Arch. Math., Vol. 56, 454-464 (1991)
0003-889X/91/5605-0454 $ 3.70/0 9 1991 Birkh/iuser Verlag, Basel
On the sequence p + h By JIAHAI NAN
Introduction. Let x be a sufficiently large positive number, h ( + 0) a fixed even number, P, Pl, ... primes. Set Ch =
I-I ( 1 - - ( P - - 1 ) -2) 1~ (P --1) (p -- 2) -~p>2
2
The work to determine the number of primes or almost primes in the sequence p + h, p < x, is closely connected with the well-known Prime Twins Conjecture. In 1973 Chen published his famous result of "1 + 2" type [2], which immediately leads to
Chen's Theorem. I{P:P < x , p + h = P l or PlP2}I > 0.67 ChX l n - 2 x . Five years later he improved the coefficient 0.67 to 0.7544 [3] and 0.81 [4] separately. By the Rosser-Iwaniec sieve [7] and invoking a deep result - due to Bombieri, Friedlander and Iwaniec [1] - about primes in arithmetic progressions to large moduli (see L e m m a 2 below), recently Fouvry and G r u p p improved the coefficient in Chen's Theorem to 1.42 (cf. [5] Theorem 4). On the other hand, for r => 3, the author proved in [9], among other things, [{P:P<=x,p+h=Pa
""P,-1
o r p l " " P r , Pl < " " < P , } l
> (0.965/(r - 2)!) c h x I n - 2 x (In In x) r- 2 which missed the presumably correct order only by a factor of In In x. The purpose of this paper is to show the following Main Theorem. F o r a n y r >: 2, I{P:P ~ x, p + h = Pl "'" P r - ~ or Pl " " Pr, Pa < "'" < P,}[ > (1.94/(r -
2)!) ChX l n - 2 x (ln In x) r-2 .
Vol. 56, 1991
On the sequence p + h
455
Lemmas. Let d denote a finite set of integers, Id l the n u m b e r of elements in d , and a set of primes. Suppose Id l ~ g and for squarefree d, (A1) (A2)
]~r I =
09(d ) d
~
S + rd, ~ d = { a : a e d , d [a}, 09 (d) is multiplicative, 0 < o9 (p) < p, p ~ ~ ;
09 (p)/p = In (In z2/ln Zl) + O (In- 1 zl ) for z 2 > z 1 > 2.
zl <=p
Forz>=2,1etP(z)=
Iq
p,S(d;~,z)=l{a:aed,(a,P(z))=l}[.
p
L e m m a 1. Assume (A 0, (A2). Then (cf [6] (1.2) (1.3)) for all 2 < z < D 1/2, (1)
S(d;~,z)<=XV(z){F(s)+e}+
Z
Rf(d,D),
j--
S(d;~,z)>__XV(z){f
(2)
(s)-e}-
~
R[(d,D),
j<-J(e)
where s = In D/ln z, J (e) depends only on e, Rj ( d , D) =
Z
2i (d) r d with a well-factorable
diP(z) d
2;(d) of level D and order 2 (for a precise statement see [7]), (3)
V (z) =
1-[ (1 - 09 (p)/p) = c (09) e - ? l n - ~ z (1 + 0 (In- ~ z)). piP(z)
? is the Euler constant, c(09) = I~ (1 - 09(p)/p) (1 - 1/p) -1. F and f are defined by p
f F (s) = 2 e?/s, f (s) = O if O < s < 2, ((sF(s))' = f ( s -- 1), (sf(s))' = F(s -- 1) if s > 2.
(4)
F r o m (4) it is easy to deduce (5)
F(s)=2er/s
(6)
f(s)=2e ~ln(s-1)/s
(7)
F(s)=
,,( S
if
0
I +
if
2_
.,). -
2
d
if
3
and (8,
f(s)=2;r(ln(s-1)+s-~ldttilln(x-1)dx 3T
) ~
if
4
-"
L e m m a 2 (ef. [1] T h e o r e m 10 and [5] L e m m a 7). Let a # O, ~ > 0 and Q = x4/V-t For any well-factorable 2 (q) of level Q and any A > 0 we have Z (q, a) = 1
2 (q) Or (x; q, a) -- li x/q, (q)) ~ .... a x l n - A x.
