Acta Mathematica Sinica, New Series 1991, Vo1,7, No .2, pp. 135- 170
Three Primes Theorem I n t e r v a l (V)*)
in a
Jia Chaohaa ( ~ . . ~ @ ) Institute of Mathematics, .Academia Sinica Received February 12, 1990 In this paper, we shall prove that for a sufficiently large odd number N. the equation N - N0.6+e
N=PI+P2+P3'
"T
N +NO.6+~:
"
(i=1 , 2 , 3 )
has solutions.
w1. Introduction In 1937, I . M . Vinogradov proved a well-known three primes theorem. Pan Chengdong and Pan ChengbiaottJgot the result in a short interval that for a sufficiently large odd number N , the equation
. N=p
L+pE+P3
,
N - A < p i ~ < ~N + A
(i=1,2,3)
(1)
has solutions, and the number of solutions T ( N ) has an asymptotic formula
T(N)-
A2 3 C 3 ( N ) log 3N '
(2)
91
where A = N ~ ,
C3(N)=p~u(l
1 (p-l)2)p~I~[u(l+
1 (p-l)3)
" 13 + t
On the basis of the work of [ 1] , Jia Chaohua c2"31 obtained that A = N ~-.~ , 2 A = N T +' respectively, and formula (2) holds. Afterwards, using the complex integral m e t h o d , Pan Chengdong and Pan Chengbiao [41 provedthat (2) holds for 2 A = N T l o g c N . Using the sieve m e t h o d , Jia Chaohua ES'6jproved that A could be taken as N ~ N~ respectively. For a survey of the history and methods regarding this problem, one could refer to w in [5]. In this paper, we shall prove Theorem 1. For a sufficiently large odd number N, the equation
N=pl+p2+p3, N 3
N~
< Pi <<- TN + NO.6+8 t
(3)
(i= 1 , 2 , 3 )
9 * ) The Project Supported by National Natural Science Foundation of China
136
Jia Chaohua
has solutions, where ~ is a sufficiently small positive number. In order to prove Theorem 1, we first prove the equivalent Theorem 2. Assume P is sufficiently large, 8 is a sufficiently small positive number, A = P ~176 Then for an even number n, 2 P < n < ~ 2 P + A , except for
0 ( \
A
I exceptional values
log 2P /
O(n)=
we always have 1 )) C(n)
~
.=pl+p2
A
log2 p ,
(4)
P
where
1
(I-~
1 5
Recently, Zhan Tao I71 proved formula (2) for et = N T log c N,As a by-product, we get Theorem 3. Let B be a sufficiently large positive number. Then except for 23
O
log eP
exceptional values, the even numbers in ( P ,
P+P~
) are all
Goldbach numbers. This is an improvement on the result of Ramachandra ISj. For any result in the problem of primes in a short interval,.there is a corresponding result in the problem of the exceptional set of Goldbach numbers in a short interval. In the following, c,c~, c2, "" denote positive constants which have different values at different places. ~ is a sufficiently small, positive constant, p, q all de A2 p note pri rues. Always assume A = p0.6+30% ~= pl+2o""~ , T = ~ , To = 4 ~ ([ e ] P +
r ) , T '~4rc([elA+ T ) , T~=P ~~, B= - -1. m ~ M denotes that there are positive constants c~, c2 such that cl M < m ~ c.2M. E,= { O'Oe [
1 ,1- +) q ~< l o g ~9 P ,
E2={ 0 0. ~
I.
1 . ,1
. 1 ),
q ~
,
O= - a- + 0 c , q
lel~< l~a
O= a + ~ , logn4P < j e [ ~ A
E3=[
1 ' I-1)-E,-E2
2 = {n'P
(a,q)=l,
"
(a,q)=l, 1
} '
Three Primes Theorem in a Short Interval (V)
137
Here the author thanks Professor Wang Yuan and Professor Pan Chengbiao for their concern and encouragement.
w
Some Lemmas Lemma 1. If z is a character mod q, Zo is the principal character mod q and ( a , q ) = 1, then
z(r)e
(q) __._s
<~ x/ff ,
r= I
zo(r)e
(ar) ~
=p(q).
r-- 1
See page 24 and 28 in [9] . Lemma 2. Let d,(n )= ~ 1, x ' < y <~ x . Then t'l~nl " " n f
~,
d , ( n ) ( ( y ( l o g x ) '-t
x-y
See Theorem 2 in [10] . Lemma 3. I f F :(u) and G(u) are monotonous functions, G(u) ~ G and IF '(u)l >>.F, then ~b
j G ( u ) e ( F ( u ) ) d u ( ( F~ a
See Lemma 1.2 on page 28 in [11]. Lemma 4. I f z is a character m o d q , ~ la(m)12((MlogCT, then m~M
~od Z(
f r:+r
q) d T 2
~
a(m) z ( m )
2dt ((
m t+it
Im-M
(M+ qT) log ~' qT.
See Theorem 3 on page 38 in [9] . Lemma 5. I f Z is a character m o d q, then
L
+it,z
d t ( ( ( q T + q T 2 z/3)(qT) ~
X (rood q)
See M . Jutila [12]. Lemma 6. Suppose
w(u)
1 u
(uw(u) ) ' = w ( u -
l~
1)
u>2.
Then w(u) >1 0 . 5 . See Lemma 19 in [5].
w3. Bilinear Mean Value Theorem of Bombieri's Type p 1-60~ 1 1 Lemma 7. Suppose q ~ ~, ( a , q ) = l , [al ~< , M ~ < ~ T7/6 , H<~T T , qz a ( m ) , b ( h ) = O ( 1 ) . We have
138
Jia Chaohua
a ( m ) b ( h ) e ( a m he (l )~ m h l ) q
G(~)= P-A
9m ~ M , h
~ H
(m/d, q)=l
#(q) ~ q
a(m)
.,~M
m
(re,q)" I
Proof.
G(~ )=
e
rffil
(q)
oP-Ae(~u)du+O
,
~.,
~"
a(m)b(h)e
e(otmhl) m
a(m)b(h')e(otmhl) .
Z (r) Z (mhl)
P-A < ,.hl~P+,t
r (q)
m --M, h - H
1
9
q
P-A
~
(r ,q) ffiI
--~
(h,q)-I
r-I (r,q) •l
~,,
h
--
h-H
X(rood q)
(~(r)etaq.))~a(m)z(m)b(h)z(h)z(l)e(~mhl)" e-A
~P (q) zC~dq) •l
Let F ( s , z ) = ResF(s)
,- i
9
(5)
m-u.h-n
~u ,._
u'-(P-A)" s
a(m)z(m) m*
"h~- n
b(h)x(h) h~
= Eo ( u - ( P - A ) )
9 L (s,z),
~p(q----~) q .,~
b(h) h~_n h '
a(m) m
(m,q)ffiJ
(h , q ) = 1
w h e r e E o = l , X=Zo; Eo--O, ZCZo. Using Perron's formula and moving the integral line, we get
~
a(m)x(m)b(h)x(h)z(l)
P - A < rnhl ~ u m-M, h-H
1 2rci
f
l+~+iTI
l+~-irl
im~
a ( m ) z (m) )(h~n b (h )z (h ) h" )L(s,z)" u m~
';(+
+ O ( P" ) = -~n
r, F
+it, x
cp(q)
+ E o ( u - ( P - a ) ) ~
q
E P-A
E m-M
(m,q)11
)
I
uT§
u ~- ( p - A ) ~
ds
I ~+it
.
