (1)
k1;k2
(x)+D (2)
k1;k2 (x);
where
D (i)
k
1
;k
2 (x)=
\
E
i S
k
1 ( )S
k
2
( )S( 2 )d ; i=1;2:
Consequently,
(8) E E
1 +E
2
;
where
E
1
=
XX
k1;k2H
(k
1 k
2
;2)=1
D
(1)
k1;k2 (x)
0 (x)
'(k
1 )'(k
2 )
Y
pj(k
1
;k
2 )
p 1
p 2
; (9)
E
2
=
XX
k
1
;k
2
H
(k
1 k
2
;2)=1 jD
(2)
k1;k2 (x)j:
(10)
TheproofoftheTheoremfollowsfrom(3),(8){(10)andfromtheinequalities
E
1
x
2
(logx) A
; E
2
x
2
(logx) A
:
4.1. Theestimate of E
1
. Wehave
(11) D
(1)
k
1
;k
2 (x)=
X
qQ q 1
X
a=0
(a;q)=1
I(a;q);
where
(12) I(a;q)= 1=(q )
\
1=(q ) S
k
1
a
q +
S
k
2
a
q +
S
2
a
q +
d :
If
(13) qQ; (a;q)=1; j j
1
q
thenwehave
(14) S
2
a
q +
= h(q)
'(q)
V( 2 )+O(xe
p
logx
);
where
h(q)= q
X
m=1
(m;q)=1 e
2m
q
=
q
(q;2)
'
q
(q;2)
'(q)
(the proofis similarto thatof [6,Lemma 3,X℄).
Consider S
k
(a=q+ )fora, q, satisfying(13). Wearenotableto nd
some hypotheses whi h have notbeenproved yet). We shall nd,however,
an asymptoti formulawithan errortermwhi his smallon average.
We have
(15) S
k
a
q +
=
X
1mq
(m;q)=1
m 2((k;q)) e
am
q
T( );
where
T( )= X
x<p3x
p 2(k)
pm(q)
e( p):
Using the elementary theory of ongruen es one may easily prove that
ifthe integersk, m, q satisfy(k;2) =(m;q) =1 and m 2((k;q))then
there existsan integer f =f(k;m;q) su h that(f;[k;q℄)=1 and su hthat
for any integer n the ongruen e n f([k;q℄) is equivalent to the system
n 2(k),nm(q). Hen e we have
T( )= X
x<p3x
pf([k;q℄) e( p):
We de ne
(t;h)=max
yt max
(l;h)=1
X
py
pl(h) logp
y
'(h)
:
UsingAbel'sformulawe obtain
T( )= 3x
\
x
X
x<pt
pf([k;q℄) logp
d
dt
e( t)
logt
dt+
X
x<p3x
pf([k;q℄) logp
e(3 x)
log3x
= 3x
\
x
t x
'([k;q℄)
+O((3x;[k;q℄))
d
dt
e( t)
logt
dt
+
2x
'([k;q℄)
+O((3x;[k;q℄))
e(3 x)
log3x
= 1
'([k;q℄)
3x
\
x
(t x) d
dt
e( t)
logt
dt+2x
e(3 x)
log3x
+O((1+j jx)(3x;[k;q℄)):
3x
\
x e( t)
logt
dt=V( )+O(1)
to get
T( )=
V( )
'([k;q℄) +O
Q
q
(3x;[k;q℄)
:
We substitute this expressionfor T( ) in (15) and we ndthat under the
ondition(13)wehave
(16) S
k
a
q +
=
k (a;q)
'([k;q℄)
V( )+O(Q(3x;[k;q℄));
where
(17)
k
(a;q)=
X
1mq
(m;q)=1
m 2((k;q)) e
am
q
:
Anexpli itformulaforthequantity
k
(a;q)isfoundin[7,p. 218℄. Itimplies
that
(18) j
k
(a;q)j1:
Furthermore, weshall usethetrivialestimates
(19)
S
k
a
q +
x
k
; jV( )j x
logx
; jh(q)j 1:
From(14),(16),(18),(19)andthewell-knownestimate'(n) n(loglogn) 1
we get
S
k
1
a
