Rend. Sem. Mat. Univ. Poi. Torino
Voi. 53, 3 (1995)
Number Theory

G. Coppola - M.B.S. Laporta
ON THE REPRESENTATION OF EVEN INTEGERS
AS SUM OF TWO ALMOST EQUAL PRIMES

Abstract. In this paper we generalize the Chudakov - van der Corput - Estermann
Theorem on the exceptional set in the binary Goldbach problem to a result on thesame problem with "almost equal" primes. Actually, we prove that the equation
Pi +P2 = 2n is satisfied by almost ali 2ra € [N, 2N] when the primes pi and p-2 lie
in the interval [n — U, n + U], with U = n5^8+£. Furthermore, we explicitly estimate
the number of representations of these 2n as sums of such primes.
1. Introduction
In the binary Goldbach problem, from the Chudakov-van der Corput- Estermanntheorem [11, ch.3] it follows that the number E(N) of even numbers n not exceeding N
for which n is not the sum of two primes satisfies E(N) <C NL~A, where L — log N and
A > 0 is an absolute Constant.
Vinogradov's three primes theorem [12] states that every sufficiently large odd
integer N can be represented as N = pi + p2 -f P3, with primes p,-, i = 1,2,3. Several
authors proved that this theorem stili holds under the further restriction \pi — Af/3| <C N9+£(6 < 1), i = 1,2,3 (see [2], [8], [1], [9], where respectively 6 = 63/64, 160/183, 2/3,91/96).
Assuming the Generalized Riemann Hypothesis (GRH), Wolke [13] established thesame result with \p{ - N/S\ < N^2(\ogN)7+£, i = 1,2,3.
Subsequently, improving the previous unconditional results, Zhan Tao [14] obtainedthe theorem for 6 = 5/8. Moreover, his method also yields an asymptotic formula for thenumber R(N) of representations. Using the circle method, the problem in [14] reduces
to obtaining nontrivial estimates for exponential sums over primes in short intervals. In
some cases, the estimates depend on density estimates in short intervals. In other cases,
Heath-Brown's identity and some analytic techniques based on mean values of the Dirichlet
L-functions are applied.

246

G. Coppola - M.B.S. Laporta

In this paper, using Zhan Tao's result, we show how one could obtain an analogous
result for the corresponding binary problem(1.1)

2n = p i + p 2 ,n-U <pi,p2 <n + U.

More preci sei y, let
Ì2(2n) = i2(2n,[7)=^' A{h)A(k),h+k=2nn-U<h,k<n + U

1'p-l\frf\P\np>2

^r/>

2\U)

/

where A is von Mangoldt's function. Then we have
THEOREM

1. Lets>0and A> 0 be arbitrary constants and let AT5/8+£ <U<N. Then]T| R(2n) - 2Ué(2n) \2 <£,A

NU2L~A.

N<2n<2NTheorem 1 implies
Let E > 0 and A > 0 be arbitrary constants . Thenfarali
2?7, E [JV, 2JV] but 0(NL~A) exceptions, the equation (LI) is solvable for U — n5/8+s
and we have
COROLLARY.

R(2n) = 2Ue(2n) +0(UL~A).

We remark that further refìnements of Zhan Tao's result have been obtained by Jia
P]-[5], with U = 777/12+£ in [5]. However, he uses sieve methods, which lead only to alower bound for the number of representations. Jia deduees his result by establishing thatfor an even integer n such that 2P < n < 2P + U with 0(U log - " P) exceptions, one has(1.2) |{(pi,p 2 ) : n = pi+p 2 , P < Pi < P+U, P'-U < p2 < P+/7}| > 6 ( 7 2 ) — ^ - ,log" P
where P is a.suffìciently large number, e is a suffìciently small positive number, andU = p 7 / 1 2 + 4 0 e . Recently, Mikawa [7] has independently established the same result of [5]without appealing to a theorem of the form (1.2). Then, adopting the approach of Mikawa,the second author [6] has obtained the following geneialization of the results concerning
the binary problem.
Let us consider the linear equation(1.3)

n = Api + Bp2ìn/2 - U < ApuBp2< n/2 + Ut247

On the representation of even integers as a sum of two almost equal primes

where A and B are any positive integers such that (A, JB) — 1 and n E A — {n : (AB, n) • =1, n = A + B( mod 2)}. Then we have
T H E O R E M 2.