~
1
dt
2 +it a(m) m
E h-H
b(h) h
~
(h,q)ffil
a(m) X (m)b (h )z (h)z (l)e(amhl)
+ o(p"
) ,
(6)
Three Primes Theoremin a Short Interval (V) =
e(otu)d d P-A
~ a(m) m
(m,q)*l
(IC
a(m)x(m)b(h)'L(h)x(l)
P - A < rahl ~ u
=Eo~cP(q) q ',.~u +0
139
e (~u )du
,d P - A
~ b(h) ~e+, e(~u)du h-H h oe-A
(h,q)=l
;? ( + ) F
+ it , x u-
T91
G(a)=~#(q) q
~ a(m) ,,-u m
logP
IC
~
dt
I) ($) +0
,
b(h) fe+a h e(ocu)du
~"
h-H
(m,q)=l
(
-~+it
d P-A
(h,q)=[
e(otu)du
;?(+ F
+it,x
) ' I) o(~) u--f+~'dt
+
(7)
By Lemma 3, we know 1
~
~q
~
(<
e(eu)du
F
Z(roodq) d P - A
1
Z{m~ q)
l (qp)T
TI
~
l (roodq)
dt
)C'(t
F -~" + i t , x dt9
(s "
_~.+.
TI
f~'(,
~
+it, Z u
t]
+
~r
)J
u-Te ~ u + - ~ logu du
.IP-A
) 0
( min A,
min
P
e-A
It+2n=ul
r~( 1 +it,Ol~t:~,+~
)
(8)
Let T0=4rr( Ict IP+ T), T ' = 4 n ( I~IA+ T). When It+ 2rc~PI > kT ', P A < u < ~ P + A , because T '>~4zrlalA, we have [t+2rc0~ul~>lt+27r~Pl-121r~ 9(P-u)l~> k T ' - 2 r c l ~ l A > ) k T '. By Lemmas 4 and 5, we know ~.,(( +2 7t
kT'
11 (qp)T
~rO f, T '
<< Alog 1P max ~ (qp )-f :r2 ~T0
(rs
,+2.w*
f~ ,ITs
"
ira- M
J('-T+it,Z)
AF
dt
k-'--~ P I F ( Tl + it, x )1 dt )
a~m,~,m, rl~b~h,~,h,ll i~ ~ L (1 +it ~)l~t m T +it
~
h T +it
"2"
140
Jia Chaohua AlogPt m a x / E / (qp)T r2~rokz Jr2
. : ~2~, I E
((
(rf,7
y~ b(h)z(h) h~n h + +it
a(m)z(m), m T +it
m-U
,
i)2dt Tl
§
dt
) (~f'"~(+)1'+ it, Z ,IT2
dt
)'
I I I 4 1 a A ( M + q T ')T (qT ,)T( q-~ T T p~4,+qT ,)T p2,(( p8, 9 I (qp ) -f
((
(9)
Using Lemmas 4 , 5 and noticing H 2 ~
~
(qP)-f X~modq~
max T
~
TO~V~TI
I(
Z q)
IL(+
((
() P q
i
T logP.
/:?
((
~P
)
1
max --~ ro.
_
h~+i`
Itl F -~ +it,
dt
a(m).__._:._z(m)
mT +~
m~M
h T +it
+it, z ) [ d t
(~;i1
a(m)z(m ) 2
dt
IL-~
~
1
,,-M
//fi'/ ~
m-f +"
I) dt
+it,
-f
dt
1 z l 1 ~-ro~v<.rlmaxT ( M + q V ) - f ( H : + q V ) ~ ( q V ) T P U ( ( A P - 8 '
(lo)
By (7), (10), we know q
,,_~
m
~ h~H (h , q ) - I
(m ,q) = 1
h
e(~u) du+ 0
9
dP-A
Lemma 8 . Under the assumption of Lemma. 7, we have a
i) I f O= ~ + o t e E3, then
S(~)=
~
a(m)b(h)e
q
amhl q
P - A < m h l ~ P+A
ii) I f O = a
()
+~eE2, then S(~)(( A log -np;
iii ) I f O= qa__q.+ ot~ Ei , then
et~mhl)((Alog_np " '
D
Three Primes Theorem in a Short Interval (V) S(~) =
a(m) m
~
d
,,,--u
d2lq
qldld2
141
~. b(h) h-n h
(m'q)=dl
e+A .Jp-a e(otu)du+O 7
"
(h'q)=d2
Proof. i) When 0 ~ E3, q > l o g ~ P . If one of (m, q), (h,q) >q2,, then we might as well suppose (m, q) >qU. From d3 (ds) ~< d3 (d) d3 (s) and Lemma 2, we know q
P - A < rt~d< P +A m-M,h~H (m,q)>q 2e
((
~q
2
<~
1 ~< ~,
P-A
d>q2r
~
d[q d> q2r
~, 9
2
.d3(r)
P-A
d3 (d)d3 (s)
dlq 9 p.L_A_ < s ~ p + A d>q2r d d d
(( d3 (q)log2P ~
A
A
-"d- (( ~
q"
dlq d > q2,t
((Alog
-Bp.
Hence,
X
S(~) =
a(m)b(h)e(a-~-~ql) e(otmhl)+ O( A l o g - n P )
p - A
dl
(m,q)=dt,(h,q)'d2
= S, (a) + 0 (Alog-nP). Let m=m, d~, h=h, d2, q~
S,
~ dllq
dl
q
q
, M , = - dMs , H , =
~
H . Then d2
a(m~d~)b(htd2)
d2lq P - A
(ml,ql)=l,(hl0q2)=l
.e where
(ad'mlhll) q / e ( czdld2mlh,l),
d '=
dl d2 , q (q,dld2) ' q = (q,dtd2)
q lq~, ( m ~ , q , ) = l = ) ( m , , q ' ) = l ,
(hl,q:)=l
' ---)(h~,q')=l,
(ad ', q ')= 1.
= P-A dl d2
2
a(m,d,)b(hld2)e
P+A dl d2 mI--Ml,hl-H1
( ad'm h ; q
e(ad~d2mlh,l).
9(ml , q l ) - I , (hl,q2)=l
The part of S2(~z)which satisfies ( l , q ')is greater than q 2 ' = O ( ~ 5 ~ ) .
Its
142
Jia Chaohua
contribution to S ( ~ ) is equal to O S o , we have to investigate
s,(=)=
E
X
a(mld,)
dlq ' ml - M I d < q 2~ ( m l , q l ) ' l
P-A ddl d2
~ hi-
b(h~d: ) HI
(hi , q 2 ) l I
( ad'mlhlll ) P+A e q7 e(otddl4mlhlll), E
(ll, q *) ffi I !
whereq"=
q d
'
l l= /
d
By Lemma 7 , we know
s3(~)=
y~
u(q.")
dlq' d< q2~
a(mt dt) m~
E ,.,-u,
q
(ml,ql)-I
E. hi-HI
(hl,q2)-I
b (hld~ ) hi
P+A
P-...._..LA dd l d2
e(add~d2u)du+O
.
Because q/, > q,-6~, then
$3 (~) ((
A
A
"--7
+ p7----7 ,
qT
S(a)(( --
A
+ Alog
l
-rip(( Alog _rip.
qT ii) When 0 e E : ,q~
'
s(,)= Z Z dllq
d21q
l~
I A
< [a[ ~<
Z P-A
qz
a(mldt)b(hld~ ) P+A dl d2
M l .h I -H
I
Onl 'ql ) - 1 , (h I 'q2 )*l
9 e ad
'm h l l .) ~, q
e(adld~mth,l),
where
d, dz , q dl= (q, d,d~) , q = (q, d, d2) "
SI(~) =
E
.
a(mtdl)b(h,d~ )e(
P-A
P+._._~4 dl d2
I,q2)-I
ad 'ml ht l ) q -~ e (adl d~ re, h,!)
Three Primes Theorem in a Short Interval (V)
E
143
E
dlq '
a(mldl)b(hl d2 )
P-A
e(~ddl dzml hill),
q~
9 e t
u q where q d By Lemma 7, we know that
#(q ") Sl(~X)= a'~q'
q"
a(mld,)
)"
b (hi d2)
ml
ml ~MI (mI , q l ) = l
h,
hi - H ! (h 1 , q 2 ) = l
P+A ddl d2
f
J P-A ddld2
a(m)
E m--M
E h- H
m
(m,q)=dl
9
e(eu)du+O dP-A
(m.q)~dl
0 q
!
