On an equation with prime numbers
by
A. Kum hev (Columbia, S.C.) and T. Nedeva (Plovdiv)
1. Introdu tion. B. I. Segal ([13℄, [14℄) was the rst to onsider in
1933 additiveproblemswith non-integerdegrees. He studiedtheinequality
(1) jx
1 +x
2
+:::+x
k
Nj<"
and theequation
(2) [x
1
℄+[x
2
℄+:::+[x
k
℄=N;
where > 1 is not an integer, and proved in both ases that there exists
k
0
( ) su h that the orresponding problem has solutions if k k
0 and
N is suÆ iently large. Later Deshouillers [4℄ and Arkhipov and Zhitkov
[1℄ improved Segal's result on (2). One may also mention the papers of
Deshouillers [5℄ and Gritsenko [7℄, where the equation (2) intwo variables
was onsidered.
In 1952 I. I. Piatetski-Shapiro [12℄ onsidered (1) with x
1
;:::;x
k re-
stri ted to prime numbers. Let H( ) denote the least k su h that the
inequality (1) with xed " > 0 has solutions in prime numbers for every
suÆ ientlylargereal N. Piatetski-Shapiroproved that
limsup
!1
H( )
log
4:
He also proved that H( )5 for1 < <3=2. The theorem of Goldba h{
Vinogradov [16℄ motivates the onje ture that for lose to 1, H( ) 3.
This was proved byD. I. Tolev [15℄. He showed that if1 < <15=14 and
"=N
(1= )(15=14 )
log 9
N thenthe quantity
D(N):=
X
jp
1 +p
2 +p
3 Nj<"
logp
1 logp
2 logp
3
1991 Mathemati sSubje tClassi ation: 11P05,11P32.
Resear hofthe rst-namedauthorsupportedbyBulgarianMinistryofEdu ationand
S ien e,grantMM-430.
is positive for a suÆ ientlylarge N. Re ently Y. C. Cai[3℄ improved the
upperboundfor to 13/12.
In [10℄ Laporta and Tolev onsideredthe orresponding equation of the
type(2). For1< <17=16 theyprovedanasymptoti formulaforthesum
R (N):=
X
[p
1
℄+[p
2
℄+[p
3
℄=N logp
1 logp
2 logp
3 :
In thepresent paperwe improvethe rangeof they obtained.
Theorem 1.Assume that 1< <12=11 and Æ >0 is arbitrary small.
Then for any suÆ iently large integerN we have the asymptoti formula
R (N)= 3
(1+1= )
(3= ) N
3= 1
+O(N 3= 1
exp( (logN) 1=3 Æ
)):
We also improve theresult from [3℄. We obtain an asymptoti formula
forthesumD(N). Sin etheproofis similarto theproofof Theorem1,we
omitit.
Theorem 2.Assume that 1< <11=10 and Æ >0 is arbitrary small.
Thenfor any suÆ ientlylarge real N and "N
(1= )(11=10 )+
for some
>0 we have the asymptoti formula
D(N)=2"
3
(1+1= )
(3= ) N
3= 1
+O("N 3= 1
exp( (logN) 1=3 Æ
)):
Therangeof inbothproblemsdependsontheestimateofanexponen-
tialsum over primes. In [10℄ and [15℄ Vaughan's identityand theexponent
pair(1=2;1=2)areused. WederiveTheorem1fromamorepre iseestimate
of this sum (Lemma 5 below). To prove it we use the identity of Heath-
Brown[8℄,vanderCorput'smethodasdes ribedinChapters2and3 of[6℄
and theestimateof a doubleexponentialsum dueto Kolesnik[9℄.