q +
S
k
2
a
q +
S
2
a
q +
=S
k1
a
q +
S
k2
a
q +
h(q)
'(q)
V( 2 )+O
x 3
k
1 k
2 e
p
logx
=S
k
1
a
q +
h(q)
'(q)
k
2 (a;q)
'([k
2
;q℄)
V( )V( 2 )
+O
Qx 2
qk
1
(3x;[k
2
;q℄)
+O
x 3
k
1 k
2 e
p
logx
= h(q)
k
1 (a;q)
k
2 (a;q)
'(q)'([k
1
;q℄)'([k
2
;q℄) V
2
( )V( 2 )+O
x 3
k
1 k
2 e
p
logx
+O
Qx 2
qk
(3x;[k
1
;q℄)
+O
Qx 2
qk
(3x;[k
2
;q℄)
:
Forthe integralI(a;q) de nedby(12), we nd
I(a;q)= h(q)
k1 (a;q)
k2 (a;q)
'(q)'([k
1
;q℄)'([k
2
;q℄) 1=(q )
\
1=(q ) V
2
( )V( 2 )d (20)
+O
xQ 2
k
2 q
2
(3x;[k
1
;q℄)
+O
xQ 2
k
1 q
2
(3x;[k
2
;q℄)
+O
x 2
k
1 k
2 e
p
logx
:
We also have
(21)
1=(q )
\
1=(q ) V
2
( )V( 2 )d = (x)+O(q 2
2
)
(theproofisanalogoustothatin[6,Lemma4,X℄).Using(18){(21) we nd
that
I(a;q)= h(q)
k
1 (a;q)
k
2 (a;q)
'(q)'([k
1
;q℄)'([k
2
;q℄)
(x)+O
q 2
2
'(q)'([k
1
;q℄)'([k
2
;q℄)
(22)
+O
xQ 2
k
2 q
2
(3x;[k
1
;q℄)
+O
xQ 2
k
1 q
2
(3x;[k
2
;q℄)
+O
x 2
k
1 k
2 e
p
logx
:
Set
b
k
1
;k
2 (q)=
q 1
X
a=0
(a;q)=1
k
1 (a;q)
k
2 (a;q);
(23)
k1;k2 (q)=
h(q)b
k1;k2
(q)'((k
1
;q))'((k
2
;q))
' 3
(q)
: (24)
From (11), (22){(24) and thewell-knownformula
'([k;q℄)'((k;q))='(k)'(q)
we get
D (1)
k
1
;k
2 (x)=
(x)
'(k
1 )'(k
2 )
X
k1;k2
(q)+O
2
(logx) X
qQ q
2
[k
1
;q℄[k
2
;q℄
(25)
+O
xQ 2
X
(3x;[k
1
;q℄)
k
2 q
+O
xQ 2
X
(3x;[k
2
;q℄)
k
1 q
+O
x 2
k
1 k
2 e
p
logx
:
Considerthefun tion b
k
1
;k
2
(q). From(18) and (23)wehave
(26) jb
k
1
;k
2
(q)j'(q):
ItisnotdiÆ ulttosee thatb
k1;k2
(q) ismultipli ativewithrespe tto q and
thatforprime p we have
(27) b
k
1
;k
2 (p)=
8
>
>
>
<
>
>
>
:
p 1 ifp-k
1 ,p-k
2 ,
1 ifpjk
1 ,p-k
2 ,
1 ifp-k
1 ,pjk
2 ,
1 ifpjk
1 ,pjk
2 .
We alsohaveb
k1;k2
(4)=0. Thereforethefun tion
k1;k2
(q)de nedby(24)
is multipli ative with respe t to q and
k
1
;k
2 (p
l
) = 0 if l 2. We apply
Euler's identity (see [4, Theorem 286℄) and also (19), (26), (27) and the
de nitionof
0
. After some al ulationswe get
(28)
X
k1;k2
(q)=
0 Y
pj(k1;k2) p 1
p 2 +O
X
q>Q (k
1
;q)(k
2
;q)
' 2
(q)
:
From(25), (28)and thetrivialestimate
(x) x
2
log 3
x
we obtain
D (1)
k
1
;k
2 (x)=
0 (x)
'(k
1 )'(k
2 )
Y
pj(k
1
;k
2 )
p 1
p 2 +O
x 2
X
q>Q (k
1
;q)(k
2
;q)logq
k
1 k
2 q
2
(29)
+O
xQ 2
X
(3x;[k
1
;q℄)
k
2 q
+O
xQ 2
X
(3x;[k
2
;q℄)
k
1 q
+O
2
(logx) X
qQ q
2
[k
1
;q℄[k
2
;q℄
+O
x 2
k
1 k
2 e
p
logx
:
Using(9)and (29) we nd
(30) E
1
xQ 2
1 +
2
(logx)
2 +x
2
3 +x
2
e