(h, q)=d2
(7)
b(h) [e+A
a(m) m
E
E
h
b(h)
A
e ( otu)du + 0
,
q
=I,
dP-A
(h,q)=d2
A
-q~>l.
l @ q l d ~ d 2 , so
S(~)=
( ,ff_~ ,ff~
~
a (m ) m
q q m-M qldld2 (m,q)~dl
~
h~H (h,q)=d2
b(h) ) f p+Ae(~u)du+O(-~6~) h /dP-A
S(~)((q'log2 P " [----( + ~
((Alog-nP.
(11)
iii) It follows from formula (11). [,emma 9. When 0 ~ E2 u E3, S(0)=
~ P-A
a(m)e(mO)((
A log np
Then for 2P < n <~2P+ A, except for 0 ( log ~ A2P ~ exceptional values, we always /
144
Jia Chaohua
have 1
~0
S(O)
=
L
~
S(O)
~
I
e(pO)e( -nO)dO
P
e(pO)e(-nO)dO+O
logY_lp
.
P
Proof. By Parseval inequality, we know that
S(O) 2P < n ~ 2P+A
((
e(pO)e(-nO)dO
P
~
IS(O )l 2 2LaE 3
e(pO) dO
P
A2 (( 1og2Bp
E
~
2wE3
2
;I
E
I
e ( p O ) dO((A31og-2nP.
P
1 is the number of n which satisfies
n
S(O)
~
e(pO)e(-nO)dO >1 logn_~p ;
P
2~E3
then ~'1~<
1og2a-2 A2
Except
for
Ire
E
2 P < n ~ 2P+A
B
these
o
2t~E 3
( ~o?e) A
s(o) P
A log2p 9
exceptional values, we always have
1
Io S ( O)
~
e ( pO )e (-nO)dO
P
A)
=re S ( O ) ~
P
1
e(pO)e(-nO)dO+O( logn-lp
Lemma 10. When q <~log n2P, I~1 ~<
[q
9
log ~
----7-- , we have
e<~e+a e ( ( q + ~ ) p )
it(q) (o (q)
1 9
logP e< ~r ~ P + A
e(~r)+O(A e x p ( - c x / ~
)).
See Lemma 10 in [5]. p i-6o~
Theorem 4.
1
If M <~ T T/~ , H <~T Y, a ( m ) , b (h ) = O (1), then except for
Three Primes Theorem in a Short Interval (V)
(A)
O log2p
145
exceptional values, we always have
a(m)b(h)( m ~ M, h - H (m,n)~(h,n)=l
A)o(
1
1-
~(mh)
mhl+p ~ n P - A < mhl <~P+ A P
log P
log -iv p
Proof.
1=
~
a(m) b(h)
m-M,h-H (m,n)-(h,n)=l
=fo ~
rahl+p = n P - A < n.gtl <~P+ A P < p<~ P+A
a(m)b(h)e(Omhl)
E
~,,
e (pO )e( -nO)dO.
P
P-A
S(O,n)=
~
a(m)b(h)e(Omhl)
P-A < m-M, (m, n)=l
<~P+A h-H 9 (h,n)=l
a(m)b(h)e(Omhl) P - A < mid ~ P+ A m-M,h-H
~ #(d~) ~ I.t(d2) d l In dllm
d2ln d2[h
,~/~(d,),/~. #(d2) e_A<~.
=
~ dll
+ y*
, 2In
=s,(o,,)+&(o,,),
dlln,d21n
g
B
dl ~
where ", " denotes that one of dl ,d2 > log T P . By Cauchy inequality and Lemma 2, we get
&(O ,n) 2
<~
E3
~,
e(pO)e(-nO)dO
P
IS2(O,n)12dO "
e(pO) dO
~., P
((d2 (n)
s
~ * dlln,d2ln
((d:(n)
~, * dl]n.d2Jn
I
a (m)b (h)e (0 mhl)l: dO . A
P-A
~
di(r) "A.
P-A
By Theorem 2 in [ 10], we get that when y > x ~,
146
Jia Chaohua
d ] ( r ) (( y (logx) 8 . x-y
d2(n)
Z*
Z
dlln,d21n
((d2(n)Ad](n)
~,
A
dlln,d2ln
A
((dJ(n) A
d](r) "A
P-A
log ~ P
[ d~T d2]
d3(n) <~log4p.
]rE
S,(O,n)
~
P
2uE3
~"
dl < log 4 p
A ) exceptional values, we lo~-2p
~ e(pO)e(-nO)dO=O e
fE2uE3
[rE
B d2~log"~'P
(A) 's
Iog~P
" ,
e(pO)e(-nO)dO[ ~
2uE3
P
log ~ -sp
Hence
S2(0,n)
log8
dJ (n)A'
9d2 (n )logSP ((
By Lemma 2, we know that except for O have
.~l. p
a(m)b(h)e(Omhl)
P-A
P
e(pO) e(-nO) dOI.
By i) and ii) of Lemma 8, we know that when 0 E E2 u E3,
a (m)b (h) e (Omhl)(( 2 P-A
A log n p
9
m~M,h~H dl Ira, d2 Ih
By Lemma 9,
f
except
for
exceptional
a (m)b (h) e (Omhl)
E 2vE3
P- A ,~mhl~ P+A m~ M . h - H dllm.d2 Ih
S~(O,n)
fe 2 ',.,'E3
~
P
values, we have
2 e(pO)e(-nO)dO (( P
e(pO)e(-nO)dO((
log ~_~p A
A log n- Lp ,
Three Primes Theorem in a Short Interval (V)
147
Combining all of these, we know that except for O ues,one always has
S(O,n) ~ 2uE3
( ) A log2P
e(pO)e(-nO)dO=O
exceptioncal val-
s
log W P
P
,
l=fe I S(O,n, P
S(O,n,=(~_~ ~_~ ~ q
q
,,,~ ~
qldld2
=d~ q ~ qldl d2
q
a(m,
~
m
h- n
On,q)'dl On.n)=l
~ m~
M (m,q)=dl
b(h) )l'e+Ae h
. e-A
O(--~)A (o~u)du+
(h,q)~d2 (h,n)=l
a(m) m
b(h)
~ h~
h
H {h.q}=d2
~ e(~)+ O P-A<~<~P+A
(__~)
(h, n) =1
(re,n} =1
By Lemma 10, we know that when 0 e E l ,
e(pO)= tz(q) 1 e
A log2 p )
exceptional values, we always have log B 4p
I= E q{logB2p
~
,o,B4eS
all
{a,q)=l
A
(q +oc, n)
E
P
e
+or p ((q))
e( q a ~1 q<~l~
P (a.q) =1
(q)
log B 4p
~T((
Io B4p A
~ ~
a(m)
b(h) )
q q m~ M qldld2 {m.q)=dl
h- H (h,q)=d2
{m.n)=l
(h,n)=l
~ e(~s)" lz(q) ~ e(~r)e(_not))d~+O( A ) e-A<,~e+A ~p(q)logP e<,,~e+A log l'~-P
'
9
148
Jia Chaohua logB 4p A
f _mlog B4p
~" e(~) ~ e(~r)e(-n~)d~=A+O( .e-A
I=
(
A
log s4P
q
).
a(m)
(a,q)=l
e(--~-))
d~lq ~21q
m~M
qidl d2
(m'q)=dl
m
(m,n)=l
b(h) h
hE -H
9log P
log -rs p
(h,n)~l
qldtd21rr~, (mh, n ) = 1 =~ ( q ; . n ) = 1,
a=l ,q) = 1
e-
=#(q),
(a I=
pZ(q) cp(q)
~ ~ q~l~
P
a(m) m
q q .,-u ~-~ ~-~ ~ q[dld2
(m'q)~dl
(m .n )" I
h
h~n
log P
log "~ P
"
(h , q ) = d 2 (h ,n ) = 1
Now we investigate
fl=
1~2 ( q ) ~ r ( q ) . ~q d~q m - M
Z z
q~log B p
qldld2
~
m
h~H (h'q)=d2
(m'q)*dl
(m ,n )= t
b ( h.__._~) h
(h ,n ) =I
.~ ('q )
a (m)b (h) mh "
E , m-- M h--H
a (m )
E
q~(q) '
q ~ log B2p (m,q)=dl , (h,q)=d2
(ink ,n )- 1
qldl d2
#Z(q)
a(m)b(h)
E
E
m~ M
On,q)=dl, (h,q)=d2 qldld2 .
