Other proofs of the Katai and Montgomery{Vaughan result have re ently been obtainedbyGoldston[3℄ and Languas o{Perelli[9℄

A onditional result on Goldba h numbers

in short intervals

by

A. Languas o(Genova)

1. Introdu tion. DeneaGoldba h number (G-number)tobeaneven

number representable as a sum of two primes, and write L = logX. The

rst result on erningthe existen e of G-numbers in short intervals is due

to Linnik [11℄ who proved, assuming the Riemann Hypothesis (RH), that

for any ">0 and X suÆ ientlylarge, the interval [X;X +L 3+"

℄ ontains

aG-number. Linnik'sresultwasimprovedbyKatai[8℄and, independently,

byMontgomery{Vaughan [14℄ who showed that the interval [X;X +CL 2

℄

ontains a G-number forsome onstant C and X suÆ iently large. Other

proofs of the Katai and Montgomery{Vaughan result have re ently been

obtainedbyGoldston[3℄ and Languas o{Perelli[9℄.

The mainaim ofthispaperistostudythedistributionofG-numbersin

short intervals under the assumption of RH and Montgomery's pair orre-

lation onje ture, aform of whi hassertsthat

(1) F(X;T)

1

2

TlogT forX!1

uniformlyforX

"

T X,forevery xed ">0,where

F(X;T)=4 X

0<

1

;

2

T X

i( 1 2)

4+(

1

2 )

2

and

j

,j =1;2, run overthe imaginarypart of the nontrivialzeros of the

Riemannzeta-fun tion (s).

It is easy to seethat

(2) F(X;T)Tlog

2

T

uniformlyinX. Moreover,Montgomery[13℄(seealsoGoldston{Montgomery

1991 Mathemati sSubje tClassi ation: Primary11P32.

[4℄) proved, againunder RH,that

(3) F(X;T)

1

2

TlogX forX!1

holdsuniformlyforX T.

In thefollowingwe willdenotebyMCthehypothesis (1)and byWMC

aweakerformofitwhererepla es. AsaslightgeneralizationofWMC,

we state thefollowing

Hypothesis. Let 2[1;2). For any ">0 the estimate

(4) F(X;T)T(logT)



holds uniformlyfor X

"

T X.

We willdenote thehypothesis above byWMC(). Observe that WMC

=WMC(1) and that, for2,(4) isimpliedby(2). Ourresult is

Theorem. Let  2 [1;2) be xed and 1=(2X)   1=2. Assume RH

and WMC() uniformly in the range 2XT X. Then



\

 



 X

Xn2X

((n) 1)e(n) 



 2

dXL



+min



L 2

(log2) 2

;XL

+2



;

where is the von Mangoldt fun tionand e(x)=exp(2ix).

WeremarkthatthistheoremisananalogueofTheorem3ofLanguas o{

Perelli [10℄. As an appli ation we an obtain the following result on the

distributionof Goldba h numbersinshortintervals.

Corollary. Let2[1;2) and H=CL



, where C >0 is a suÆ iently

large onstant. Assume RH and WMC() uniformly in the range X=H 

T X. Then,for all suÆ iently large X,the interval [X;X+H℄ ontains

a G-number.

We remark that our Corollary an be obtained using the method of

Goldston[3℄.

We also re all that, under RH and MC, Goldston [3℄ proved that the

interval[X;X+CL℄ ontainsaG-numberandthatFriedlander{Goldston[2℄

provedthattheinterval



X;X+ C

(loglogX) 2

logloglogX



ontainsaG-numberassuming

RH together with a strong form of MC and a suitable form of theElliott{

Halberstam onje ture.

Our se ond aim is to prove the following result, whi h may have some

independentinterest,on themean-squareofthesingularseriesoftheGold-

Proposition.Let

S(n)= 8

>

>

<

>

>

: 2

Y

p>2



1

1

(p 1) 2



Y

pjn

p>2



p 1

p 2



if n is even,

0 if n is odd.