2. Notation. Sin e for 1 < < 17=16 Theorem 1 is proved in [10℄,
we an assume that 17=16 < 12=11. In this paper > 0 is a xed
small number depending only on ; P = N 1=
; ! = P 1
; p, p
1
;::: are
primes; 2 (0;1); " is an arbitrary smallpositive number, not ne essarily
thesame indi erent appearan es. We use[x℄,fxgand kxkfortheintegral
partofx,fra tionalpartofx andthedistan efrom xtothenearestinteger
respe tively. (n) isvon Mangoldt'sfun tion. Moreover,
e(x)=exp(2ix);
f(x) g(x)meansthat f(x)=O(g(x));
f(x)g(x) meansthatf(x) g(x) f(x);
xX meansthatx runsthrougha subintervalof[X;2X℄;
f(x ;:::;x ) g(x ;:::;x ) meansthat
j1+:::+jn
x j
1
1 :::x
j
n
n f(x
1
;:::;x
n )=
j1+:::+jn
x j
1
1 :::x
j
n
n g(x
1
;:::;x
n
)(1+O())
forall n-tuples(j
1
;:::;j
n
) forwhi hit makessense.
We usesumsof two types,whi hwede ne inthefollowingway:
type Isums:
XX
mM;nL
mnX a
m
F(mn);
type II sums:
XX
mM;nL
mnX a
m b
n
F(mn);
wherethe oeÆ ientssatisfythe onditionsa
m
m
"
,b
n
n
"
.
We de ne
=exp((logN) 1=3 Æ
):
We alsoset
S( )= X
pP
logpe( [p
℄);
R
i
=
\
i S
3
( )e( N)d (i=1;2)
where
1
=( !;!) and
2
=(!;1 !).
3. Some preliminary results
Lemma 1.Let D bea subdomain of the re tangle f(x;y)jXx2X,
Y y 2Yg (X Y) su h that any line parallel to any oordinate axis
interse ts it in O(1) line segments. Let , be real numbers, 6= 0,
+ 6= 1, + 6= 2, and let f(x;y) be a real suÆ iently many times
di erentiable fun tion su h that f(x;y)
Ax
y
throughout D. Setting
N =XY,F =AX
Y
, we have
X
(x;y)2D
e(f(x;y))
(NF)
"
(F 1=3
N 1=2
+NY 1=2
+N 5=6
+NF 1=4
+NF 1=8
X 1=8
+ 2=5
F 1=5
N 9=10
X 2=5
+ 1=4
NX 1=4
):
Proof. This isa version of Theorem1 of [9℄. The proof maybe found
in[11℄.
Lemma 2. Let 3 < U < V < Z < X and suppose that Z 1=2 2 N,
2 2 3
valued fun tionsu h that jF(n)j1. Thenthe sum
X
nX
(n)F(n)
maybede omposed intoO(log 10
X) sums, ea h eitheroftype Iwith L>Z,
or of type II with U <L<V.
Proof. This isLemma 3of [8℄.
Lemma 3.Let x notbe an integer, 2(0;1), H3. Then
e( fxg)= X
jhjH
h
( )e(hx)+O
min
1;
1
Hkxk
where
h ( )=
1 e( )
2i(h+ ) :
Proof. See Lemma12 of[2℄.
In the following lemma we estimate the number N() of quadruples
(h
1
;h
2
;n
1
;n
2
) forwhi h h
1
;h
2
H,n
1
;n
2
N and
j(h
1 + )n
1 (h
2 + )n
2 j:
Lemma 4. Supposethat 6=0, 2(0;1),>0,H3and N islarge.