(( l~
q~(q )
q >log B2 P
h-H (ink .n ) - 1
P
Y" m-M,h~H
mh
LL L It
d ~(m) dZ(h)
(( log -B2+., p m~M,h~H.
mh
(( log-s p .
(12)
Three Primes Theorem in a Short Interval (V)
149
Hence,
E a(m)b(h)mh d~.d~2h
f~=
m--
qJd~d:E ~O(q)+0
(1) lognP
. (13)
M
h- H (mh , n ) = l
(m,q)-d I, (h,q)=d2
When/~2 ( q ) = l , qlmh, there must be qldld2. If not, q'l,d,d: and #2(q) = I, then there is a prime number p such that p lq and p~.d~d:. p lq =>plmh. Assume p l m. Then p l ( m , q ) = dl. This is a contradiction. Hence, when #2 (q) = I , qld~d2 <=>qlmh.
a(m)b(h)
E
2=
mh
/z2(q) ( 1 q~(q) + O l~
qt,,~
m~ M h- H (mh ,n ) ffi I
)
(ra ,q)=dl (h ,q)=d2
E
a(m)b(h)mh
m-- M
q~(mh )
Y'. gZ(q) ) +( O 1 qt~
n'l~ M h- H (mh ,n ) " 1
q~(q)
+0
lognP
I~
.
h- H (mh,n)~l
By (12) and (14), we know that
a(m)b(h) m-- M h- H (m,n)=(h ,n) =1
a(m)b(h) =
~-.I h - n (~,.)=(h,.)-I
~., ralg+p=n P-A
,4+0
~o(mh)
logP
(
A) n
log ~- P
9
So far, Theorem 4 is finished. [] Remark. When a (m) = O(d~ ( m ) ) , b (h) = O(dk (h) ), Theorem 4 still holds.
w
lwani~'s Sieve Method Assume ,~r
{ a" a = n - p , P
l-I p '
1,
S(J,z)=
p <= p 9~
a~.~ ( a , P ( z ) ) =1
D (n)equals the number of solutions of the equation
n=pl+t~, P
S ( J , z) ~
~ 11
(_V_)
I
S(od~, q)
(15)
150
Jia Chaohua II
t> log Plog z f
+O
log z
log 2p
(16)
9
/ f v (p) >> P', then v (p)) A_
i
p 15 < p ~ (2P)']" (p,n)'l
C(n)A log P
I1
p u
1
y,
(log
plogv(p) F , p "~" < p ~ (2P)-2-
-~0-
log~p)
)+o(
eC(n) A), log 2 p
, (17)
where F (u) , f(u ) satisfy
{
2
F(u)=--~ , f ( u ) = O ,
1 ~
( u F ( u ) ) ' = f ( u - 1 ) , ( u f ( u ) ) ' = F ( u - 1 ) , u > 2,
and p,1--I,( 1
C(n)=
( p - l1) : )fl-[ ( 1+ ( p -1I ) ) "
A Proof. In [13] , we take X= logP ' d co(d)= ~o(d) ' 0
# (d)q=O, (d,n) = 1 others,
,
r (d) = I.~al
X q~(d) By Theorem 2 on page 164 and formula (40) on page 171 in [9], we know that W(z)=I-Jz (1 =2i-i(1 p:.2
1 )~ (P-l) ~
co(p))p
p-~I e -~ p - 2 " logz ( 1 + 0 ( ~ ) )
p>2 -?
=C(n)
e
(1+0(5)).
By Theorem 4, we have
a(m) m
l,t-60~ < - ~7/6 J (m, n ) - I
~, b(h)r(..d',mh)<< , h < T'2" (h'n)~l
AB - 2
log ,o p
Three Primes Theorem in a Short Interval (V)
151
p J-6o,
Taking D =
T2/3 , by Theorem 3 in [13] and the continuity of f ( u ) , we
get
S(..~, z)-
E
I 3
>f W(z)Xerf
logz,]
S("n/~' q)
O
log2p
II
C(n)A
( l o g p'iT ) ( e C ( n ) A ) logPlogz f " logz +O. Iog2p
9
By the upper bound sieve method, we get
_ _S(~'q)=O(
I
II
.
log J) 9
)
p<$
So, formula (16 ) follows. By Theorem 3 in [13] , we know that
S(..%, v(p)) <~( l + O ( e ) ) +
E m<
plog v (p) log P F
h
p T 7/6 (m,n)=l
log v (p)
b (h)r(,~, mhp).
a(m) E
p 1-60t:
(')
log 19
C(n)A
I
(h,n) ~ 1
By Theorem 4, we know that
E
E
7.2_ 1 p 15 < p ~ < ( 2 p ) ' T
a(m) E
p1-60~ m<
~
p T 7/6 (m,n)~l
{P ' n ) ~ l
b(h)r(..~',mhp)=O(
AB - 2)
log --iU P
1 h< T 2 (h,n)=l
Hence,
S(,-~p, v(p) ) 7
1
p l"~
p~-tl
C(n) A ~<
log P
1 ~
7 I p 15 < p~< ( 2 P ) ' ] "
We obtain formula (17).
plog v (p) F
log log v (p)
+O
log2p
. [-]
152
Jia Chaohua Lemma 12. When F(u) and f ( u ) are defined in (18), we have
2 u '
F(u)= ,
l~
(!
F(u) = 2
log ( t - 1 ) dt"~ t
+
3~u~5,
)
f(u) = 2log(u-I)
2~u~4.
U
See page 176 in [9] . Lemma 13. If. ~ = { n : P < n <~ 2P } , then S(~,
z)~> ( 0 . 5 + 0 ( ~ ) )
P mlog z Proof. By Lemma 4 on page 240 in [9] , we know that
((m)), log ~
S(~,,, z)=
w
logz
+0(~)
mlogz
where w(u) is defined in Lemma 6. By Lemma 6, w(u) >1 0.5, hence
Sf~,., z)/> (0.5 + O(~))
P
D
m log z
w5. Balog's Method
In this section, we use the method of Balog [141 to estimate the trigonometric sum in E3. I Lemma 14. Assume log B2 P < q ~< z, I ~ [ ~< qz , T 0 = 4 r ~ ( I a I P + T ) , T ' = 4~z( lalA+ T ) , T~ = P ~~, P~
M , H <~ po.6, MH= P , a(m)= O(dkfm) ) , b(h) = O(dk(h) ), and Z is a character rood q. Then we have 1
~ I
(qp)T
Xr163 tiT2
~
a(m)g(m),§
~.
rn-M
mT
h--H
(( l o g - S P ,
b(h)Z(h), h T+it
I
dt
IT2 ~< To,
(19)
-2V"
( p1) ~ ~
-'-V-xo~ J~ I~=~M a(m)X(m)• '
1[~h n b(h)X(h), +~ Id t
hT
m
To <~ V <~ TI.
<< A log-BP, Proof. By Lemma 4,
(qp)T
z
o
2
,, ~
mT+it
2 h-n
b(h)~(_h) I dt
h§ §
(20)
Three Primes Theorem in a Short Interval (V)
153
I l
1
<<
~;( I / 2 + T ' ;t 0 2
( q e ) TI
m-
E
a(m)z(m)_T__ [:dt -f m T +it M
r2+r' z
b(h) x(h)
2
2
o
~
1
h- ,v
5-
dt
h T+u 1
1 1 ( M + q T ,)l (H+qT ')-f log"P ( ( l o g - S P 9 (q p)T
<<
hence formula (19) follows. Formula (20) can be proved in the same way. [] Lemma 15. Let ~, q, To, T ', Tl be defined in Lemma 14, M ~< p0.4,
M L I L z = P , a(m)=O(dk(m)). 1
, (qp)T
f T2 +T '[
E X*Xo , J T 2
l~,, mT
•
1~
E
12-L2
E o
((Alog-aP,
mT
t~
Idt
"~(ll)
,
re- M
Z(12) I +it
(21)
a(m)g(m)
#~z "j~ z ~
X (ll)
E II-LI
T2I~
I
V
a (m) z (m)
E re- M
((log -Bp, (__~)' ~
Then
+it
E
--v--
/I-L]
l T+it
1
12E -- L2
x(l~) l 1__+it Id/ (22)
T0~< V~< T,.