Then

X

nX S(n)

2

=2X Y

p>2



1+ 1

(p 1) 3



1

4 L

2

+O(L 5=3

):

The Propositionis a sharper version of Lemma 2 of Goldston [3℄, and

its proof isbased onthe argument inFriedlander{Goldston[2℄.

A knowledgments. We wishto thank Prof. A.Perellifor his en our-

agement. We also thank therefereeforhis positive suggestions andfor the

arefulrevision ofthispaper.

2. Some lemmas. Now we state two lemmas whoseproofsfollow the

linesof Languas o{Perelli [10℄ (seealso Heath-Brown [6℄).

Lemma 1. Writing

(X;T;v)= X

0< T X

i

e iv

we have

F(X;T)= 1

\

1

j(X;T;v)j 2

e 2jvj

dv:

Lemma 2. Let  = (X) and  = (X) be real numbers satisfying

  <   C for some absolute onstants ;C > 0. Let T > U  0.

Then

X

\

X 



 X

0< T y

i 



 2

dyXF(X;T)

and

X

U< T X

i

T 1=2

( max

UuT

F(X;u)) 1=2

:

The next lemma isan analogueof Lemma 2inwhi hweinsert a fa tor

1=%inthe summationon the .

Lemma 3. Let % =1=2+i ,  =(X) and  =(X) bereal numbers

satisfying   <   C for some absolute onstants ;C > 0. Let

X

\

X 





 X

U< T y

i

% 





 2

dyX



F(X;T)

T 2

+

F(X;U)

U 2

+ 1

U 1=2

T

\

U

F(X;u) du

u 5=2



and

X

U< T X

i

%





1

T max

UuT

F(X;u)



1=2

+ T

\

U ( max

Uvu

F(X;v)) 1=2

du

u 3=2

:

Proof. By partialsummation

X

U< T X

i

%

 1

T 



 X

U< T X

i 



 +

T

\

U 



 X

U< u X

i 



 du

u 2

:

These ondestimateabove followsimmediatelyfromthese ondestimatein

Lemma2. Fortherstestimatewehave,bytheCau hy{S hwarzinequality

and Lemma2,

X

\

X 





 X

U< T y

i

% 





 2

dy

 1

T 2

X

\

X 



 X

U< T y

i 



 2

dy+ X

\

X 





 T

\

U 



 X

U< u y

i 



 du

u 2







 2

dy

 X

T 2

(F(X;T)+F(X;U))+ X

\

X



T

\

U 



 X

U< u y

i 



 2

du

u 5=2



T

\

U du

u 3=2



dy

 X

T 2

(F(X;T)+F(X;U))+ 1

U 1=2

T

\

U

X(F(X;u)+F(X;U)) du

u 5=2

X



F(X;T)

T 2

+

F(X;U)

U 2

+ 1

U 1=2

T

\

U

F(X;u) du

u 5=2



:

3. Proof of the Theorem and of the Corollary. Writing Selberg's

integral

J(X;2X;H)= 2X

\

X

j (t+H) (t) Hj 2

dt;

where (t) = P

nt

(n), we get, by Gallagher's lemma (see, e.g., Mont-

(5)



\

 



 X

Xn2X

((n) 1)e(n) 



 2

d

 2



X

\

X 1=(2) 









t+ 1

2



(X)



t X+ 1

2







 2

dt

+J



X;2X 1

2

; 1

2



+ 2X

\

2X 1=(2)

j (2X) (t) (2X t)j 2

dt



+X

2

+

= 2

(I

1 +I

2 +I

3

)+O(X

2

+);

say. Weremarkthatthe termO(X

2

+) in(5)arises fromtheO(1) term

intheestimate P

a<n<b

1=b a+O(1)appliedinthe above integrals.

