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o

### p. 301-313

<http://www.numdam.org/item?id=JTNB_1998__10_2_301_0>

© Université Bordeaux 1, 1998, tous droits réservés.

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

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mean

### integers

par ART016ARAS DUBICKAS

RÉSUMÉ. Soit 03B11 = 03B1, 03B12,..., 03B1d l’ensemble des

d’un

entier

03B1 de

### d,

n’étant pas une racine de l’unité. Dans cet article on propose de minorer

## \$\$Mp(03B1)=p~1 d03A3i=1|log|03B1i||p

### où p&#x3E;

1.

ABSTRACT. Let 03B1 be an

of

### degree

d with

conju-gates 03B11 = 03B1, 03B12, ..., 03B1d. In the paper we

### give

a lower bound for

the mean value

## \$\$Mp(03B1)=p~1 d/d03A3i=1|log|03B1i||p

when 03B1 is not a root of unity and p &#x3E; 1.

1. INTRODUCTION.

Let a be an

number of

d &#x3E; 2 with

as its minimal

### positive. Following Mahler,

the Mahler measure of a is defined

The house of an

### algebraic

number is the maximum of the modulus of its

- -

-Put also

(3)

for

deviation" of

### conjugates

from the unit circle. Denote

### for p &#x3E; 0

Our main concern here is the lower bound for this mean value when a is an

### = 1)

which is not a root of

In

D.H. Lehmer

### 

asked whether it is true that for every

### positive

e there exists an

### algebraic

number a for which 1

1 + c. In its

form Lehmer’s

### problem

has been reformulated as whether it is true

that if a is not a root

then

### M(a) &#x3E;

ao = 1.1762808... where cxo is the root of the

In

C.J.

that if a is a

then

### M(a)

&#x3E; 8 = 1.32471... where 0 is the real root of the

poly-1. This result reduces Lehmer’s

to the case

of

with minimal

the

P.E.

and

used Fourier

to prove that

&#x3E; 1 +

In

C.L. Stewart

that

&#x3E; 1 +

### Although

this result is weaker than the

### previous

one, the method used has become very

### important

and led to further

M.

and

M. Waldschmidt

### 

obtained Stewart’s result via the

deter-minant.

In

E. Dobrowolski

### 

obtained a remarkable

of these

results

### showing

that for each £ &#x3E;

### 0,

there exists an effective

such that

for d &#x3E;

### d(~)

D.C. Cantor and E.G. Straus

### 

in 1982 introduced the

determinant to

Dobrowolski’s

and to

### replace

the constant 1 - ê

2 - ê.

R. Louboutin

was able to

this constant to

E. M.

### 

obtained Louboutin’s result

a version of

### Siegel’s

lemma due to Bombieri and Vaaler.

P. Voutier

showed that

### (1)

holds for all d &#x3E; 2 with the weaker constant

In

### 1965,

A. Schinzel and H. Zassenhaus

### conjectured

that there exists

an absolute

such that

&#x3E; 1

### + ~y/d

whenever a is not a

(4)

we have

where d &#x3E; In

both

and the

### con-jectures

can be considered in terms of the lower bound for

notice that

for

If

=

### 1,

then

Louboutin’s result can be written as follows

### Taking

p =

oo, we can write the

in the

### following

form

The function p --~

is

Hence the

### inequality

cp where 1 p oo and cp &#x3E; 0 lies between the

### conjecture

of Lehmer p = 1 and

form of the

### conjecture

of Schinzel

and Zassenhaus p = oo

also

for a

### problem

which lies between

these two

### conjectures).

We have noticed above that the

### conjectural

value for ci is 2

### log

aa. It would be of interest to find out whether it is true

that

&#x3E; The

holds for the

2. We

### conjecture

that the answer to the above

is

so that

Coo =

### log

2. In this

paper, we take up the

determinant

### )

and fill the gap between

and

### (Theorem 2).

One can also consider the mean value of

of an

### integer

(5)

The lower bound for

where a is a

was consi-dered

I. Schur

C.L.

C.J.

In

M.

solved Favard’s

that

:= maxi,j

lai-aj

### I

&#x3E; 2 - c for an

of a

The author

that

&#x3E;

c. The

of

an upper bound

for

is known as a

In this

we

### apply

the lower bound for

to estimate

from below

### (Theorem 3).

2. STATEMENT OF THE RESULTS.

The notations are the

Let

### G(x)

be a real valued function in

such that

### G(l) =

0. Let also the derivative of

### G(x)

be continuous and

in the interval

Put

Put also for

Let a be a

i.e. d =

a2m =

a2m-1 =

where

1.

### Suppose

also that a is not a root of

With these

### hypotheses,

our main result is the

### following:

Theorem 1. For every e &#x3E; 0 there exists such that we have

whenever d &#x3E;

The constant

### do (e)

and the constants

### d5 (P)

used below are effective.

we

I =

J =

L =

(6)

### Corollary

1. For every &#x3E; 0 there exists

### d1 (c)

such that

whenever d &#x3E;

This

### obviously implies

Louboutin’s result. On the other

### G(x)

= 1 - we have I = 1 -

J =

L =

Hence

We can

in the

above

and so Theorem 1

the

### Corollary

2. For every - &#x3E; 0 there exists

such that

d &#x3E;

we have

where

2

the

### (2),

since T~ = 1. The

### following

theorem fills the gap between

and

### (4).

Theorem 2. Let 1 p oo and e &#x3E; 0. Then there is

such that

d &#x3E;

### d3 (e)

we have

-where the constant

is

We are not

the

of

the maximum in

### (9)

for a

fixed p from the interval

notice that if

and p = 2 then

and

we

&#x3E; 6.2679.