456
J. KAN
ARCH. MATH.
Lemma 3. Assume (A0, (A2). Then (cf. [8] (6) (7) (8) (9) with ~ = 1) S ( ~ ; ~ , z) < X V(z) ( f ( s ) + O(ln-1/aD)} + R o, S ( d ; ~ , z) > X V(z) {f(s) + O(ln -1/3 D)) - R D, [ral, V(z), F, f are the same as in Lemma 1.
where s = In D/ln z, R D = d
L e m m a 4 (cf. [10]). L e t 0 be a f i x e d number with 0 < 0 < 1, n (y; a, d, l) = ~ 1, f (a) ( ~ 1) be a real function. For any given A > 0 and any small e > O, ~P<=y'"P=-ira) Z
max max
d<=x 1 / 2 - e
y~x
a<=x~-Za.n)=o(
(l,d)=l
f ( a ) ( l r ( y ; a, d, l)-li(y/a)/~o(d))
xln-nx.
1
Hereafter we always take ~ ' = ( p + h: p < x), ~ = {p: p f h } , ~o(p) = p/(p - 1), p ~ h . T h e n it is easy to see that (A 0 a n d (A2) are b o t h satisfied.
Theorems and their proofs. Theorem 1. Let 6 be a f i x e d number with 0 < 5 < 1. For any r > 3, [{p: p < x, p + h = pl " " p , _ l or pl ".. p,,pr > " " > pl > e x p ( l n ~x)}[ > (1.94 (1 - 6)~-2/(r - 2)!)ChX In - 2 x (In l n x ) ~-2 . P r o o f. F o r d e a r n e s s , the p r o o f is divided into five parts. 1. W e i g h t e d s i e v e. Let v = (In x) l - ~, u = In In x. F o r any fixed a > 3, if r > 3, we have (if h < 0, then the second expression in the sequel s h o u l d be d r o p p e d ) [ { p : p < x, p + h = p l . . . p r _ l (9)
or p~ " ' ' p r , pr > " " > Pl > e x p ( l n ox))[
>](p:p
Pr>'">pl>exp(lnax)}]
> S - $1/2 - $2/2 - O ( x l n - 3 x), where
s =
Z
s ( d , , . . . , , _ ~ ; ~,~...,~_~, x TM)
xl/v < Pl < " "" < Pr- 2 < xl/u
(recall
S ( J a ; ~q, z ) = [ { a : a ~ d d , (a, Pq(z)) = 1}[, d d=(a:a~d,dla},
~q=(p:p~,pXq},
FI
P~(z)=
p),
p
Sl=
Z Xl/v
$2=
Z
X1/u
E
"''pr-2
~
IV
x1/~<__p
ZZ xl/~pr-l
Z p=pl""pr+l-h Pr < Pr + 1 < x / ( P r -- 1 Pr)
Vol. 56, 1991
On the sequence p + h
457
with pi~/h,i=
1. . . . . r + 1.
The reason is as follows. First of all, we m a y disregard those a's (a = p + h) for which (a, h) > 1; for then necessarily (a, h) = p, so that the number of such elements a is at most v (h) (v denotes the n u m b e r of distinct prime factors) = O (In x), and can be a b s o r b e d into the error term. Next, since Z I~r E x/p~xl-1/~xln-3x, we only consider those p>xl/v
p>x1/v
squarefree a's (a = p + h, p < x) counted in S. If f 2 ( a ) > r + 2 (t2 denotes the total number of prime factors), say, a = P i ' " P t , t>r+2, with x l / V < P l < ' " < P , - 2 < x ~ / U , xl/~ < p ~ - ~ < p , < p , + ~ < ' ' ' < p t < x / ( p l . . . p t _ ~ ) , then clearly p,_~ < Pr < Xl/a" SO a must be counted in S~ for at least 2 times and is then subtracted from S. If f2(a) = r + 1, say, a = p~ ...p~+~ with x 1/~ < pl < . . . < P~-2 < x~/~, x~/~ < P,-~ < p~ < p~+~ < x/(p~ . . . p,), then it is not difficult to know that such a must be counted in $1 for at least 2 times, or in b o t h S~ and S 2 separately. Hence a is subtracted from S too. N o w the remaining a's are those with r - 2 < f2 (a) < r. But obviously I{P + h : p < x , p + h -- Pl'" "Pr-2}l '~ X2(r-2)/u ~ X l n - a x .