Let e and E be arbiirary positive constants, and N be a sufficienti}'
large integer. Then far every n E AN — [N} 2N] fi ^4 with 0(N log - j B N) exceptions,the equation (1.3) is solvable far U = n7'1-2**, far 1 < A, B < logc N and (A, B) = 1,
where e = c(E) > 0.
2. Notation
A - an arbitrary positive Constant,
B,C,D- positive constants depending on A,e - an arbitrarily small positive Constant,
V,Y - positive integers such that N7/l2+£< Y < V,N,U - positive integers such that N5/8+s<U<N,H = UL~B:
NAll'T~

N-1U2L~D.

LJ =P = LC,ni,n2, ...,IIT- integers such that N/2 < ni < 112 < ... < UT < N,S(a;V,Y)=^

A(h)e(ha),V-Y<h<V

Sn(a) = S(a; n + U, 2*7),

Tn(q) =^e(hq).

n-U<h<n+UfamA= U Ù J*.°> andq<p( L°-u*m«=11 ,U)

L°\1LO\m.

(a,q) = l<?

*X> »=i£•a=l1i f x = Xo,«*= n0 i i ffX^^ XrO ,IWI = sài/»-»i€!
n( a ,<,) = !

r(x) = Yl x(°) e ( " ) ' Gauss'sum,
a=l^ '248

G. Coppola - M.B.S. Laporta*l>(x>x) = X]A(w)x(")n<.x

3. Outline of the method
We have that
Sn(a)2e(-2na)da.

R(2n) = /Jo
Then we have|fl(2n)-2t/6(2n)| 2 <E + E>

E
N<2n<2N

ITI

Qft

whereJJ =mY]/

»rvo_^01r

itti

N<2n<2N]£=]£/

QH

N<2n<2NJ

Sn((x)2e(—2na)d(x

5 n (a) 2 e(-2no;)da-2C/6(272)^In order to prove Theorem 1 it suffices to show thatY,<NU2L~A,m(3.1)^2 < NU2L-A(3.2)

m4. The major arcs
Let a e I?ia- Then we haveSn(a)=J2Mh)e(ha) =n-U<h<n+UA(fc)e ( - ) e(ft,) + 0(L 2 ) =E(/,.,<,) = !

aòEe(T)
fc=l\EA(/»KM + O(£2)

1 / n-U<h<n+U
h = b ( raod g)

4r E

E

A(fc)X(A)e(ft,) E *(% (7) + °(^) =6=1

On the representation of even integers as a sum of two almost equal primes

249+ 0(L2)^ y £ x ( " M x ) ^ n ( x , » ? ) + ^fn(ri)whereWn(x, »/) =EA ( % ( / i ) e ( M - 6xTn(»?).

n-C/<J»<n + C/

Now we use the following short intervals version of the Siegel-Walfìsz theorem (see[10].). If V 7 / 1 2 + e < Y < V and q<P = Lc, thenfor every % ( mod 5) and Ci > 0.
If £/" > n 5 ./ 8+e , then by partial summation, from the above result and the estimate
T(X) <C <p, we obtainS„(a) = 4 4 T „ ( V ) + 0 ( C / P 1 / 2 i D - c I ) j
uniformly for 77 £ fg, q < P and (a,</) = 1.
Then we write/^jje ( - ^ )6T5n(a)2e(—2na)do! =jf T n fo) 2 e(-2n,)d, + 0 ( [ / P L - 2 ™ ) =UL -D2S^ ( ~ T 0 £ M <-^)dfl + P(UPL-^^) + o(Therefore, we have/Sn(a)2e(-2na)da= 2!76(2n, P) + 0(TX) + 0(T2),

where?<Pr ua=l

Tl=UP3L~2Cl+3D,'V

T2 ='/UL~D+1.