Using theexpli itformula (seeDavenport [1℄,Ch.17)

(x)=x

X

j jK x

%

% +O



x(logxK) 2

K



+O



logxmin



1;

x

Kkxk



;

wherekxk=min

n2N

jx nj,withK =XL 2

 1=2

,we get, by(5),

(6)



\

 



 X

Xn2X

((n) 1)e(n) 



 2

d 2

(J

1 +J

2 +J

3

)+O(X);

where

J

1

= X

\

X 1=(2) 





 X

0< K

(y+1=(2))

%

X

%

%







 2

dy;

J

2

=

2X 1=(2)

\

X 





 X

0< K

(y+1=(2))

%

y

%

%







 2

dy;

J

3

= 2X

\

2X 1=(2) 





 X

0< K (2X)

%

y

%

% 





 2

dy:

We rst onsider the terms in J

1 +J

2 +J

3

where 0 <  2X, and

showthat they make a ontribution

(7) 

1

X

2 X

\ 



 X

y i





 2

dy:

UsingtheCau hy{S hwarz inequalitywith0<V <W we get







 X

0< U W

%

V

%

% 





 2

= 



 W

\

V X

0< U u

% 1

du 



 2

jW Vj W

\

V 



 X

0< U u

i 



 2

du

u :

Applying this estimate in J

1

;J

2 and J

3

we obtain (7). By Lemma 2 and

WMC() the right hand sideof (7) is  F(X;2X)=

2

 XL



=, whi h,

by(6), makesthe ontributionXL



.

Now we onsider the ontribution from the terms 2X <  K in

J

1 +J

2 +J

3

. We see immediatelythatthis ontributionis

X X

\

X 





 X

2X< K y

i

% 





 2

dy+ X









 X

2X< K X

i

% 





 2

+ 





 X

2X< K (2X)

i

% 





 2



:

By Lemma3and WMC() thersttermisXL



=. The otherterms

on the right ome from J

1 and J

3

. By Lemma 3 and WMC() they on-

tribute  XL 2+

=; alternatively they are bounded by I

1 +I

3

, whi h by

theBrun{Tit hmarshtheoremisL 2

=(

3

(log2) 2

). Theseestimates, om-

binedwith (6), givethe se onderrorterm intheTheorem.

NowweprovetheCorollary. LetH=[CL



℄,whereC 1isa onstant.

Dene

L()= 



 H

X

m=1

e( m) 



 2

= H

X

m= H

a(m)e( m);

wherea(m)=H jmj,

R (n)= X

h+k=n

(h)(k); S()= X

Xn2X

(n)e(n);

T()= X

Xn2X

e(n); E()=S() 2

T() 2

:

We have

X+H

X

n=X H

a(n X)R (n)= 1=2

\

1=2 S()

2

L()e( X)d (8)

= 1=2

\

1=2 T()

2

L()e( X)d

+ 1=2

\

E()L()e( X)d =A+B;

say. It iseasy to prove that

(9) A=

X+H

X

n=X H

a(n X) X

h+k=n 1=H

2

X+O(H 3

):

Now we pro eedto estimate B. Using

T()min(X;1=jj) forjj 1=2

we get

(10)



\

 jT()j

2

d



X 2

 if0<<1=X;

=X+O(1=) if1=X1=2:

Hen e,usingtheidentity

f 2

g 2

=2f(f g) (f g) 2

;

theCau hy{S hwarz inequalityand(10), we have

(11)



\



jE()jd



X



\



jS() T()j 2

d



1=2

+



\



jS() T()j 2

d

provided1=X  1=2.

Sin e

(12) L()min(H

2

;1=jj 2

) forjj1=2;

we have

(13) B H

2 1=H

\

1=H

jE()jd+ 1=2

\

1=H jE()j

d

 2

:

Fromthe Theoremand (11)weget

(14) H

2 1=H

\

1=H

jE()jdH 3=2

XL

=2

:

By partialintegration,theTheorem and(11) we obtain

(15)

1=2

\

1=H jE()j

d

 2

H 3=2

XL

=2

:

Hen e from (13){(15) we have

(16) B H

3=2

XL

=2

;

(17)

X+H

X

n=X H

a(n X)R (n)H 2

X

providedthat C is suÆ ientlylarge. Thusthe Corollaryfollows.