Then
N() HN 2
+H 3=2
N log (2HN):
Proof. We follow the approa h of D. R. Heath-Brown [8℄. We de ne
thequantity
N(;a;b)=#f(h
1
;h
2
;n
1
;n
2 )jh
1
;h
2
H; (h
1
;h
2
)=a; n
1
;n
2
N;
(n
1
;n
2
)=b; j(h
1 + )n
1 (h
2 + )n
2 jg
whi h we are going to estimate. If h
1
;h
2
H, n
1
;n
2
N and j(h
1 +
)n
1 (h
2 + )n
2
jwe have
n
1
n
2
h
2 +
h
1 +
HN
;
h
2
h
1 h
2 +
h
1 +
1
H
;
hen e
(3)
h
2
h
1
n
1
n
2
1
H +
HN
:
We also have
(4)
n
1
n
h
2 +
h +
1=
HN
:
From (3)and (4), arguingason pp. 256{257 of[8℄, we obtain
N(;a;b)
HN
H
2
N 2
a 2
b 2
+min
H 2
a 2
; N
2
b 2
+ HN
2
a 2
b 2
:
Sin e
N() X
a2H X
b2N
N(;a;b) ;
theproofof thelemmais omplete.
4. The main lemma
Lemma 5.Suppose that X >P 9=10
,H=X 1
and
h
( ) are omplex
numbers su h that j
h
( )j (1+jhj) 1
. Then, uniformly with respe t to
2(!;1 !), we have
T( )= X
jhjH
h ( )
X
nX
(n)e((h+ )n
) X 2 %
for some suÆ ientlysmall %>0, depending only on .
Proof. We use Lemma 2 with F(n) = e((h+ )n
) to redu e the
estimationof T( )to theestimationof thesums
T
i ( )=
X
jhjH
h ( )
X
i
(i=1;2)
where P
1 ,
P
2
are type I and type II sums, respe tively. We hoose the
parameters U,V,Z asfollows:
U =X
2 2+2%
=256; V =4X 1=3
and
Z = 8
<
: [X
(16 16)=3+3%
℄+1=2 if17=16 <14=13,
[X
(13 13)=3+3%
℄+1=2 if14=13 <13=12,
[X
(20 21)=2+5%
℄+1=2 if13=12 <12=11 .
Let us onsiderT
2
( ). We have
(5) T
2
( ) max
!2 jT
(1)
2
()j+(logX) max
2JH jT
(2)
2
( ;J)j
where
T (1)
2
()= X
mM X
nL a
m b
n
e((mn)
);
T (2)
2
( ;J)= X
h ( )
X X
a
m b
n
e((h+ )(mn)
) :
Firstwe estimate T (2)
2
( ;J). We obtain
T (2)
2
( ;J) X
"
J X
mM X
X
(h;n)2I
q
d(h;n)e((h+ )(mn)
)
wherejd(h;n)j1,Q>1 isaparametertobede nedlaterandforq Q,
I
q
=f(h;n)jhJ; nL; 5(q 1)JL
<Q(h+ )n
5qJL
g:
So,usingtheCau hyinequality,we get
jT (2)
2
( ;J)j 2
X
"
MQ
J 2
XX
h
1
;h
2
J
n
1
;n
2
L
jj5JL
=Q
X
mM e(m
)
where=(h
1 + )n
1 (h
2 + )n
2
. Weestimatetheinnermostsumtrivially
ifjjM
,and usingtheexponent pair (13=40;11=20) otherwise. From
Lemma4 we nowobtain
jT (2)
2
( ;J)j 2
X
"
MQ
J 2
(MN(M
)
+ max
M
5JL
=Q (
13=40
M
(9+13 )=40
+ 1
M 1
)N())
X
"
(J 1=2
M 2
LQ+J 13=40
M
(49+13 )=40
L
(80+13 )=40
Q 13=40
+J 1
M 2
L 2
Q+J 7=40
M
(49+13 )=40
L
(40+13 )=40
Q 27=40
):
We hooseQviaLemma2.4of[6℄and the onditionsonJ,M and Limply
(6) max
2JH jT
(2)
2
( ;J)j X
2 %+"
:
Let us nowestimate T (1)
2
(). Using theCau hy inequalityand Lemma
2.5of [6℄ we get
jT (1)
2 ()j
2
X
"
M 2
L 2
Q +
ML
Q X
qQ X
nL
X
mM
e(((n+q)
n
)m
)
where Q L is a positive integer. We apply the exponent pair (13=40;
11=20) to theinnermostsum and hoose Qvia Lemma2.4 of[6℄ to obtain
jT (1)
2 ()j
2
X
"
(M 2
L+ 13=40
M
(49+13 )=40
L
(67+13 )=40
+
13=53
M
(75+13 )=53
L
(93+13 )=53
)
and usingthe onditionson M,L and we get
(7) max
!2 jT
(1)
2
()j X
2 %+"
:
The needed estimateforT ( ) follows from(5){(7).