(7)

Theorem 3.

a is an

### integer

which is not a root

then

### for

every p &#x3E; 0 there exists

such that

d &#x3E;

we have

In

Theorem 1. Let

be a continuous

function in i

=

1

such

=

and let

### = f f (y)dy.

Put

o x

Define

Consider the determinant

where the matrix consists of N -

+

+ ... +

u, =

### 1, 2, ... ,

d. Here pr is the r-th

number

Recall that a is

and a2m

Then see

### )

the determinant D is

where the first

is taken over

m and 0 u v s

u = v, then i

The second

is taken over

=

m;

u, v =

### 0, l, 2, ... ,

s. Let us denote these

(8)

We first consider

We have:

we have for the

### Combining

these results we find

Now from each term

in the first

we take

This is the

of our

### argument.

Write the determinant D as

(9)

Denote yl =

y2 -

yrrt =

Then D can be

in the form

where

is a

### polynomial

in y2 , ... , The power sj is

### Using

the maximum modulus

and the

we have

where

=

= ... = = 1. Now

we find

### )

On the other hand

(10)

Since

and

(11)

For a

d we have

This

D

### Proof of

Theorem 2. If a is not

then

result

### 2 log 0,

and the theorem follows from

Let

a be

Then

and

Holder’s

we have

where

+

### llq

= 1.

Note first that for a

a

For d

to

we have

Hence

(12)

### Proof of

Theorem 3. We have

If a is

### reciprocal,

then the inner sum

for

and zero

for

Hence

3 we have

if d is

### large enough

and the statement of Theorem 3 follows.

### Suppose

now that a is not

If

then

for d &#x3E;

### d5 ~p) .

Hence it is sufhcient to consider the case when

### I ao

=1. Let a1, a2, ... , ar be the

of a

### lying strictly

outside the unit circle. Put

(13)

Then

We shall show now that the last

is

### greater

than

where 0 =1.32471....

if

then

the function is

in the interval

and

theorem

Put for

p = zd and r =

We are

### going

to prove that

for z &#x3E; 0 and 0 y 1.

= 0 and

with our

### hypotheses

(14)

REFERENCES

 D. Bertrand, Duality on tori and multiplicative dependence relations. J. Austral. Math. Soc. (to appear).

 P.E. Blanksby, H.L. Montgomery, Algebraic integers near the unit circle. Acta Arith. 18

(1971), 355-369.

 D.C. Cantor, E.G. Straus, On a conjecture of D.H.Lehmer. Acta Arith. 42 (1982), 97-100.

 E. Dobrowolski, On a question of Lehmer and the number of irreducibile factors of a

polynomial. Acta Arith. 34 (1979), 391-401.

 A. Dubickas, On a conjecture of Schinzel and Zassenhaus. Acta Arith. 63 (1993), 15-20.

 A. Dubickas, On the average difference between two conjugates of an algebraic number. Liet. Matem. Rink. 35 (1995), 415-420.

 M. Langevin, Solution des problèmes de Favard. Ann. Inst. Fourier 38 (1988), no. 2, 1-10.

 D.H. Lehmer, Factorization of certain cyclotomic functions. Ann. of Math. 34 (1933),

461-479.

 R. Louboutin, Sur la mesure de Mahler d’un nombre algébrique. C.R.Acad. Sci. Paris 296

(1983), 707-708.

 E.M. Matveev, A connection between Mahler measure and the discriminant of algebraic numbers. Matem. Zametki 59 (1996), 415-420 (in Russian).

 M. Meyer, Le problème de Lehmer: méthode de Dobrowolski et lemme de Siegel "à la Bombieri-Vaaler". Publ. Math. Univ. P. et M. Curie (Paris VI), 90, Problèmes

Dio-phantiens (1988-89), No.5, 15 p.

 M. Mignotte, M. Waldschmidt, On algebraic numbers of small height: linear forms in one

logarithm. J. Number Theory 47 (1994), 43-62.

 A. Schinzel, H. Zassenhaus, A refinement of two theorems of Kronecker. Michigan Math.

J. 12 (1965), 81-85.

 I. Schur, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Math. Zeitschrift 1 (1918), 377-402.

 C.L. Siegel, The trace of totally positive and real algebraic integers. Ann. of Math. 46

(1945), 302-312.

 C.J. Smyth, On the product of the conjugates outside the unit circle of an algebraic integer. Bull. London Math. Soc. 3 (1971), 169-175.

 C.J. Smyth, The mean values of totally real algebraic integers. Math. Comp. 42 (1984),

663-681.

 C.L. Stewart, Algebraic integers whose conjugates lie near the unit circle. Bull. Soc. Math. France 106 (1978), 169-176.

 P. Voutier, An effective lower bound for the height of algebraic numbers. Acta Arith. 74

(1996), 81-95. Artfras DUBICKAS Department of Mathematics, Vilnius University, Naugarduko 24, Vilnius 2006, Lithuania

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