p
logx
;
where
1
= X
k
1
;k
2
H X
(3x;[k
2
;q℄)
k
1 q
;
2
= X
k
1
;k
2
H X
qQ q
2
[k
1
;q℄[k
2
;q℄
;
3
= X
k1;k2H X
q>Q (k
1
;q)(k
2
;q)logq
k
1 k
2 q
2 :
Consider
1
. We have
1
(logx) X
kH X
(3x;[k;q℄)
q
=(logx) X
hHQ
(3x;h)(h);
where
(h)= X
kH X
[k;q℄=h 1
q
= X
dQ X
kH X
[k;q℄=h
(k;q)=d 1
q
X
dQ X
q0(d) 1
q
log 2
x:
Hen e
1
(log 3
x) X
hHQ
(3x;h):
Nowweusethe de nitionsofH,Qand theBombieri{Vinogradovtheorem
(see [2, Chapter28℄, forexample) andwe nd
(31)
1
x
(logx) 12A+72
:
We now treat
2
. We have
2
= X
d
1
;d
2
Q d
1 d
2 X
k
1
;k
2
H X
(k
1
;q)=d
1
(k2;q)=d2 1
k
1 k
2
Q X
d
1
;d
2
Q d
1 d
2
[d
1
;d
2
℄ X
k
1
;k
2
H
k10(d1)
k
2
0(d
2 )
1
k
1 k
2
Q(log 2
x)
;
where
= X
d1;d2Q 1
[d
1
;d
2
℄
= X
dQ X
d1;d2Q
(d1;d2)=d d
d
1 d
2 (32)
X
dQ 1
d X
d1Q=d 1
d
1 X
d2Q=d 1
d
2
log 3
x:
Hen e
(33)
2
Qlog 5
x:
To ompletetheestimateof E
1
we have to onsider
3
. Obviously
(34)
3
= X
d
1
;d
2
H d
1 d
2 X
k
1
;k
2
H X
q>Q
(q;k1)=d1
(q;k )=d
logq
k
1 k
2 q
2
=
4 +
5
;
where
4
= X
d
1
;d
2
H
[d1;d2℄>Q d
1 d
2 X
k
1
;k
2
H X
q>Q
(q;k
1 )=d
1
(q;k2)=d2
logq
k
1 k
2 q
2
;
5
= X
[d
1
;d
2
℄Q d
1 d
2 X
k1;k2H X
q>Q
(q;k
1 )=d
1
(q;k
2 )=d
2
logq
k
1 k
2 q
2 :
We have
4
X
d
1
;d
2
H
[d
1
;d
2
℄>Q d
1 d
2 X
k
1
;k
2
H
k
1
0(d
1 )
k
2
0(d
2 )
1
k
1 k
2
X
q>Q=[d1;d2℄
log (q[d
1
;d
2
℄)
q 2
[d
1
;d
2
℄ 2 (35)
(logx) X
d
1
;d
2
H
[d
1
;d
2
℄>Q 1
[d
1
;d
2
℄ 2
X
k
1
H =d
1 1
k
1 X
k
2
H =d
2 1
k
2 1
X
q=1
(1+logq)
q 2
(log 3
x) X
h>Q 1
h 2
X
[d1;d2℄=h
1 (log 3
x) X
h>Q
3 (h)
h 2
log
4
x
Q :
Forthe sum
5
we nd
5
(log 3
x) X
[d
1
;d
2
℄Q 1
[d
1
;d
2
℄ 2
X
q>Q=[d
1
;d
2
℄ logq
q 2
log
4
x
Q
where
is de nedby(32). Consequently,
(36)
5
log
7
x
Q :
Finally, ombining(30), (31), (33){(36) and usingthe de nitionsof Q and
we get
E
1
x
2
(logx) A
:
4.2. Theestimate of E
2
. It is lear that
E
2
X
k1;k2H
\
E2 S
k1 ( )S
k2
( )S( 2 )d
:
Usingthede nitionof S
k
2
( ) weget
E
2
X
k1;k2H X
x<p3x
p 2(k
2 )
\
E2 S
k
1
( )S( 2 )e( p)d
X
k1;k2H
X
(x+2)=k
2
<r(3x+2)=k
2
\
E2 S
k1
( )S( 2 )e( 2 )e( rk
2 )d
X
k
1
H X
n3x+2
X
k
2
H
X
(x+2)=k2<r(3x+2)=k2
rk2=n
1
\
E
2 S
k1
( )S( 2 )e( 2 )e( n)d
X
kH X
n3x+2
(n)
\
E
2 S
k
( )S( 2 )e( 2 )e( n)d
:
By Cau hy'sinequalityweget
E
2
X
kH X
n3x+2
2
(n)
k
1=2
X
kH k
X
n3x+2
1 1=
\
1=
f( )e( n)d
2
1=2
;
where
f( )=
S
k
( )S( 2 )e( 2 ) if 2E
2 ,
0 if 2E
1 .
