J
OURNAL DE
T
HÉORIE DES
N
OMBRES DE
B
ORDEAUX
X
IAODONG
C
AO
W
ENGUANG
Z
HAI
On the distribution of p
αmodulo one
Journal de Théorie des Nombres de Bordeaux, tome
11, n
o2 (1999),
p. 407-423
<http://www.numdam.org/item?id=JTNB_1999__11_2_407_0>
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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
On
the
distribution
modulo
onepar XIAODONG CAO et WENGUANG ZHAI
RÉSUMÉ. Dans cet
article,
on donne une nouvellemajoration
dela
discrépance
D(N):=
de la suite
lorsque
5/3 ~
03B1 3 et03B1 ~
2.ABSTRACT. In this paper, we
give
a newupper-bound
for thediscrepancy
D(N):=
|03A3 p~N {p03B1}~03B3 1201403C0(N)03B3
{p03B1}~03B3
for the sequence when
5/3 ~
03B1 3 and 03B1~
2.1. INTRODUCTION.
In 1940 I. M.
Vinogradov
[14]
considered the distribution of the fractionalparts
of the sequence where p runs over the set ofprime
numbers andf
apositive
constant. This celebrated work motivated the interests of many authors toinvestigate
the distribution ofpet
modulo one for fixed a >0, a V N(see
[1,2,3,4,6,7,9,10]).
It is well-known that the sequence(pet)
isuniformly
distributed modulo one as p runs overprime
values. To make thisprecise,
we consider thediscrepancy
I ,
where denotes the fractional
part
of real t. Leitmann[10]
showed thatManuscrit reçu le 21 septembre 1998.
This work is supported by NSF of China (Grant No. 19801021) and NSF of Shandong
as N tends to
infinity,
for some J > 0. R. C. Baker and G. Kolesnik[1]
showed that(1.2)
holds withfor a >
1,
a ~
N. This is thesharpest
result forlarge
a atpresent.
We can do much better for small a, but one can’t find a unified method as a
varies, just
as Baker and Kolesnikpointed
out in their paper[1]:
"Of course, one can do much better for small a,
although
theoptimal
technique
variesconsiderably
as a moves over an interval such as 1 a 2".By Vaughan’s identity
and mean value estimates for Dirichletpolynomial, Balog
[2]
proved
thatfor
1/2
a 1.Throughout
the paper, e denotes anarbitrarily
smallpositive
constant,
and m - M means that M m 2M. Harman[7]
proved
that(1.3)
holds for a =1/2
independently.
Baker and Kolesnik[1]
also proves that
for a =
3/2
by
Van derCorput’s
method(essentially).
The aim of this paper is to use a new
technique
tostudy
the case5/3
a3, a ~
2. Previous methodsonly yield
weak results in this interval.We shall combine the double
large
sieveinequality
due to Bombieri and Iwaniec[4]
and Heath-Brown’s method[8]
to prove thefollowing
Theorem 1. The estimate
holds where
2. A SPECIAL SPACING PROBLEM
In
1989, Fouvry
and Iwaniec[5]
first used the doublelarge
sieveinequal-ity
to estimateexponential
sums with monomials. If theexponential
factoris
large comparable
to thevariables,then
it oscillates toorapidly
and can’t be controlledby using
the doublelarge
sieveinequality directly.
In orderto overcome this
problem, Fouvry
and Iwaniecappealed
toWeyl’s
method,
which reduces theoscillatory
behaviour of theexponential
factor.Conse-quently
it causes seriousproblems
about thespacing
of theresulting points.
SeeProposition
2 ofFouvry
and Iwaniec[5]
and Theorem 2 of Liu[11].
Sometimes if the
exponential
factor is verylarge,
we have to use theWeyl’s
shift two times. Thus it causes a newspacing problem.
In this section we shall
investigate
thespacing problem
for thepoints
..