Y. x l / u < p j < . . . < p r _ 2 < 2~l/u
So we get (9). 2. A 1 0 w e r b 0 u n d f 0 r S. To estimate S from below, we apply L e m m a 1 with X - co ( P i ' " ' Pr-2) li x, D = x 4 / 7 - ~ / ( p i . . . Pr-2), z = x TM, and Pl"" "Pr-2 V(z) --
H PIPPl""P~-
(1 - o 2 ( p ) / p ) . 2 (z)
F r o m co(p) = p/(p - 1), p X h , it follows that c(co) = 2Ch and V(z)>
H
(l--l/(p--l))----2che-Tctln-lx(l+0(ln-lx)).
piP(z)
M o r e o v e r by the continuity of f, since P l ' " ' P r - 2 < x~-2)/~ < x,~ (e 1 denotes a very small positive number), f (s) = f (In (x 4/7 - 8 / ( p l . . . Pr- 2))/In (xi/~)) = (1 + o (1)) f (4 ~/7). Thus by (2) (3) and (8) (assume 7 < ~ < 10.5), (10)
S>(I+o(I))7ChXln-2x
1 n(4~/7--1)+
4~/7 - 1 d x x - : i l n ( y ! y
-;!
where 2:=
~E
xl,o~, ......
1 pr_2~xl,o ( p l -
1)..
(pr-2 -
11'
1) "~
dy) Z - R,
458
J. K A N
ARCH. MATH.
and R denotes the complex remainder term (I1)
Z
~
2 j ( P i ' ' "P,-2 m)
~2
X llv < P l < "'" < P r - 2
9 (~z(x; P i ' " "Pr-2 m, -- h) - lix/q~(pi'"" pr_2 m)). By arguments of elementary combinatorics and the Prime N u m b e r Theorem, S = (1 + o(1)) (12)
2
1/
(r - 2)! - O
~ \ x l / V < p
(
2
xilv < p
l/p
2)}
-- (1 + o (1)) {ln r-2 (v/u)/(r - 2)! - O (1)} = (1 + o (1))in "-2 (v/u)/(r - 2)!
To estimate R, it should be noticed that, in the multiple sum (11), a fixed q = P ~ ' " P ~ - 2 m may be repeatedly counted, i.e. counted for more than one time. This is because, among all the prime factors of q, we m a y take r - 2 of them to be Pt," "', P~-2, while q/(p~ "" "Pr-2) to be m; and there may be more than one way for the suitable choice (i.e. satisfying all the conditions of summation in (11)). But the number of ways for the choice is at most
r-2
~
r-2
~ln-
xprovidedq
repeatedly counted for at most 0 (ln r - 2 x) times. So by L e m m a 2 with A > r + 1, (13)
R ~ In " - 2 x . x l n - A x ~ x l n - 3 x .
F r o m (10) (12) (13), it follows that (14)
S ~ (1 § o(1)) 7 C h X l n - 2 x (1 4~/~-ldx~-'lln(y~-l) 9 n ( 4 c ~ / 7 - 1) + ! ~ !
3. U p p e r
bound X =
d
y)
l n r - 2 ( v / u ) / ( r - 2)!
o f S1. In L e m m a 1 let
o)(Pi'"Pr-2P)
Pi'"Pr-zP
lix,
D
=
xg/7-e/(pl'"
"Pr-2P), z = x 1/~,
and
V(z) =
[I
(1 - l/(p - 1))9
P l P p l . . " P r _ 2 (Z)
Since x 1Iv < Pa <
"'"
< Pr-2
<
xi/U, we have by (3), r--2
V(z)
=
II
(1 -
piP(z)
1/0, -
1)) I 1
(1 -
1 / 0 , , - 1)) -1
i=1
= (1 + o(1)) 2 c h e - ~ l n - i x ( 1
+ O(ln-1 x)),
and by the continuity of F, F (s) = f (ln (x 4/7-~/(pl... Pr- 2 p))/ln (xl/~)) = (1 § o (1)) F (4 ct/7 - ~ In p/ln x).