Let usnow onsiderT
1
( ). We have
(8) T
1
( ) X
"
max
jj2(!;H+1) X
mM
X
nL
e((mn)
)
:
If LX
(57 49)=23+3%
we estimate thesum over n using theexponent
pair (8=41;26=41) to obtain
(9) jT
1
( )j X
2 %+"
:
Otherwise we rst use the Cau hy inequality and Lemma 2.5 of [6℄ to
thesum on theright-handsideof (8)and obtain
jT
1 j
2
X
"
M 2
L 2
Q +
ML
Q X
qJ X
nL X
mM
e(f(m;n;q))
wheref(m;n;q)=((n+q)
n
)m
,J Q=2 andQ Lisaparameter
tobe hosenlater. ThenweapplythePoissonsummationformula(Lemma
3.6of[6℄)tothesumsovermand nsu essivelyandAbel'stransformation:
X
q X
m;n
e(f(m;n;q))
= X
q;n X
2
f(m
;n;q)
m 2
1=2
e(1=8+f(m
;n;q) m
)
+O(MLJF 1=2
+LJlogX)
MF 1=2
X
q;
X
n e(f
1
(;q;n))
+XJF 1=2
+LJlogX
MF 1=2
X
q;
X
2
f
1 (;q;n
)
n 2
1=2
e(1=8+f
1 (;q;n
) n
)
+MF 1=2
JFM 1
(LF 1=2
+logX)+XJF 1=2
+LJlogX
MLF 1
X
q;;
e(g(; ;q))
+F
1=2
JlogX+LJlogX+XJF 1=2
whereF =JM
L 1
,f
1
(;q;n)=f(m
;n;q) m
,
g(; ;q)=f
1 (;q;n
) n
0 (q)
1=(2 2 )
1=2
=(2 2)
F;
0
isa onstant dependingonlyon ,=J=L, FL 1
,FM 1
.
Hen e
X
"
jT
1 j
2
X 2
Q 1
+X 2
F 1
Q 1
X
qJ
X
FM 1
X
FL 1
e(g(; ;q))
(10)
2 1=2 1=2
If X 1=2
L < X
(57 49)=23+3%
we estimate the sum over ; in (10)
usingLemma 1 with X =FM 1
,Y =FL 1
and f(x;y) =g(; ;q). We
get
X
"
jT
1 j
2
X 2
Q 1
+F 1=3
X 3=2
+XF 1=2
L 1=2
+X 7=6
F 2=3
+X 3=2
F 3=5
J 2=5
L 4=5
+XF 3=4
M 1=8
+J 1=4
X 5=4
F 3=4
L 1=2
+X 2
F 1=2
+XL:
Now we substitute the expressionfor F in thelast estimate and hoose Q
via Lemma2.4 of[6℄. We obtain(9).
IfZ L<X 1=2
weinter hangetherolesofand andprove(9)again.
This ompletes theproofof thelemma.