In this section all constants
implied by "4",
"« "depend only
on a. -2Let M >
100, Q
M/10,A
> 0. We set T =and G =
log(2MQr).
Thus for m -M, q - Q
and rfixed,
we havet(m, q; r) -
T. We use to denote the number ofquadruples
with
m, iii - M;
q,;¡ rv Q
and rfixed,
such thatOur aim is to prove the
following:
Remark 1. In Theorem
2,
the condition reQ
can bereplaced by
rC(a)Q,
whereC(a)
is a small constantdepending only
on1,
thenthis condiction can be removed.
Remark 2. In
applications,
the contribution of(2.4)
isalways
absorbedby
the contribution of(2.3).
Since(2.3)
isproved only
for reQ,
we state(2.4)
forcompleteness
of the Theorem.Proof.
Clearly
(2.2)
implies
thatnamely,
(2.6)
whereUsing Taylor’s
formula we havefor ,Q ~
0,
Ixl
1/2
Moreover,
where
Using
(2.7)
again
and after asimple calculation,
one hasFrom
(2.11)
we see thatBy
substituting (2.14)
into(2.11)
we find the firstapproximation
Again by substituting
(2.15)
into(2.11)
weget
a moreprecise
We assume that AM
1/4,
for otherwise Theorem 2 follows from(2.16).
If AM
1/4,
we claim from(2.16)
thatSince is
non-decreasing
in we havewhere
A2
= AM +M-’(r2/3
+~2/3 ~ ,
Now we use the method
of Fouvry
and Iwaniec[5 ,p319-321]
to estimateBo (M,
Q, A2;
r~.
For S’ > 0 we have theidentity
For fixed
(q, q-),
the number of latticepoints
counted inBo(M,
Q,
A2; r)
is b oundedby
where S =
ê(4.d2)-I.
Using
ourassumption
and Lemma 4.8 of Titchmarsh[12],
the innermost sum in(2.20)
equals
toThus
by
the mean value theorem we havefor some qo rv
Q,if r
for some smallC(a)
> 0. It should be remarked that if a >1,
then this condition can be removed.We
always
have « This is trivial if!g - $1 >
Q/20
sinceC(q,
q;
r)
«Q~ .
Now suppose!? - g)
(~~20.
For0,
we have
Inserting
this into theexpression
ofC(q,
q;
r)
weget
the desiredinequality.
and if
IlAsil
3sIBIM-2,
thenby
Lemma 4.4 of[12]
we haveCombining
(2.20)-(2.25)
we havewhere
and
= 1, 2)
can be estimated in the same way asFouvry
andIwaniec
[5] (page
320),
and we haveThe estimate
(2.3)
follows from(2.26)-(2.28).
To prove the estimate(2.4),
we needonly
to considerwhere
A 3
=AM +
M-3(r4
+Q4).
Since theproof
is almost the same as3. SOME PRELIMINARY LEMMAS
In this section we
quote
some lemmas needed for ourproof.
We shall usethe work of Heath-Brown
[8]
on thedecomposition
of sumswith 2P. Heath-Brown subdivides the sum
(3.1)
intoexpressions
ofthe form
with X
Xl
2X, Y
2Y,
«P-,
b(y)
« P’ for everyfixed e > 0. The above sum is
usually
called"Type
I" sum ifb( y) -
1 orb(y)
=log
y; Otherwise it is called
"Type
II" sum.Lemma 1. Let
f (n)
(n =
1,2,...)
be acomplexe
valuedficnction.
Let P >2, P1
2P. Let u, v, z bepositive
numberssatisfying
Then the sum
(3.1)
may bedecomposed
into(log
p)6
sums, eachof
whichis either
of Type
I with Y > z orof Type
II with u Y v.Lemma 2. Let Xl, X2, ... , XJ be real and H > 1. Then the
discrepancy of
sequences is
defined by
~
satisfies
Lemma 3. Let L >
K, Q
>0,
and let Zk becompleac
numbers. We then haveLemma 4. Let X and
Y
be twofinite
setsof
realnumbers,
X C[-X, X],
Y
C[-Y, Y].