We nowapplyBessel'sinequalityto obtain
E
2
x 1=2
(log 2
x)
X
kH k
\
E
2 jS
k
( )S(2 )j 2
d
1=2
(37)
x 1=2
(log 2
x)
X
kH k
X
x<p
1
;p
2
3x
p
1
p
2
2(k)
\
E
2
jS(2 )j 2
e((p
1 p
2 ) )d
1=2
=x 1=2
(log 2
x)
1=2
; say :
We have
= X
kH k
X
jrj2x
r0(k)
XX
x<p
1
;p
2
3x
p
1
p
2
2(k)
p
1 p
2
=r 1
\
E
2
jS(2 )j 2
e( r)d
(38)
x X
kH X
jrj2x
r0(k)
\
E
2
jS(2 )j 2
e( r)d
0 00
where
0
= X
kH
\
E2
jS(2 )j 2
d
;
00
= X
kH X
1jrj2x
r0(k)
\
E2
jS(2 )j 2
e( r)d
:
Obviously
(39)
0
H 1
\
0
jS(2 )j 2
d Hx
logx :
By theCau hyinequalitywe nd
00
X
kH X
1r2x
r0(k)
\
E
2
jS(2 )j 2
e(r )d
= X
1r2x
X
kH
kjr 1
\
E
2
jS(2 )j 2
e(r )d
X
1r2x
(r)
\
E2
jS(2 )j 2
e(r )d
X
1r2x
2
(r)
1=2
X
1r2x
1 1=
\
1=
g( )e(r )d
2
1=2
;
where
g( )=
jS(2 )j 2
if 2E
2 ,
0 if 2E
1 .
We again applytheBesselinequalityto obtain
00
x 1=2
(logx) 3=2
\
E
2
jS(2 )j 4
d
1=2
(40)
x 1=2
(logx) 3=2
sup
2E
2
jS(2 )j
1
\
0
jS(2 )j 2
d
1=2
x(logx) sup
2E
2
jS(2 )j:
Usingthede nitionsofQ, and E
2
we an proveinthesamewayasin
[6, Theorem3,X℄that
(41) sup
2E
jS(2 )j
x
(logx) 2A+7
:
From (37){(41) we obtain
E
2
x
2
(logx) A
:
The Theoremisproved.
Finally, the authors would like to thank the Ministry of S ien e and
Edu ationof Bulgariafor nan ial supportunder grant MM{430.
Referen es
[1℄ J.BrudernandE.Fouvry,Lagrange'sFourSquaresTheoremwithalmostprime
variables,J.ReineAngew.Math.454(1994),59{96.
[2℄ H. Davenport, Multipli ative Number Theory (revisedbyH. Montgomery),2nd
ed.,Springer,1980.
[3℄ H. Halberstam and H.-E. Ri hert, Sieve Methods, A ademi Press, London,
1974.
[4℄ G.H.HardyandE.M.Wright,AnIntrodu tiontotheTheory ofNumbers,5th
ed.,OxfordUniv.Press,1979.
[5℄ D.R.Heath-Brown,Threeprimesandanalmost-primeinarithmeti progression,
J.LondonMath.So .(2)23(1981),396{414.
[6℄ A. A.Karatsuba,Prin iples of Analyti Number Theory,Nauka,Mos ow, 1983
(inRussian).
[7℄ H.Maier andC.Pomeran e,Unusually largegaps between onse utive primes,
Trans.Amer.Math.So .322(1990),201{237.
[8℄ D.I.Tolev,Onthenumberofrepresentationsofanoddinteger asasumofthree
primes, one of whi h belongs to an arithmeti progression, Pro . Steklov Math.
Inst.,toappear.
[9℄ J. G.van der Corput,
Uber Summenvon Primzahlen und Primzahlquadraten,
Math.Ann.116(1939),1{50.
[10℄ I. M. Vinogradov, Representation of an odd number as a sum of three primes,
Dokl.Akad.NaukSSSR15(1937),169{172(inRussian).
DepartmentofMathemati s
PlovdivUniversity\P.Hilendarski"
\TsarAsen"24
Plovdiv4000,Bulgaria
E-mail:tpenevaul .uni-plovdiv.bg
dtolevul .uni-plovdiv.bg
Re eivedon24.3.1997 (3150)