4. Proof of the Proposition. Let

S= Y

p>2



1

1

(p 1) 2



:

We have

X

nX S(n)

2

=4S 2

X

2nX Y

pjn

p>2



p 1

p 2



2

(18)

=4S 2

X

2nX Y

pjn

p>2



1+

2p 3

(p 2) 2



=4S 2

X

nX=2 X

jjn f(j);

where

f(j)= 8

<

:

 2

(j) Y

pjj

2p 3

(p 2) 2

ifj isodd,

0 ifj iseven.

Then, hangingthe orderof summation in(18), we obtain

X

nX S(n)

2

=4S 2

X

jX=2 f(j)



X

2j



(19)

=2S 2

X 1

X

j=1 f(j)

j

2S 2

X X

j>X=2 f(j)

j

2S 2

X

jX=2

f(j) 4S 2

X

jX=2 f(j)P



X

2j



;

whereP(u)=u [u℄ 1=2.

By straightforward omputations weget

2S 2

1

X

j=1 f(j)

j

=2 Y

p>2



1

1

(p 1) 2



2



1+

2p 3

p(p 2) 2



(20)

=2 Y



1+ 1

(p 1) 3



:

Next, we willshow

(21)

X

jU

f(j)= 1

8S 2

log 2

U+BlogU +O(1);

whereB isa onstant. By partialsummation thisimplies

(22)

X

j>U f(j)

j

= 1

4S 2

logU

U +O



1

U



:

The Propositionfollowsfrom (19){(22) togetherwith theestimate

(23)

X

jX

f(j)P(X=j)L 5=3

:

Now we prove (21). Writing

H(s)= 1

X

m=1

f(m)m s

= Y

p>2



1+

2p 3

p s

(p 2) 2



we see that H(s) is an analyti fun tion for Res =  > 0 and, using the

Perronformulawith errorterm(see,e.g., Lemma3.12 ofTit hmarsh[15℄),

we obtain

X

jU

f(j)= 1

2i +iZ

\

iZ H(s)

U s

s

ds+O



U

1

X

m=1

jf(m)j

m

(1+Zjlog (U=m)j)



;

where">0 isaxed onstant, ="+1=logU <1=4 andZ willbe hosen

lateron.

The errorterm anbe estimatedusing

f(m) d(m)

m X

hjm d

2

(h)

h

(see (30)of Goldston[3℄), whered(m) is thedivisor fun tion,and the las-

si alestimates P

vm d(v)

q

m(logm) 2

q

1

(see,e.g., Theorem5.3of Hua

[7℄) and d(m)  m

"

(see, e.g., Theorem 315 of Hardy{Wright [5℄). So we

have

(24)

X

jU

f(j)= 1

2i +iZ

\

iZ H(s)

U s

s

ds+O



U

Z



:

Now we observe that

H(s)=



1 1

2 s+1



2

(s+1) 2

g(s);

where

g(s)= Y



1 1

p s+1



2



1+

2p 3

p s

(p 2) 2



;

whi h onverges absolutelyand isanalyti for > 1=2. So H(s)U s

=s has

a triplepoleat s=0 withresidue

1

8S 2

log 2

U +BlogU+O(1):

Consider a re tangular ontour with right side s = +it, t 2 [ Z ;Z℄,

and left side 1=4+it, t 2 [ Z ;Z℄. The ontribution of the top, bottom

and left sides of the ontour an be estimated using (+it)  t 1=6

for

1=2 (see,e.g., Tit hmarsh [15℄,p.115). Hen ewehave

X

jU

f(j)= 1

8S 2

log 2

U +BlogU+O(Z 2=3

U

+Z 1=3

U 1=4

)+O(1):

ChoosingZ =U 3

,we obtain(21).