Thenfor
anycomplexe
functions
u(x)
andv(y)
we haveLemma 5.
1, m - M, q - Q.
Leta(~ 0,1)
be a realnumber,
t(m~ q) - (m
+q)a - (m -
MCt-1Q,
and let the numberof
latticepoints
(m, ml, q, ql )
such that m,ml "JM, q7 q, - Q,
andSuppose
t7 is asufficiently
smallpositive
constant.If Q
the
implied
constantrelying
at most on 77 and a.Lemma 1 and 2 are Lemma 1 and 2 of Baker and Kolesnik
[1]
respectively.
Lemma 3 is Lemma 2 of
Fouvry
and Iwaniec[5].
Lemma 4 is the doublelarge
sieveinequality,
seeProposition
1 ofFouvry
and Iwaniec[5].
Lemma 5is Theorem 2 of Liu
[11].
4. ESTIMATIONS OF EXPONENTIAL SUMS We first estimate the bilinear forms of
type
with MN - X and ocm «
1, b,
Proof.
TakeQ,
=[X28].
By
Cauchy’s inequality
and Lemma 3 weget
for some 1 «
Qi
GQl,
wherewith
It reduces to show that
Take
Q2
= + 10.Using
Lemma 3again gives
for some 1 C
Q*
CQz,
whereand
t(n,
q2;ql)
is definedby
(3.1).
No loss of
generality,
we suppose thatQ*
CQ2.
It suffices for us to show for fixed ql thatIf
0~
»Q*,
we shall estimate the sum over ql.for some 1 C
Q~
GQ3,
whereIt is easy to show that
Q2
=o(N 3/4)
andQ3
=O(M3/4).
Let F =FQ*Q*
hM’N’ and G = It is easy to check that G » 1.
By
Lemma 4we obtain that
where ,A is the number of
quadruples
(m,
m,
q3,4’3’)
such thatfor
M; Q3, f3 "-I Q~,
and where B is the number ofquadruples
(~ ~) 92 ?~)
such thatfor
n, fi - ~V;q2)92 ~
~2.
ThenA, B
can be estimatedby
Lemma 5 andTheorem
2, respectively;
And weget
thatCombining
(4.12)-(4.14)
weget
thatNow it is easy to check that under the conditions of
Proposition
1 wehave
Proposition
2.Suppose
that 2 a3,
0 8 ~339 ,
0 8 ~13s ,
0
e -
1269~ ~
and 08 ~ ~ - 1,
hxe.
Then(,~.2)
holdsfor
Proof.
TakeQl
=[X28]
and Lemma 3gives
us , 1 -,for some 1 C
Q*
CC~1.
Let
Sl
denote thetriple exponential
sum in theright
side of(4.18).
TakeQ2 =
+ 10 and Lemma 3gives
for some 1 «
Q2
«Qz.
Let
82
denote thequadruple
sum in theright
side of(4.19).
TakeQ3
=[lo~~]2
~- 10 and Lemma 3gives
for some 1 «
(Q3
«Q3,
whereNo loss of
generality
we suppose that~1
«Q~;
Otherwise we estimate the sum over qi . Thus we havewhere
x31
denotes the number ofquadruples
such thatfor
0~,
and where~2
is the number ofquadruples
(n,
n,
93,q3 )
such thatCombining
(4.23)-(4.25)
we obtainunder the conditions of
Proposition
2. NowProposition
2 follows from(4.18)-(4.22)
and(4.26).
0Proposition
3.Suppose
Proof.
Weapply
Heath-Brown’s method[8].
