The distribution of square-free numbers of the form $[n^c]$

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

X

IAODONG

C

AO

W

ENGUANG

Z

HAI

The distribution of square-free numbers of the form [n

c

_]

Journal de Théorie des Nombres de Bordeaux, tome

10, n

o

_{2 (1998),}

p. 287-299

<http://www.numdam.org/item?id=JTNB_1998__10_2_287_0>

© Université Bordeaux 1, 1998, tous droits réservés.

L’accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les condi-tions générales d’utilisation (http://www.numdam.org/conditions). Toute uti-lisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

The

distribution

of

_square-free

numbers

of the

form

_[nc]

par XIAODONG CAO et WENGUANG ZHAI

RÉSUMÉ. Nous montrons que pour 1 c

61/36 =

1.6944...,

la suite

_[nc]

_{(n = 1,2,...)}

contient une infinité d’entiers sans facteur

carré ;

cela améliore un résultat antérieur dû à

_Rieger

_qui

obtenait l’infinitude de ces entiers pour 1 c 1.5.

ABSTRACT. It is

_proved

that the _sequence

_[nc]

_(n

=

1, 2, ...)

contains infinite

_{squarefree integers}

whenever 1 c

61/36 =

1.6944..., which _improves

_Rieger’s

earlier _{range 1} c 1.5.

1. INTRODUCTION

A

_positive

_integer

n is called

squarefree

if it is a

_product

of different

primes.

Following

a _paper of Stux

[15],

_Rieger

showed in

[11]

that for all

real c with 1 c

1.5,

the

equation

holds,

which is an immediate _consequence of Deshouillers

[4].

Here

[t]

denotes the fractional

_part

of t and E is a

_positive

constant small

_enough.

It is an _easy exercise to _prove that

for 0 c 1. When 1 c

2,

one still

_expects

(1.2)

to

_hold,

but if c =

2,

Inc]

is

_always

a _square, so that = 0.

It is worth

_remarking

that Stux

_[15]

has shown that tends to

_infinity

for almost all

positive

real 1 c 2

(in

the sense of

Lebegue

measure),

however this result

_provides

no

specific

value of c.

Manuscrit reçu le 26 février 1998.

M ots- clés. Square-free number, exponential sum, exponent pair.

This work is _{supported by} National Natural Science Foundation of China _(Grant N° _19801021).

The aim of this _paper is to further

_improve

_Rieger’s

_range

1 c 1.5

by

the method of

exponential

sums.

Basic

_Proposition.

Let 1 c

2,7

= ~,

and x &#x3E; 1. Then we have

with

where

_0(t)

= t -

_{[t] - 1/2}

and

_p(n)

is the well-known M6bius

_function.

Using

the

_simple

one-dimensional

_{exponent pair,}

we can _prove

imme-diately

from the Basic

_Proposition

that

Corollary.

Let 1 c

1.625,

then

four 6

&#x3E; 0

Combining

Fouvry

and Iwaniec’s new method in

[5]

and Heath-Brown’s new idea in

[6],

we can _prove the

_following

better Theorem .

Theorem. Let c be a real constant such that 1 c

61/36,

then

Notations. W

_g(x)

means that

f (x)

=

0(g(x)),

_{m -} M _means

c1M

m

c2M

for some _{constants cl, c2} &#x3E; 0. We also use notations

L =

log(x), e(x)

=

exp(27rix)

and = B -

[0]

-

1/2.

To

_simplify

_writing

logarithms,

we will assume that all

_parameters

are bounded

by

a _power of

x.

_Throughout

the _paper we allow the constants

implied by

‘O’ or _‘G’ to

depend

on

only arbitrarily

small

_positive

number E and c when it occurs.

2. PROOFS OF BASIC PROPOSITION AND COROLLARY Proof of Basic

_Proposition.

It is well-known that

_(see

_[9])

and

Obviously,

In’]

is

_square-free

if and

_only

n

where

¿From

(2.3), (2.1)

and

_partial

summation we can

_get

the Basic

_Proposition

at once.

The

_proof

of

_Corollary

will need the

_following

two lemmas. Lemma 1 is

well-known

_(see

_[1]),

Lemma 2 is contained in Theorem 18 of Vaaler

_[16].