This lemma amounts to Proposition 2 of [7] in essence. Lemma 16. Suppose O= fiE_+ ~ E3 , M ~ (2P)-~ , a(m)= O(1) , and q a (m)= 0 if m has a prime factor which (( P~. Then, we have V(~)=
re-~
. - ~ < m~ ~<.+~
Proof. (rap, )-1
(rap,q)> 1
By Lemma 2, we know that m-E M
a(m)
<<
~
P-A < mp<~P+A (rap, q) >t
~q
e (--~)
~
P-A
pt dlr <<
~',
~"
dlq d>p~ P-A
e (~mp)
d(r)
d(q) d(s )
154
Jia Chaohua
d2(q) d(q)AlogP elq ,~. e' - 1d (( p------z-A log P ( ( A log -Bp.
<<
Arguing as in Lemma 7 , we get
o,m, ittE~ M
P-A
logP << ~
e-a
E ~#~0
~ < mp~
a(m)z(m) 7(p)e(~rnp) + A l o g P P+A ' q
(23)
And then, we investigate
1
2= ~q
~xo e-A
(24)
m-- M
By H e a t h - B r o w n identity (refer to [14] or [7] ),
a(m)z(m) A (h)z(h)
E
P - A < mh<~u m- M
=
a(m) z(m) b~ (h~) z(hx)'"b2o(h2o) Z(h2o ),
E
P - A < mh l . - ' h 2 0 ~ u m-
M,hj-
Hj
where log h , 1 ,
bj(h)=
j = 1, 2 ~
It(h), I
M'HI'"H2o=P, Hi ~ P~(i= 11 , .-. , 20). As in Lemma 7 (also refer to [7] ), we reduce the estimation of << A log-n-2P to the estimations of
1
~
1
(qp)-f
F
art((log-nP,
+it,z
[T2]<~ To
(25)
T0~< V~< T~,
(26)
z*xoJr2 m V
F x
dt((Alog-nP,
+it,x
o
where
a (m)z (m)
2O
F (s, Z) =M(s, g). 1-IHj(s, Z), M(s,z) = j'I
m~M
Hj(s,z) = • h~ nj
bj(h) X (h) h"
m s
Three Primes Theorem in a Short Interval (V)
155
Using the method of Balog [14], we reduce F(s, Z) to one of the following two forms 9
f(u ) Z(u) g(v) Z(v) u" ", ~" v" - U ( s , Z) V(s, Z ) ' where - v
a) F ( s , z ) = ~~ ~v
f(u) =O(d,(u)), g(v)=O(dk(v)), U, V <~p0.6, UV=P. b) F(s, Z) =
~v u-
f(u) Z ( u ) us
f ( u ) = O ( d k ( u ) ) , U <~P~
t,-E Ll
Z (ll)
t2-E L2
Z(12)
"
where
=P and we allowt~'.~z2 Z(121____)_72---- I .
We discuss four different cases. Case 1 . po.4 < M ~< po.5. 2O
Let M ( s , z ) = U ( s , z ) ,
I-IHj(s, Z ) = V(s,z). Then it becomes type a). j-I
Hence in what follows, M ~< p 0 . 4 c a n be assumed. Case 2. There are at least t w o H j ' s (assume j = 1 , 2 ) > P ~ . Let H~(s, z )= ~ It-
~(l~.__~) H2(s, z) = LI
l~
'
~
Z(I~)
1 2 - L2
The product of
l~
the rest by M (s ,;() is taken as U (s, Z). Then it becomes type b). Case 3. There is just one H~ > p0.2. If Hi >p0.4, let H~(s, Z ) =I1--L1 ~ Z(I~ l~ ) , Hz(s, Z) ~
1; the product of the
rest by M ( s , X) is taken as U(s, Z). Then it becomes type b). If H~ ~< p0.4, on the basis of H ~ ( s , Z), multiply the remaining factors successively so as to make the length just exceeds P~ M ~< p0.4, this can be achieved). This product is taken as U(s, X) and the product of the rest by M ( s , Z) as V(s, ~). Then it becomes type a). Case 4. All Hy ~< p0.2. On the basis of M ( s , Z ) , multiply H i ( s , Z) successively so as to make the length just exceed p0.4. This product is taken as U(s, Z) and the product of the rest as V(s, Z). Then it becomes type a). When F(s, Z) becomes type a), by Lemma 14 we know that formulas (25) and (26) hold. I f F ( s , Z) becomes type b), when p0., ~< U~< p0.6, let v=l~lz; then it becomes type a). When U ~< p0.4, by Lemma 15 we know that formulas (25) and (26) hold. So << A Iog-B-2P.
Hence,
,og X#XO z I
z
P-A
V(~) << A log-B P . Remark. For a(m)=O(dk(m)), Lemma 16 still holds.
[-]
156
Jia Chaohua
w 6. Application of Zero Density Estimations in the Short Interval L e m m a l 7 . Assume q <<. logn2P, ( a , q ) = l
,lot[ <~ 1 , To = 4 Z c ( l ~ l P + T ) '
qz T'=4rc(l~lA+ T), T~=P ~~ a(m), b ( h ) = O ( 1 ) . If m, h have a prime factor which<< P', then a(m)=b(h)=O. Let X be a character modq, p=]~+/y a zero of L(s, z) in 0 ~
a(m)x(m)
r~<,,~r2+r'
m-~U
mP
((exp(-c~) 2
V
ra
hP
h~H
,
T21 ~< To
(27)
a(m)g(m)l[ b(h)z(h) Ipp mp 2 h" h- H
1
V
b(h)z(h)[pa-,
~M
( ( A e x p ( - c lo.,/-i~gp ) .
(28)
To~< V~< T~,
we have ~=
~
a(m)b(h)
m-- M h~ H
#(q) cp (q)
~ P-A mh
a (m)b (h) m~ 2M mhlog P h~ n mh
e ( ( q + ~)mhp) P+A mh
e(~r)+O(Aexp(-c~ "-A <~'~P+A
))-.
Proof. q
2=
2e
-~-)
(s,q)=l
1
(q
~
a (m)b (h)e (o~mhp)
P - A < mhp ~ P+A m h p ~ s(mod q)
(q))
a (m ) g (m ) b (h )z (h )z (p )e (o~mhp). P - A < mhp ~ P+A
(29) And then we investigate = EI
~_,
a(m)x(m)b(h)x(h)A(r)z(r)e(otmhr)
P-A
=
e(~u)d dP-A
~
a(m)x(m)b(h)x(h)A(r)z(r
"
P-A
Let
F(s, Z) = ,~~.% ~
a(m)z(m) m"
h-~ H
b(h)z(h) h"
L (s, x) L
"
Three Primes Theorem in a Short Interval (V)
157
Using Perron's formula and moving the integral line (refer to H e a t h - B r o w n [15]), we get
a(m) z(m)b(h ) z(h ) A (r) z(r) P - A < mhr<~u
fl l+'+iri
1
27ri
+~-iTl
a(m) b(h) m h ~- H h
~u m--
a(m)x(m).) (,~ b(h)z(h) mP i~
m~M
ds+ O ( P ' )
S
=Eo(u-(P-A)) I;~l~
u,-(p_A),
F ( s , Z) "
(uP-(P-AY)p
+o(P'),
Eo =0 ' X#Zo 9 a(m)b(h) fp§ = 9 E l E0 .,-~M mh .Je-A e(~u)du
where E0 = 1 . X=Xo ;
h~H
-,.~,~r, ~2=
(
a(m)z(m))( b(h)z(h))f ,§ (-~-) mP h~-u -~ " .Je-a up-le (~ ) du+ O "
.,~- M
,rl.