Suppose
1Q
HN is aparameter
to bespecified
later. For a fixedq(1
q
Q),
defineThen we have
Cauchy’s inequality gives
where A = and
(*)
denotes a sum overhi , h2 rv H, nt, n2 rv N
such that
Applying
theexponent
pair
(1/14’
we have14
14
- - ,
By
the sameargument
as in Heath-Brown[8],
weget
that-20
Finally
takeQ
= andProposition
3 follows. 05. PROOF OF THEOREM 1
Let
Dp
denote thediscrepancy
of the sequenceip a :
p-Pl.
By
aSuppose
P » LetHo
=[P6],
where 6 isgiven
in Theorem 1.Applying
Lemma2,
we obtainI I
for some
Applying
Lemma 1 withf (n)
=e(hna),
we findI
where
S(h)
is either aType
I sum withor a
Type
II sum withIf
S(h)
is aType
II sum, thenby Proposition
3(take 0
=8)
we haveIt suffices for us to show that for any
S(h)
ofType
I we haveWe prove
(5.5)
by Propositions
1 and 2. Case1. 3
3 - -a 26
26*In this
interval,
we have J -Z6.
We take 0= 26
inProposition
1 and we find that(5.5)
holds for For ~r »p23/39 ,
using
the
exponent
pair
( i4 ,
on the sum over y andestimating
the sum over ~trivially,
weget
Case 2
.
a -
°_
In this
interval,
we have ð =5 40a .
By Proposition
1 we have(5.5)
holdsfor
P2*~
« y «P 8 i5 5 .
We use theexponent
pair
( 1 , 11 )
again
forIn this
interval,
we have I =5J.0.
By Proposition
2 we see that(5.5)
holdsfor
P 2 -
« Y «P ~5 .
For Y »P 75
we use theexponent
pair
(30’ 2s )
and(5.5)
still holds.C 4 317 347
Case4..
°In this
interval,
we have ð =9i33
and(5.5)
can beproved
as the samein Case 3.
In this case, we have j =
5-,, .
By Proposition
2 we know that(5.5)
holds for
p!-6
« Y «P 2 .
ForP 2 «
Y «pl-24~,
(5.5)
also follows fromProposition
2 with X and Yinterchanged.
For Y »pl-246 ,
we usethe
exponent
pair
( 30 , 30 )
again.
Case 6.59
a 3.In this case, we have 5 -
339
and(5.5)
could beproved
as the same inCase 5.
Combining
all theabove,
we see that(5.5)
holds in any cases. Thiscompletes
theproof
of Theorem 1.Acknowledgement.
The authors thank the referee for his valuablesug-gestions.
REFERENCES
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356 (1985), 174-193.
[2] A. Balog, On the fractional part
of p03B8,
Arch. Math. 40 (1983), 434-440.[3] A. Balog, On the distribution of
p03B8 (mod1),
Acta. Math. Hungar. 45 (1985), 179-199[4] E. Bombieri and H. Iwaniec, On the order of
03B6(1/2
+ it), Ann. Scuola Norm. Sup. Pisa. 13(1986), 449-472.
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[6] S.A. Gritsenko, On a problem of I. M. Vinogradov (in Russian), Mat. Zametki 39 (1986), 625-650
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[9] R.M.Kaufman, The distribution of ~p, Mat. Zametki 26 (1979), 497-504
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[11] H.-Q Liu, On the number of abelian groups of a given order(supplement), Acta Arith. 64
(1993), 285-296.
[12] E. C. Titchmarsh, The Theory of Riemann Zeta Function, Oxford, 1951.
[13] R. C. Vaughan, The Hardy-Littlewood Method, Cambridge, 1981.
[14] I. M. Vinogradov, Special Variants of the Method of Trigonometric sums (in Russian), Izda.
Xiaodong CAO
Beijing Institute of Petro-chemical Technology Daxing, Beijing 102600
P. R. China
E-mail: biptiaoQinfo.iuol.cn.nat
Wenguang ZHAI
Department of Mathematics
Shandong Normal University
Jinan, Shandong 250014
P. R. China