Vol. 56, 1991
On the sequence p + h
459
Therefore by (1) we have S 1 ~ (1 -~- o(1))
(15) Z
1 ~. ~'/-
2ChXl n - 2 x
gl/~ <:p
e -~ a
where RI=
Z
E
E
xl/V
Y~
2~ (Pl " " " P,- 2 pm) 9 (Tr (x; P l " ' " P , - 2 pro, - h)
m
- li x/~o (pl " "p,_ 2 pm)). By an a r g u m e n t similar to that for R, and by L e m m a 2 with A > r + 2, (16)
R 1 ~ In r-1 x . x l n - A x ~ x l n - 3 x.
By (5), (7) and the Prime N u m b e r Theorem, for a with 7 -< a -< 10.5, Y~
F (4 a/7 - a In p/ln x)/p
xl/u ~p
2e ~ ( 1 E 4 ~/7 ~1/. =
+ xl/.<=p
I2
d
f xl/3
= (1 + o (1)) (7/2) e~/~ { S dt \~,/. t In t (1 - 7/4 In t/In x)
~,/7-3/~ dt 4.,/7-1-~lnt/lnx In (y--1)dv'~ + xl/~ S t l n t ( 1 - 7/41nt/lnx) y 2 V
(17)
(1 + O (1)) (7/2)
eT/0~
,.flEa dy 4/7 - 3/, dy 4~/7 -1 -,y In (x - 1) d f ] \ ~ y ( 1 - 7 7 / 4 y ) + 1/, ~ y(1 - - 7 / 4 y ) ~2 -x
= (1 + o (1)) (7/2) e~/a (ln (4 ct/7 -- 1) + In (7/5) \
4,/7-1 +
!
~dt t(~-7/4t)
t~l ln(x - 1 ) d x ) 2
x-
"
F r o m (15), (16), (17) and (12), it follows that /
$1 =< (1 + o ( l ) ) 7 C h X l n - Z x ( l n
(4a/7 -- 1) + ln(7/5)
\
(18)
+
4,/7--1 adt t-1 ln(x -- 1 ) d x ) ! t(ct- 7/40 2 x9 in , - 2 (v/u)/(r - 2)!
460
I KAN
4. U p p e r b o u n d o f
ARCH. MATH.
Consider the sets
S 2.
g = {e: e = Pl "" "Pr, xl/V < Pl < "'" < Pr-2 < %l/u, X1/g ~ Pr-1 < xl/3
<=p, < (x/pr_l) 1/2} and s
= {/:/= ep-
h, ep < x , e ~ g } .
Clearly ]g] ~ x z/3+(~-2)/" ~ x z/3+`~ and e > x I/'+x/3, e e g. So that 1{I: l e 5f, 1 < xl/~'+~/3}] ~ x 2/3+el . Moreover, $2 = I ( P : P = ePr+l - h, ep,+l < x, e e C , p, < p~+~ < x / ( p , _ lp~)}l < ]{P: P = epr+~ - h, eP~+l < x, e e g}l = the number of primes in 5e. Thus (19)
$2 < S ( ~ ; ~ , x
~/3+~/~) + 0 ( x 2 / 3 + ~ ) , ~ = { p : p X h } .