Lemma 1. Let

_{Ig(m) (x)1 ’"}

YX’--

_for

1 X ~ 2X and m =

1, 2, ....

. Then

where

_{(K, A)}

_{is any}

_exponent

_pair.

Lemma 2.

_Suppose

J &#x3E; 1. There is a

_function

0* (x)

such that

By

Lemma 1 and Lemma 2 we

immediately

obtain

Lemma 3.

_{Let y}

&#x3E;

_{0, X}

&#x3E;

_{1, 0 Q}

_1,

_{g(n) = (n + o,)’Y.}

Then

8c-5

By

Lemma 3 with

_{(r~, A)}

=

( 7, 7 )

and

simple

splitting

_argument

_we have

By

Lemma 3 with

_{(~, A) =}

_(1,

_{3 )}

we have

Now the

_Corollary

follows from

_{(2.6), (2.7),}

_(2.8)

and the Basic

3. SOME LEMMAS

Lemma 4. Let 0 a

b

2a. Let be

holomorphic

on an _open

convex set I

containing

the real line

_segement

[1, b/a~.

Assume also that

(1)

M on

_I,

(2)

is real when x is

real,

(3)

f"(x) -cM

for

some c &#x3E; 0. Let

_f’(b)

=

a,

f’(a) =

{3.

For each

integer

v in the _range

a v

(3,

define

xv

by

f ~(xv)

= _v. Then

For the

_proof

of Lemma

_4,

see Heath-Brown

[6],

Lemma 6.

Lemma 5.

_{Suppose Ai,}

_Bj,

_ai and

_bj

are all

_positive

numbers.

If Q1

and

_Q2

are real with 0

Q2,

then there exists some _q such that

and

This is Lemma 3 of Srinivasan

_[14].

Lemma 6. Let 0 M N

_AM,

and let a"L be

complex

numbers

with 1. Then we have

See Lemma 6 of

_Fouvry

and Iwaniec

_[5].

Lemma 7. Let a, be

given

real numbers with

a(,31 - 1)/32

~

0 and

a ~

N. Let

_1,

0 and

Lemma 7 can be

proved

in the same _way as the

proof

of Theorem 2 of

Baker

_[1].

The idea of the

_proof

is due to Heath-Brown

_[6].

Lemma 8. Under the conditions

_of

Lemma

_7,

_if

we

further

_suppose that

F »

_M,

then

Proof.

This is Theorem 3 of Liu

_[8],

which is

_proved

_{essentially by}

the

_large

sieve

_{inequality developed by}

Bombieri and Ivaniec

_[3].

But the term

in Liu’s result is

_superfluous,

since

then

Proof. By

Lemma 4 and

_partial

summation we

_get

for some cl , c2 &#x3E; 0.

Applying

Lemma 6 to the variable

_n,

we obtain that for some

(h,

1

Now we use Lemma 7 to estimate the above sum, with

M, H,

F/N

in

place

of

_{M, Ml , M2.}

This

_completes

the

_proof

of Lemma 9. D

Lemma 10. Under the conditions

_of

Lemma

_9,

we have

Proof.

Applying

Lemma 8 to estimate the sum in

(3.3),

with

M,

H,

F/N

in

_place

of

_{M, Ml, M2,}

we can obtain the bound

(3.4)

if we notice

4. PROOF OF THEOREM

Lemma 11. We have

where

_{(K, A) is}

_{any exponent}

_pair.

Proof. By

Lemma 2 we

_get

for _any J &#x3E; 0

I c(h) I G 1, bl (d)

«

_{1, S(H, M, N)}

is defined in Lemma 9. D

Write

_lllh(y)

=

e (h(y7 - (y -~ 1)y)) -

1.

_{By partial}

_{summation and}

Lemma

_1,

we have

Now take

_(K,

_{A) =}

_(2/7,4/7)

in

_(4.5)

we have

¿From

(4.4)-(4.6)

we

_get

where in

_S(H,

_M,

_N)

the coefficient of d is

_b(d)

or

_b1

(d).