5. Proof of Theorem 1. Itis easy to seethat
R (N)= 1
\
0 S
3
( )e( N)d =R
1 +R
2 :
TheintegralR
1
isstudiedbyLaportaandTolev[10℄,pp. 928{929. They
proved that if1< <17=16 then
R
1
= 3
(1+1= )
(3= ) N
3= 1
+O(
1
N 3= 1
)
but the same argument shows that this asymptoti formula holds for 1 <
<3=2. Hen e thetheorem follows fromtheestimate
(11) R
2
1
P 3
:
It is notdiÆ ultto prove that
R
2
PlogP max
2
2
jS( )j:
To prove (11) itremainsto show that
max
2
2
jS( )j 1
P 2
:
We have
S( )= X
nP
(n)e( n
)e( fn
g)+O(P 1=2
):
So,it issuÆ ient to prove thatforX satisfyingP 9=10
<X P,
S
1 ( )=
X
(n)e( n
)e( fn
g) 1
X 2
:
UsingLemma3 withx=n
and H = X 1
weobtain
S
1 ( )=
X
jhjH
h ( )
X
nX
(n)e((h+ )n
)
+O
logX X
nX min
1;
1
Hkn
k
:
The estimationof theerrorterm intheabove equalityis standard(see [8℄,
pp. 245{246). Hen e (11) followsfrom Lemma5.
The proof of Theorem1is omplete.
A knowledgements. We would like to thank D. I. Tolev for intro-
du ing us into this problem and the regular attention to our work and to
M. B. L.LaportafortellingusaboutCai'spaper[3℄.
Referen es
[1℄ G. I. Arkhipov and A. N. Zhitkov, On the Waring problem with non-integer
degrees,Izv.Akad.NaukSSSR48(1984),1138{1150(inRussian).
[2℄ K.Buriev,Additiveproblemswithprimenumbers,thesis,Mos owUniversity,1989
(inRussian).
[3℄ Y.C.Cai,Onadiophantineinequalityinvolvingprimenumbers,A taMath.Sini a
39(1996),733{742(inChinese).
[4℄ J.M. Deshouillers, Probleme de Waringave exposants nonentiers,Bull.So .
Math.Fran e101(1973),285{295.
[5℄ |,Un problemebinaire entheorieadditive,A taArith.25(1974),393{403.
[6℄ S. W. Graham and G.A. Kolesnik, Van der Corput's Method of Exponential
Sums,LondonMath.So .Le tureNoteSer.126,CambridgeUniv.Press,1991.
[7℄ S. A. Gritsenko, Three additive problems, Izv. Ross. Akad. Nauk 56 (1992),
1198{1216(inRussian).
[8℄ D. R. Heath-Brown, The Pjate ki-
Sapiro prime number theorem, J. Number
Theory16(1983),242{266.
[9℄ G.A.Kolesnik,Onthenumberofabeliangroupsofagivenorder,J.ReineAngew.
Math.329(1981),164{175.
[10℄ M.LaportaandD.I.Tolev,Onanequationwithprimenumbers,Mat.Zametki
57(1995),926{929(inRussian).
[11℄ H.-Q.Liu,On square-fullnumbersinshortintervals,A taMath. Sini a(N.S.)65
(1993),148{164.
[12℄ I.I.Piatetski-Shapiro,OnavariantoftheWaring{Goldba hproblem,Mat.Sb.
30(1952),105{120(inRussian).
[13℄ B.I.Segal,OnatheoremsimilartotheWaringtheorem,Dokl.Akad.NaukSSSR
1(1933),47{49(inRussian).
[14℄ |, The Waring theorem with fra tionaland irrational degrees, Trudy Mat. Inst.
Steklov.5(1933),73{86(inRussian).
[15℄ D.I.Tolev,Onadiophantineinequalityinvolvingprimenumbers,A taArith.61
[16℄ I.M.Vinogradov,Representation ofanoddnumber asthesum ofthree primes,
Dokl.Akad.NaukSSSR15(1937),291{294(inRussian).
DepartmentofMathemati s Hrabretz22, apt.11
UniversityofSouthCarolina Plovdiv4019,Bulgaria
Columbia,SouthCarolina 29208
U.S.A.
E-mail:koumt hemath.s .edu
Re eivedon17.7.1996 (3024)