Then, by classical arguments [11, ch.3], we getE| 6(2n, P) - 6(2n) | 2 <N<2n<2N

where>2 e lo=la=lr w /NP~lL\

250

G. Coppola - M.B.S. Laporta

Finally we obtain
Y,=YJ

U2+<Yl\<3{2n,P)-e(2n)\2
N<2n<2NJ2< NU2P~lL2

Sn(a)2e(-2na)da-2Ue(2n)f+Sn(cY)2e{-2na)da-2Ue(2nìP)[+ NU2P6L-4Cl+6D+ NU2L~2D+'2+ 2,4d

NU2L~A,<provided
O A

2D>A+ 2,>-A + 6(C + D).

5. Minor arcs
In this section we prove that (3.2) holds.
Let us consider integers 721,722, ...,77/7; such thatT

N/2 < 72i <n2yv< ... < 72T < N andcU 7 *=1with /,; = [ni — H, ni + H]. For every 72 £ li, we set
Sn(<x) - Sn,(a)= 5+(a) + S " » ,where
if n < n?: >5(0;; 72^ — U, ni ~ n),

if ??, = m,

5+ (a) - { 0,, S(®', n -f C/", 72 — ni),if n»- < 72

•S(a; ìij, + £/", w* — 72), if 72 <%72>
S~{a)=

lf ?2 ir m

< 0,< —S(a; n — U,n — ni),

Then we write
7Jm

Sn(a)2e(—2na)da<if 72; < 72.

On the representation of even integers as a sum of two almost equal primes/Jm/

Jm

Sni(a): e(—2na)da
S~(a)2e(—2na)da/+ Jm+S+(a)2e(—2na)da251+S+(a)S-(at)e(-2na)da+m/ Sni(a)S+(a)e(—2na)da-f / Sni(a)S(a)e(—2na)daJm
Jm
Usirtg the Cauchy-Schwarz inequality and Parseval's identity, we obtain
S+(a)2e(—2na)da/JmS(a)"e(—2na)da

H2L\<<H2L\m/ S+(a)SJm

Sni(a)S+(a)e(-2na)da/Jm

Sn. (a)S~(a)e(—2na)da

E < È Eva«=i neii/^m

UHL\<UHL\Sn(a)2e(-2na)dc

E<2T

Since Nò/8+£<mThen we may write(5.1)<H2L\(a)e(—2ncv)da

2E+NUHL4.

/ Sni{a) e(-2na)da< U < N, from Theorems 2 and 3 of [14], we obtain that, for any B' > 0,Sni{a)= S(a]m + U,2U)<UL-B',Va e m. -Then, by Bessel's inequality and Parseval's identity, we conclude that(5.2)

E m Sn i ( & ) e (—2?m' ) da « /m | S „ > ) | < f o < < [ / ? I - < ' - >242B

1ne liHence (3.2) follows from (5.1) and (5.2) if A + 4 < 5 < 2B' - A. - 2 and Theorem 1 iscompletely proved.
The authors would like to thank Prof. A. Perelli for useful and encouraging
discussions.

252

G: Coppola - M.B.S. Laporta

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[3] JiA C H . - H . , Three primes theorem in a short interval (III).. Sci. Sinica 34 (1991),1039-1056.
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[5] J I A C H . - H . , Three primes theorem in a short interval (VII). Acta Math. Sinica (N.S.) 10 (1994),369-387.
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"[9] C H E N G - D O N G P A N , C H E N G - B I A O PAN , On estimations of trigonometrie sums over primes inshort intervals (II). Sci. China Ser. A 32 (1989), 641-653.
[10] P E R E L L I A., P I N T Z J., SALERNO S., Bombieri's theorem in short intervals //.Invent. Math.
79(1985), 1-9.
[11] VAUGHAN R.C., The Hardy-Littlewood Method. Cambridge U.P. (1981).
[12] VINOGRADOV I.M., Selected Works. Springer Verlag (1985).
[13] W O L K E D., Uber Goldbach-Zerlegungen mit nahezu gleichen Summanden. J. Number Theory
39 (1991),237-244.
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Acta Math. Sinica, New Series, 7, N° 3 (1991), 259-272.

Giovanni COPPOLA
D.I.I.M.A., Facoltà di Ingegneria
Università di Salerno
Via Ponte Don Mei ilio,84084 Fisciano (Sa), Italy.
Maurizio B.S. LAPORTA
Dipartimento di Matematica e Appi. "R. Caccioppoli"
Complesso Universitario di Monte S. Angelo
Via Cinthia,
80126 Napoli, Italy.