To estimate S (5r ~ , x ~/~+ ~/3), we apply L e m m a 3 with X = Z li (x/e), 09 (d) = d/~o (d), ee~
#(d)~=0, ( d , h ) = l , D = x i/2-~, and z = x ~/~+~/3. Since F is continuous and (1/2)/(1/~ + 1/3) < 3/2, from (5) we have F(s) = (1 + o ( 1 ) ) 4 e r ( l / ~ + 1/3). Therefore by L e m m a 3 with e (co) = 2 Cn, S(5r ~ , x 1/~+ l/3) < (1 + O ( J ) ) 8 c h X l n - 1 X + R D
(20) < (1 + O ( I ) ) 8 c h X l n - ~ x
+ R 2 "q- R3,
where X = Z li(x/e),
and R D = a
d
alP(z)
I] { l : l e S f ' d ] l ) l - ~ - ~ e ~ 1g = d
li (x/e)
alP(z)
=
diP(z)
=<
1
5Z d
Z Z l---e~li(x/e) ep_-
Y.
Z
d
Z1 ~e
1
qg(d)
es'~28li
(X/e)
Z d
=R2+R3,
ep=-h(d)
say.
(d, h) = 1
1 (p (d)
Z ~ (e,d) > 1
li(x/e)
Vol. 56, 1991 Since
On the sequence p + h
X 1/~+1/3 ( e
Xr/(r+l), e e ~, it follows that
<
52
R2=
461
d<=D,(d,h) = i
xl/~+
5-'.
1/3
f(a)(
< 52
ap=x, ap~h(d)
1-li(x/e)fip(d))
(a, d) = 1
wheref(a)=
Z
1 ~l. HencebyLemma4withA=3,
e=a, e e 8
(21)
R 2 ~ X In - 3 X.
N o w the estimation o f R 3. Note that for squarefree q, d (q) = 2 ~(q) (d denotes the divisor function, v denotes the number of distinct prime factors), q~(q) > q/d (q). Therefore
R 3 ~ Y'. d(q)/q q<--D
•
x/(eln(x/e))
eEg,(e,q)> 1
~xln-lx
3-'. d(q)/q q< D
(22)
=xln-lx
Y'. d(q)/q qND
~x
2 Y,
1/m
m[q,m>xl/v
Z d(q)/q q~-D
1/a
a < x ~/(~+ 1),(a,q)> xl/V
~,
52
1/b
b
1/m ~ x 1-1/v 52 d2(qffq
mlq, m>xUv
q<--D
x i-i/v (lnD) 22 ~ x 1-1/v ln4x ~ x ln-3 x. It remains to calculate X. By the Prime N u m b e r Theorem and by Stieltjes' integration,
+o(l))xln-lx
(23)
dt," 99dtl
1/u 1/3 (1 - t r - 1)/2
11. 1/u
X=(l
~ I "'" I I 1/v ti tr-3 i/~
I 1/3
ti" "" tr(l -- tl . . . . .
dy dx
1/3 (1 --x)/2
= (1 + o(1))(xln-ixln~-Z(v/u)/(r - 2)!) ~ i/~
tr)
1/3 x y ( 1 - x - y ) "
F r o m (19), (20), (21), (22) and (23), 1/3 ( 1 - x ) / 2
(24)
$2<(1 + ~
2) !) I 1/~
1[3
dy dx x y ( l - x - y)
5. C o m p 1 e t i o n o f t h e p r o o f. Bringing (9), (14), (18), (24) together, we obtain (25)
[{p:p_-< x,p + h = Pl'"P,-1
or Pl""P.
Pr > "'" > Pi > e x p ( l n ~ x ) } l
> (1 + o(1))K(a)ChXln-2xln*-2(v/u)/(r-- 2)!,
r > 3,
where (assume 7 < e < 10.5)
K(a)=3.51n((4c~/7-1)(5/7))+ 7
4,/v- 1 dx x-1 ln(y - 1) dy
!
~
-
-
3.5
,,~/7-1
~dt
' lln(x-l~)dx-4
(~-7/4t) t 2
x
y
2
1/3 (1 - x)/2
~
~
1/~
1/3
dy dx
xy(l-x-y)"
462
J. K A N
ARCH. MATH.