Now use Lemma 9 to estimate the above sum and then choose a best

J E

_(0,

_+oo)

_by

Lemma

_5,

we obtain the bound

(4.3).

Lemma 12. We have

Proof.

In the

_proof

of Lemma

_11,

_using

Lemma 10 in

_place

of Lemma 9 to

estimate the sum in

(4.8),

we can

_get

Lemma 12. D

Lemma 13. We have

Proof.

In the

_proof

of Lemma

_13,

we will use

M2N

« x’.

Similar to

_(2.6),

_by

Lemma 3 with

_{(~, A) =}

_we

_get

By

Lemma 11 with I we obtain

By

Lemma 12 we have

Lemma 14. We have

take

_By

lemma 2 we have

where H runs

_{through J,}

2 , 2 , - - - ,

and

for some

Since e we have

So it suffices to bound

T1

(H,

M,

N).

Now first

_applying

Lemma 6 to the variable d and then

_using

Lemma 8

Using

the bound

_{M2N « x’,}

one has

Now we note that N

_{x s ,}

_by

_(4.16)

and

_{simple splitting}

_argument,

the

bound

_(4.13)

could be obtained at once. D

Finally,

Combining

Lemma

13,

Lemma

_14,

_{(4.1 )}

and the basic

Proposi-tion

_completes

the

_proof

of Theorem .

Acknowledgement.

The authors would like to thank the referee for his

kind and

_{helpful suggestions.}

REFERENCES

[1] R.C. Baker, The square-free divisor problem. Quart. J. Math. Oxford _Ser.(2) 45 _(1994), no.

179, 269-277.

[2] R.C. _Baker, G. _Harman, J. _Rivat, Primes _of the form [nc]. J. Number Theory 50 _(1995), 261-277.

[3] E. _Bombieri, H. _Iwaniec, On the order

_{of 03B6(1/2}

+ it). Ann. Scula Norm. _Sup. Pisa Cl. _Sci.(4) 13 _(1986), no. _3, 449-472.

[4] J.-M. Deshouillers, Sur la répartition des nombers _[nc] dans les progressions arithmétiques.

C. R. Acad. Sci. Paris Sér. A 277 _(1973), 647-650.

[5] E. Fouvry, H. Iwaniec, Exponential sums with monomials. J. Number Theory 33 _(1989), 311-333.

[6] D.R. Heath-Brown, The

Pjatecki~-0160apiro

prime number theorem. J. Number Theory 16

(1983), 242-266.

[7] M.N. Huxley, M. Nair, Power free values of polynomials III. Proc. London Math. _Soc.(3) 41 _(1980), 66-82.

[8] H.-Q. Liu, On the number of abelian groups of given order. Acta Arith. 64 _(1993), 285-296.

[9] H.L. Montgomery, R.C. Vaughan, The distribution of square-free numbers. in Recent

Progress in Number _Theory vol _1, pp. 247-256, Academic Press, London-New York, 1981.

[10] M. Nair, Power free values of polynomials. Mathematika 23 _(1976), 159-183.

[11] G.J. _Rieger, Remark on a _{paper of} Stux concerning squarefree numbers in non-linear

se-quences. Pacific J. Math. 78 _(1978), 241-242.

[12] F. Roesler, Über die Verteilung der Primzahlen in _Folgen der Form _{[f(n +x)].} Acta Arith. 35 _{(1979), 117-174.}

[13] F. _{Roesler, Squarefree integers} in non-linear sequences. Pacific J. math. 123 _(1986), 223-225.

[14] B.R. _Srinivasan, Lattice problems of many-dimensional hyperboloids II. Acta Arith. 37

(1963), 173-204.

[15] I.E Stux, Distribution _{of squarefree integers} in non-linear sequences. Pacific J. Math. 59

[16] J.D. Vaaler, Some extremal _problems in Fourier analysis. Bull. Amer. Math. Soc. 12 _(1985), 183-216.

Xiaodong CAO

Beijing Institute of Petrochemical Technology,

Daxing, Beijing 102600 P. R. China

Wenguang ZHAI

Department of Mathematics

Shandong Normal _University

Jinan, Shandong 250014 P.R. China