<<
~
['/l~ TI
min
A.
E
17l ~ TO
rain
IV +2n0~ul
P-A ~u~P+A
,,,_ EM =
h-~l b(h)z(h)) p .,e-a up-le(~u) du
~,.-M a(m)z(m))(m p
+
mp
h~~'. H
hp
E
TO< I?l ~ TI
In the same way as in Lemma 7. we can get
IT2I~T0
fl~<
max To
T2
1 TI
W
~ g
m~ M
m~
1 ~ a(m)z(m) ]1 m-
M
mp
By formulas (27) and (28), we know that ~r'~l 3t-~~2 ~ A e x p ( - c lox/~gp ),
Y', h~ H
b (h)g (h) I pa-i hp I
h ~E H
b(h)z(h) l p ~ hp
.
158
Jia Chaohua
ZI
= go "
a(m)b(h) mh
E
m~ M h - II
p+ae (a u)du+ 0 (A exp ( - c lox/-i-~g P ) ), -A
e( r)=je
P-A
e(au)du+ 0
9
-A
Hence,
a (ra )z (m )b (h )z ( h )x (p )e (~mhp ) P-A
a (m)b (h)
:eo-E
.- u
mhlog ~
E
#(q)
e ( a r ) + O ( A e x p ( - c lo.fi-~gP )),
~-,l<,,~e+x
mh
h- n
E=
e
a(m)b(h)
~(-q) m~Xu m k l o g ~ P h -- H
e-a<,,~e+a
e(~r)+O(Aexp(-c
loxfi-~gp )).
mh
In [ 3 ] , author used the square mean value theorem of Rane [16] concerning L-function and obtained an estimation of 9 density in the short interval. 1
L e m m a 18. I f a >l-~ , q <<.exp(loglog3T),
~ is a character m o d q ,
and 1
N ( a , T , Z ) is the number of zeros of L ( s , Z) in Re s >~a, [ I m s [ ~
x)-N(a,TI,x)<<
T
logcT.
In [17] , Z h a n Tao used a new square mean value theorem of L-function and obtained 1
L e m m a 19. I f T ~ T ~ 3
, -~ <~ a<<. 1, N ( a , T , q ) =
~
N ( a , T, Z),
z(~-od q)
then N ( a, Tl+ T, q ) - N ( a , where
4 3-2a
Tl, q ) < < (qT) ~7~')+~)"-') logCqT, 1 -~-~
'
2 a
3 4
3 --~- , 9
~
In [17], there is (qT)r(')(I-r on the right of the above f o r m u l a . By some simple operations, .it becomes the form of L e m m a 19. Assume a (m)
M(s)=
~u
m--
m"
b (h)
,H(s)=
~
h-
H
h"
(30)
Three Primes Theorem in a Short Interval (V)
159
The methods in Lemmas 20,21 , and 22 are all adopted from Lemmas 4 , 5 , in [ 15]. 1
7
Lemma 20. I f PT<< M<< P f f , q <~ logB2P, Z is a character m o d q , p= fl+ iy is a zero of L ( s , Z) in 0 <~ Res ~< 1, then
I M ( p ) I P ~-' << e x p ( - c lox/'~gp ) ,
IT21< To,
(31)
T 2 <[?1~ T2+T'
1 ~ iM(p)lPp<
To~< V~< Tl.
lo.,fi-o--g-P fg ),
(32)
Proof. In order to prove formula (31), it is enough to prove
~_, T2
IM(p)IP r
<< p,r162
'
and to use the zero-free domain of L (s, Z).
T2 < 1 7 1 ~ T 2 + T '
Z
T2, Z))
<< ( N ( a , T2+ T ', z ) - N ( a ,
I M ( p ) 12 ,
T2 < 171 ~< T 2 + T
'
IM(p)I 2 T2<17I~
< < M l - 2 ~ ( M + m i n ( T ', T ' T ( N ( a ,
T2+ T ', z ) - N ( a ,
T2,Z)))).
(33)
By Lemma 19, formula (31) follows. Formula (32) can be proved in the same way. [7 Lemma 21
:
9
u
I f P T < < M H < < P r~-,
p~-<<
M H
i
< < p T , q~< logn2P,p=fl+i~
is a zero of L (s, z ) in O ~ R e s ~ < l , t h e n T2 < I?1 ~< T 2 + T
T
1
~
'
[ M ( p ) H ( p ) I P ~-x< < e x p ( - c lox/'i~gP ),
I M ( p ) H ( p ) I P P < < A e x p ( - c lox/-~gp ),
IT21< To,
Zo<~ V<~T~.
V
Proof 9 ~
IM(p)H(p)l l
<<(N(a,T2+T',z)-N(a,T~,z))7
l
(~lM(p)l 2)T (~lH(p)t,)-r
l
.
Using formula (33), Lemma 19 and an argument similar to that for Lemma 20, we can get Lemma 21. 17 2
Lemma 22. I f p2/3<< MH<< p1-~, M~-<< H<< M 3/2.,q ~ log n2p, p= fl + i7 is
a zero o f L ( s , z )
in 0<. Re s<. 1,then
160
Jia Chaohua
r=<,,~r2+r' IM(p)H(P)[P#':'<< exp(-c loff-logP ),
IT21< To ,
!
To~< V~< TI-
V
~
IM(p)H(p)lP#<
lo~/-i~gp )
V
Proof. ~ I M ( p ) H ( p ) l < < ( ~ IM(p)l 2 ) + ( ~ l H ( p ) l 2 )~-. Using formula (33), Lemma 19 and an argument similar to that for Lernma 20, we can get Lemma 22. ['] w
The Asymptotic Formula and Lower Bound of Some Sieve Functions Lemma 23. For 2P
always holds true that
#2 (q) q~2(q)
E q > log B 2 p
q log Bp
a-I
(a, q)= 1
See Lemma 12 in [5]. I
Lemma 24. Assume P T < < M < < Pi2r, a ( m ) = O ( d ( m ) ) , a ( m ) = 0 when m
has a prime factor which << P" . Then for 2P < n ~ 2P+ A , except for 0
log 2p
exceptional values, it always holds true that K=
~
a(m)
mp+pl =n P-A
-
a(m)
log
m-EM
1+O
~ P < P ~ 2P ?n m
eC(n)A
I
.O~
Proof. I
K=
e(ptO ) e ( - nO ) dO.
~M
P-A < mp~P+A
P < Pl ~ P + A
By Lemma 16, we know that when 0e E3, S(0)=
~ m-
a(m) M
~
e(Omp)<
P-A
By Lemmas 20 and 17, we know that when 0e E 2 ,. S(0)<< A log-BP, and when 0 = a
q
+~eEl,
Three Primes Theorem in a Short Interval (V)
I~(q) a(m) S(O)= ~P(q) m-~u m l o g - -P m
161
~., e(ow) + O ( A e x p ( - c e-a<,<.v+A
l~ogP )).
ByLemmas9 and 10, we know that for2P< n~<2P+A,exceptforO exceptional values, it always holds true that K=
l
S(O)
~
e
e(N)e(-nO)dO+O
log~P
.
log 2P
log B 4p
E
( q)f_ ~
E
q~logB2p
e -
a=l (a,q)=l
E
I~
m -- M
a(m)
A
v
e((q
E
e
P - A < p<~ P+A rn ra
((q)+~ m,)
+ ~ ) p l ) e ( - n ~ ) d o t + O ( eC(n)A log 2P / log B4p
p2(q) ~ e -( . _ ~ _ ) f_ ~,a4 a cp2(q)
E
a=l
q ~l~
~
(a,q)-I
A
a(m) m-E U m log--P m
1
logP
~
e(~)e(-n~)d~+O
v<~,;e+a
E
e(~r)
P-A
( tC(n)A ) l o g 2P
'
iogB4p
_ A
IogB4p
~,
e(otr)
P-A
~.
e(v.s)e(-no~)d~=A+O
P
A
A -)exceptionalvalues, By Lemma 23, except for O ( log2P
cP2 (q) a-I ~
q~logB2p
e
-
(a,q)~ 1
00
q
/z2(q ) q-i cp2 (q)
~, e a-1
(- q-) +O(~o1,,)
(a,q)- I
=C(n)+O
(i)
logSP .