Note that 4e/7 - 1
I3
edt t-ft l n ( x - 1) dx (c~- 7/4 t) t J2 x
4~17-1 1 + t1) dt t~l l n ( x ! 4 a/7 - t 2 x
1) dx,
and by changing the order of integration, x i * ln(yy_- 1) dy 4~/7-1!_xzdX
= !4~/7-2
l n ( y y - 1) dy 4~/7 y +-l x1 5-dx-
4,/7-z l n ( y - 1 ) l n ( z 4a/7 - 1
3
dt
tzXln(yy_l) dy= t
4 e/7 -
4a/7-1)y+~ dy,
y 4a/7- 2 in
(y - 1)
J2
y
4e/7 - 1
dt
r+~S 4e/7-t
dy
4~/}-2 l n ( y - 1) ln(4 a/7 - 1 - y) J 2
dy.
y
So it is easy to deduce (for ~ with 7 < e < 10.5) K(e) = 3.5 (In ((4 e/7 -- 1)(5/7)) 4~/7--2
(26)
-
I
In (y - 1)ln((y + 1)(1 - y/(4~/7 - 1)))/ydy)
2
4 ~ In (2 - 3/0 dt. t-I Set e = 9.35 in (26). Numerical calculation by the computer shows K(9.35) > 1.94. Finally, recalling v = (lnx) 1-~, 0 < 6 < 1, u = In In x, we get ]{p:p < x,p + h = Pl""Pr-1
or P l " " P , , Pr > " " > Pl > exp(lnax)}[
>1.94(1--~)r-2/(r--2)!ChXln-2x(lnlnx)
"-2,
r > 3.
This completes the proof of T h e o r e m 1.
Theorem2. ]{p'p < x , p + h = p ~ ' " p , _ a or p l ' " p ~ , pl < ... < p,}] > (1.94/(r -- 2)[) ChXln -2 X (ln lnx) r-2,
r --> 3.
P r o o f. T h e o r e m 2 will follow at once if we let 5 ~ 0 + in T h e o r e m 1.
Vol. 56, 1991
On the sequence p + h
463
Theorem 3.
[ { p : p ~ x , p + h = Pl or PlP2,Pl < P2}[ > 1"94Ch x l n - 2 x " P r o o f. The procedure of this proof is similar to that for Theorem 1. So we only sketch it. Let ~r = { p + h: p __ 3, (27)
] { p : p =< x , p + h = p~ or P~P2, Pl < P2}[ > S - S,/2 - $2/2 - O(x In -3 x),
where S = S ( ~ ' ; ~ , x~/~),
S1 =
Z
S(dp; ~, xl/~),
xl/~ < p < x l / 3
and S2=
~,
~,
X I / ~ < ~ P I < x l / 3 < = p 2 < ( x / P l ) 1/2
1, p i f h ,
i = 1,2,3.
P=plp2P3-h P2 < P3 < x/(p I P2)
By means of 2., 3., 4. in the proof of Theorem 1, we can get (cf. (14), (18), (24) separately), for 7 _< ~ < 10.5, (28)
S>(l+o(1))7ChXln-Zx
n(4~/7--1)+
!
~-
!
d
$1 < (1 + o(1)) 7ChX l n - 2 x (In (4~/7 -- 1) + ln(7/5) \
(29)
+
4,/7-1 ~ dt ! t(~ - 7/4t) t ~ l l n ( x -x- - 1 )
and
(30)
s2 _-<(1 + o(1))8chxln-2x
1/3 (1-x)/2 1/~
1/3
dydx xy(l-x-y)"
Combining (27), (28), (29), (30) leads to (cf. (25), (26)) [ { p : p < x , p + h = Pl or PlP2, Pl < P2}] >=(1 + O(1))K(~)ChXln-Zx. Since K(9.35) > 1.94, the proof of Theorem 3 is completed. Finally, the Main Theorem stated in the introduction follows immediately from Theorem 2 and Theorem 3. N o t e. By a more complex weighted sieve and more calculation the coefficient 1.94 may be improved. References [1] E. BOMBIERI, J. B. FRIEDLANDERand H. [WANIEC, Primes in arithmetic progressions to large
moduli. Acta Math. 156, 203-251 (1986). [2] J. CrmN, On the representation of a large even integer as the sum of a prime and the product of at most two primes. Sci. Sinica 16, 157-176 (1973).
464
J. KAN
ARCH. MATH.
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