Now we prove (23). Forj odd we have

f(j)= 2

(j) Y

pjj



2

p 2



p 3=2

p 2



=

 2

(j)d(j)

'

2 (j)

Y

pjj



1+ 1

2(p 2)



=

 2

(j)d(j)

'

2 (j)

X

Æjj

 2

(Æ)

2

!(Æ)

'

2 (Æ)

;

where!(n) is thenumberof distin tprimefa tors of n, '

2

(p)=p 2 and

'

2

isextendedto square-free numbersbymultipli ativity.

Hen e, inter hangingtheorder ofsummation,weobtain

X

jX f(j)P



X

j



= X

ÆX

(Æ;2)=1

 2

(Æ)d(Æ)

2

!(Æ)

('

2 (Æ))

2



X

kX=Æ

(k;2)=1

 2

(k)d(k)

'

2 (k)

P



X=Æ

k



:

Usingtheargumentin(2.9){(2.13)ofFriedlander{Goldston[2℄,wendthat

theinnersum an be estimated by

X

nX d(n)

n

P(X=n);

whi h is  L 5=3

by the remark at the end of Se tion 2 of [2℄. Using this

estimate we obtain

X

jX f(j)P



X

j



L 5=3

X

ÆX

(Æ;2)=1

 2

(Æ)d(Æ)

('

2 (Æ))

2

and hen e(23) follows fromthe onvergen e ofthe series

1

X

Æ=1

 2

(Æ)d(Æ)

('

2 (Æ))

2 :

Referen es

[1℄ H.Davenport,Multipli ative NumberTheory,2nded.,Springer,1980.

[2℄ J.B.Friedlanderand D.A. Goldston,Some singularseriesaverages andthe

distribution of Goldba h numbers in short intervals, Illinois J. Math. 39 (1995),

158{180.

[3℄ D.A.Goldston,Linnik'stheorem onGoldba h numbersinshort intervals,Glas-

gowMath.J.32(1990),285{297.

[4℄ D.A.GoldstonandH.L.Montgomery,Pair orrelationofzerosandprimesin

short intervals,in: Analyti NumberTheoryand Dioph.Probl., A.C.Adolphson

etal.(eds.),Birkhauser,1987,183{203.

[5℄ G.H.HardyandE.M.Wright,AnIntrodu tiontotheTheory ofNumbers,5th

ed.,ClarendonPress, 1990.

[6℄ D. R. Heath-Brown,Gaps between primes, andthe pair orrelationof zeros of

thezeta-fun tion,A taArith.41(1982),85{99.

[7℄ L.K.Hua,Introdu tion toNumberTheory,Springer,1982.

[8℄ I. Katai, A remark on a paper of Yu. V.Linnik, Magyar Tud. Akad. Mat.Fiz.

Oszt.Kozl.17(1967),99{100(inHungarian).

[9℄ A.Languas oandA.Perelli,OnLinnik'stheoremonGoldba hnumbersinshort

intervalsandrelated problems,Ann.Inst.Fourier(Grenoble)44(1994),307{322.

[10℄ |,|,Apair orrelationhypothesisandtheex eptionalsetinGoldba h'sproblem,

Mathematika43(1996),349{361.

[11℄ Yu.V.Linnik,Some onditionaltheorems on erningthebinaryGoldba hproblem,

Izv.Akad.NaukSSSRSer.Mat.16(1952),503{520(inRussian).

[12℄ H. L. Montgomery, Topi s in Multipli ative Number Theory, Le ture Notes in

Math.227, Springer,1971.

[13℄ |,Thepair orrelationofzerosofthezetafun tion,in:Pro .Sympos.PureMath.

24,Amer.Math.So .,1973,181{193.

[14℄ H.L.MontgomeryandR.C.Vaughan,Theex eptionalsetinGoldba h'sprob-

lem,A taArith.27(1975),353{370.

[15℄ E.C.Tit hmarsh,TheTheoryofRiemannZeta-Fun tion,2nded.,OxfordUniv.

Press,1986.

DipartimentodiMatemati a

UniversitadiGenova

ViaDode aneso35

16146Genova,Italy

E-mail:languas odima.unige.it

Re eivedon30.4.1996

andinrevisedformon8.9.1997 (2973)