Hence,
x=
E
m-- M
a(m) m log P m
C(n)A + 0 ( t C ( n ) A ) \ log2P logP
(lo:,,).
162
Jia Chaohua
C(n)A -
P log
a(m)
P ,,_2
E
M
I+o(
P (
p ~
"
- -
9~
L e m m a 2 5 , I f l O - a lq<
log ~P
2P
- -
eC(n)A)
m
- -1~ , ( a , q ) = 1 , log BzP< q ~< z, a ( m ) = O ( d k ( m ) ) ,
b ( h ) = O ( d k ( h ) ), then a(m) p 0 . 4 < < m << p 0 . 6
~"
.
P - A < mh ~ P+ A
b(h)e(Omh)<< logA np
This follows from resorting to Vinogradov's method. One can refer to Lemma 4 in [5]. Lemma 26. When m, h have a prime factor which<< P" , a ( m ) = b ( h ) = 0. Suppose a ( m ) = O(d(m) ), b ( h ) = O ( d ( h ) ), and M , H satisfy 11
1
i ) One of M', H or M H lies between p0.4 and P~ P 2/J << MH<< pfr, p -rr << 1
M << p~-, or H ii ) One of M, H or M H lies between P~ and p0.6, P ~-<
A ) exceptional values, we alThen for 2P < n <~2P+ A, except for 0 ( lo~-~p ways have K=
~
a(m)b(h)
rn h p+p I - n P - A < m h p <~ P + A P < P l <~ P + A m~M.h~H
-
C(n)A Plog P
h-
1+O(~C(n)A) log 2 p
a (m).b (h)
~
P mh
H
9
2P mh
Proof. Assume that M , H satisfy i). S(0)=
a(m)b(h)e(Omhp),
~ P-A
1
K=
f ,_"s(o) y, -
e( pL O)e(-nO )dO .
P
One of M , H or M H lies between p0.4 and p0.6. By Lemma 25, we know that when O~E3 , S(O) << Alog-nP. By Lemmas 21 and 17, we know that when OeE2, S(O)<
Three Primes Theorem in a Short Interval (V)
p(q) a(m)b(h) S(O)- ~o(q) ,,,~~u m h l o g - - P ~- u mh
163
e(rtr)+O(Aexp(-c~ e-A<~<~e+A
)).
The remaining procedure is similar to that for Lemma 24. If M , H satis~ ii), Lemma 26 can be proved in the same way .
Lemma 27.
E=
Z 1 (2P) T
[3
S(~'~, p) 7
7
C(n)A
=log ~
log2p
+0
( eC(n)A'~
log2p j -
Proof.
~=
~ 1, Pq+Pl =n
where
P-A
1
7
(2p)r
~
1
factor of q > p . Because p > q T , q is a prime number. Taking m=p, by Lemma 24 and the prime number theorem, we know that
C(n)A E=
PlogP
( eC(n)A) , E , (2P)T
-~-
~ (:e)3-
p p log P
IogP
p E
1+ 0
+ O
log 2 p
logZP
7
C(n)A f 'f_5 log2p
dt
/(l-t)
+O
( eC(n)A'~
~ o g ~ /]
3
7
C(n)A
=log T
Lemma 28.
( eC(n)A.'~
logZP
E =
+0
II
D
IogZP ,].
S(~'/q,q)>~O. 13 C(n)A log ~P
E
p ' ~ < p <~ P~-5
Proof. S ( ~ ,
q ) / > ]{r" rpq e ~ , r is a prime number }l,
E >-"
E
l=y',,
pqr+Pl =n
where
11
4
P-A< pqr <<.P+A, P < p ~ P+A, P~
q<
164
Jia Chaohua
min(,
7
,
Let orem,
, r is a prime number. P
m=pq.
E'=
I
7
When p T < m ~< p i r ,
PlogP
a(m)
~
y'
by Lemma 24 and the prime number the-
1, < p ( 2.e m
1+O
log2p
m
C(n)A ( -
I
logp
II
p~O
+
E
7
4
,,
7
E
II
P
,
pcl log
P
,,)+o(
--7
pq log
is
~C(n)A) log 2p
pq
P
c,.>.(f..,f, log 2p
7
4
7
dw w(1-t-w) + 7"T
II
u T
-~--t
L-T-* -IIi T - t
wfi-t-w)
3
+ 0 ( eC(n)A )
/>0.13
log 2 p Lemma 29.
C(n)A log 2p
S (..~pq,q)~>0.212
~ = p ts-
log 2 p
P'T I
1
z~__lT
Proof. Because
C(n)a
pT <
2 >> -
( p-iT
I
pT
s
I
pT
I
S(4,,q)
I
e T,
PT
=El=E,, pqr+Pl'n
1
4
P - A < pqr <~P+ A , P
where min
I
,
, the least prime factor of r > q.
pT ~ <1 q < pr
Three Primes Theorem in a Short Interval (V) I
7
pT 1 p~-
When
165
pW /7 < q ~< --'---7- ' q < min ' p~-
=)
2F"
4
let m = r ,
h=q. PT;-
/ I
P~
I a PT p~/3<
,
-~ / '
7
I
q
po.6'~
m =)PW<
1 pT
P~176
1
pT ~ p2
<
By i) of Lemma 26, we know that the
P asymptotic formula holds. 7 (p When pw1
p0.6) , let
__PW
h=p. pT1
m=r,
P
p0. 6~ =) P 2/3<< mh < < P ~-', mZ/3<< h < < m 3/2, p0.4<< m < < p0.6. By ii ) o f L e m m a 26, P / we know that the asymptotic formula holds.
~,
C(n)A =
~
PlogP
~
,
'
~
e,-r
e~l
<
~
(o, po.6~ q < rnin
pT
V
P
)
1+0(eC(n)A)
P 2e W < , , - - Pq
log2
P
the prime factor ofr>q
-
C(n)A
~
PlogP
eff
,
S(~;q,q)+O(,C(n)A) l~ 2P
"
I.!_ ~
< q < rain
,
P2
By Lemma 13 and the prime number theorem, we know that
C(n)A
El >
~
log P
0.5 pq log q
4
+ C(n)A log P
p'~-< p < pO.3 P3
1
pT
,
pO.3
~
<
pq log q
log 2 p
q < - -
_.*
p
p2
1
C(n)A log 2 p
/> 0.212
(
0.5 f ~
C(n)A log 2p
dt t
dw
, - 'w- T + 0.5 T
~- dt
fo , .3
..:'-_: .~-3"
dw + 0 (~))
-W - 2
2
D
166
Jia C h a o h u a
Lemma 30.
.~
S~,.~pq, q)>~0.512
= P t~ < P ~ ( 2 p ) 2 II I
C(n)A log2p
I
Proof. Because
J" > pSI3
,_E
E
,_
S ( , ~ , q)
PIS< p~(2P) 2 ,
,
p5/3
=
E
I=E,'
pqr+Pl =n 7
where
11
1
~@
P - A < pqr<~ P+ A , P
P < p5/3 '
the least prime factor of r > q . I1
Let
1
m=r ,h=p. ( p l ~ ) ' 7 < q< ~ P p 2=>.
7
/3<
1
p W < p ~< (2P) 5- => po.4<< h<
C (n)A
~=
1+O(
PlogP
eC(n)A~og ~-~ )
e_._ < q ~< 2_.L? pr
C (n)A --- P l o g P
~P,q
pr
E
1+0
log2p /
P---
Pq
the prime factor ofr>q
C (n)A P log P
E
s f~,, , q ) + O (
log2p
//
p I"]'
• p5/3
/>
C (n)A log P
7
E
p l'T
0.5
pq log q
I
(2p)'T
I
P
5/3
+ 0 ( eC(n)A ) log 2 p
Three Primes Theorem in a Short Interval (V) 5
C (n)A . 0.5
dt
>I log 2 P
167
~ + o ( ~C(n)A.'~~ 0 . 5 1 2 C(n)A
it dw w -t w 2
t
log2P ,]
log 2p
5
w8. Proof of Theorem 2 By Buchstab identity, we get
'
(s (,~, ( 2 p ) T ) -
S(..~, ( 2 p ) T ) =
AlL i p6O< p ~ (2P)T
E
s(~,p)-
(2P)3 < p <
S(.~p, p )+
E
PIS
S(~q,q))
11
E
I p'-~'< p <~(2P) 3
pin
ILl
S(~q,q)
•
(34)
"~- E l - - E 2 - - E 3 "dl-E 4 9 By Lemmas 11 and 12, we know that
C(n)A
i /> By Lemma 27,
O(eC(n)A)
~f (_.~_)+
log 2 p
log 2p
C(n)A
>/O. 497
(35)
log 2p
we know that
7C(n)A z =I~
(eC(n)A)
Iog2P
By Lemmas 11, 12 and 30,
E ~=
+O
C(n)A
log2P
~<0.56
(36)
log2p
we know that 7
E
t_
s(4,p)
p i T < p ~< (2P) 2
E
S 7____ p 15
S(~pq
, q)
7 ! p i"]-
1
I
/ - - 7 - : "~< ~<
C(n)A log P
5F(5) 7 ~
,
p " ~ - < p ~< (2P) -"2"
I1
p-w
- 0.5115
C(n)A log 2 p
plog P
C(n)A ( log 2 p
1+
ff log(u-l)
2dt
u
15
t
11 - t )
53-
-0.5114
C(n)A log 2 p
168
Jia Chaohua
~< 0.266
C(n)A
(37)
log2p
By Lemmas 28 and 29, we know that
C(n)A
4 I> 0.342
log 2p
(38)
By ( 3 5 ) - (38), we know that I
S(o~, (2P) ~- )I>0.01
C(n)A log2p
(39)
So far, Theorem 2 is finished9 w
Proof of Theorem 1 In Theorem 2, we take P= N 9 p is a prime number satisfying ~N - A < 3
,4 " There are >> log A N even numbers such P~< TN ' the number of which >> logN A ) e x c e p t i o n a l v a l u e s i n Theoas n = N - p , 2P > C ( N - p ) P2
____L__a >> a__A__ log2N logZN pairs Pt,
satisfying
N + A , TN -N~ < P1<~-~-
N-p=p,§
-
A
N + A.
~
So, formula (3) has solutions. So far, Theorem 1 is proved. w10. Proof of Theorem 3 23
L e t P < n <~ P+ A , A = P ~ 9 It is enough to prove Dl(n )=
~ n-pl+P2 pI ~ A
A 1 >> C(n) log2P "
(40)
P-A < p 2 ~ P + A
Assume J =
{a: a = n - p , p<~ A } , ~ = { p : p.4.n }. P ( z ) = I - I p, t21={0: p
0 ~ [ 0 , 1 ) , 0 = - a a q + e , (a,q)= 1,q<~logB2P,lel~ logB4p},A f ~ z = [ 0 , 1 ) - f ~ l . D, (n) equals the number of solutions of the equation
n=pt+p2, 1
D, ( n ) = S ( . ~ , ( 2 P ) T ) +
O(P').
According to Iwaniec [18], we have
p~<~A,
(41)
Three Primes Theorem in a Short Interval (V)
l
S(.~, (2P) ~- ) =
E
( S(~,
169
P ~- ) -
E
s(~,p)-
E
s(~,p)-
3 16 P ~ - < p .< P ' ~ -
S(~,
E
s(~,p)
71 I p - i ~ - < p .< ( 2 P ) T
16 7...~1 p ' ~ - < p .< p 147
+
q))
S(.:r
E
(_%)-
3
,
=EI-E~-E,-E,+Es"
(42)
In using Iwaniec's sieve method for ~ ,
we shall deal with the remainder
term
a (m)b (h )r(ocef, mh )
R= m
a(m,b(h, ( m < M. h < n
(m'n)=(h'n)=l
Z
1-
mhl+p=n P-A<
1
~o
A)
logP
(mh)
"
mhl ~ P + A p<~A
Using the same method as in Lemma 7 and Theorem 4 (here, the procedure is simpler), we get
I=
~
a(m)b(h)
m
~
1
mhl+p~n. P - A < m h l ~
a(m)b(h)e(Omhl)
~ e(pO)e(-nO)dO.
P - A < m h l <~P+A m
p<~A
(re,n) ~ (h,n)=l
When 0 s E2, using Vinogradov's method, we get
e(pO ) << A 10g-2B-2p. p ~
A )exceptionalvalues, wealwayshave So, except for O ( lo~Sp
1
According to the methods in Lemma, 7 and Theorem 4, we can reduce the search for ~ l t o an estimation of
l(
~d
q)
2 ,~ u
--Z~ m
2
tt
2
h-
n
h
' .
T
+tt
E
l-- '
~
l
T
+it
dt (43)
170
Jia C h a o h u a
when q ~< logB2p. This estimation can be obtained by method in [18] (here, because q ~ logB2p, the upper bound of (43) has one more factor logcP as compared with the corresponding formula in [18], which however does not influence the conclusion). Hence,
I=
~
ra
q)
(mh)
log P
R<
1
' (44)
For Y-2, E3, E4 and Xs, we can deal with them according to the methods of Lemma 7, Theorem 4 and Lemma 17. We reduce them to the form of complex integral and zero density, then apply the method of [18]. So far, we have given the outline of the proof of Theorem 3. References [ I] [2] [3] [4] [ 5] [ 6] [ 7] [8 ] [9] [10] [II] [ 12] [ 13] [14] [ 15] [16] [17] [18]
Pan Chengdong and Pan Chengbiao, The estimation for the trigonometric sum over primes in the short interval (II), Sci. #~ Chbta, Series A, 6(1989), 641-653. Jia Chaohua, Three primes theorem in a short interval(in Chinese), Acta Math. Sin., 4(1989), 464-473. Jia Chaohua, Three primes theorem in a short interval (If), to appear in the proceedings of the meeting for memorializing Professor Hua Luogeng. Pan Chengdong and Pan Chengbiao, The estimation for the trigonometric sum over primes in the short interval (III), to appear. Jia Chaohua, Three primes theorem in a short interval HID, to appear. Jia Chaohua, Three primes theorem in a short interval (W), to appear. Zhan Tao, On the representation of large odd integer as the sum of three almost equal primes, to appear. Ramachandra, K., On the number of Goldbach numbers in small intervals, J. Indicm Math. Soc., 37(1973), 157- 170. Pan Chengdong and Pan Chengbiao, Goldbach Conjecture (in Chinese), Science Press, Peking, China, 1981. Shiu, P., A Brun-Titehmarsh theorem for multiplicative functions. J. Reiae Angew. Math., 313 (1980), 161 - 170. Min Sihe ,TheMethod of Number Theory (second volume) (in Chinese), Science Press, Peking,China, 1981 . Jutila, M., Mean value estimates for exponential sums with application to L-function, to appear. Iwaniec, H . , A new form of the error term in the linear sieve, Acta Art'th., 37 (1980), 307-320. Balog, A . , and Perelli, A . , Exponential sums over primes in short intervals, Acta M a t h . Hung., 4 8 ( I - 2 ) (1986), 223-228. Heath- Brown, D .R. and Iwaniec, H . , On the difference between consecutive primes, Invent 9 M a t h . , 55 (1979), 4 9 - 6 9 . Rane. V . V . , On the mean square value of Dirichlet L-series, J. London M a t h . Soc., 21 (1980), 203-215. Zhan T a o , Zero density theorems for Dirichlet L-functions in short intervals, to appear. lwaniec, H . and Pintz, J . , Primes in short intervals, Monatsh. M a t h . , 98(1984), 1 1 5 - 1 4 3 .
