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J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

A

RT ¯

URAS

D

UBICKAS

The mean values of logarithms of algebraic integers

Journal de Théorie des Nombres de Bordeaux, tome

10, n

o

2 (1998),

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

(2)

The

mean

values of

logarithms

of

algebraic

integers

par ART016ARAS DUBICKAS

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

conjugués

d’un

entier

algébrique

03B1 de

degré

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

algebraic integer

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

algebraic

number of

degree

d &#x3E; 2 with

as its minimal

polynomial

over Z and ad

positive. Following Mahler,

the Mahler measure of a is defined

by

The house of an

algebraic

number is the maximum of the modulus of its

conjugates:

- -

-Put also

(3)

for

the "symmetric

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

algebraic integer

(ad

= 1)

which is not a root of

unity.

In

1933,

D.H. Lehmer

[8]

asked whether it is true that for every

positive

e there exists an

algebraic

number a for which 1

M(a)

1 + c. In its

strong

form Lehmer’s

problem

has been reformulated as whether it is true

that if a is not a root

unity

then

M(a) &#x3E;

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

polynomial

In

1971,

C.J.

Smyth

[16]

proved

that if a is a

non-reciprocal

algebraic

integer

then

M(a)

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

poly-1. This result reduces Lehmer’s

problem

to the case

of

reciprocal algebraic integers

(those

with minimal

polynomial

satisfying

the

identity

P(x) -

xdP(1/x)).

P.E.

Blanksby

and

H.L.Montgomery

[2]

used Fourier

analysis

to prove that

M(a)

&#x3E; 1 +

1/52d

log(6d).

In

1978,

C.L. Stewart

[18]

proved

that

M(a)

&#x3E; 1 +

1/104d

logd.

Although

this result is weaker than the

previous

one, the method used has become very

important

and led to further

improvements. Recently

M.

Mignotte

and

M. Waldschmidt

[12]

obtained Stewart’s result via the

interpolation

deter-minant.

In

1979,

E. Dobrowolski

[4]

obtained a remarkable

improvement

of these

results

showing

that for each £ &#x3E;

0,

there exists an effective

d(é)

such that

for d &#x3E;

d(~)

D.C. Cantor and E.G. Straus

[3]

in 1982 introduced the

interpolation

determinant to

simplify

Dobrowolski’s

proof

and to

replace

the constant 1 - ê

by

2 - ê.

Finally,

R. Louboutin

[9]

was able to

improve

this constant to

9/4 -

E. M.

Meyer

[11]

obtained Louboutin’s result

using

a version of

Siegel’s

lemma due to Bombieri and Vaaler.

Recently

P. Voutier

[19]

showed that

inequality

(1)

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

1/4

instead of 1- E.

In

1965,

A. Schinzel and H. Zassenhaus

[13]

conjectured

that there exists

an absolute

positive constant -y

such that

ïar

&#x3E; 1

+ ~y/d

whenever a is not a

(4)

we have

where d &#x3E; In

fact,

both

inequalities

(1), (2)

and the

respective

con-jectures

can be considered in terms of the lower bound for

Mp(a).

Indeed,

notice that

Therefore,

for

lao I 2::

2,

If

laol

=

1,

then

Louboutin’s result can be written as follows

Taking

p =

oo, we can write the

inequality

(2)

in the

following

form

The function p --~

Mp(a)

is

nondecreasing.

Hence the

inequality

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

conjecture

of Lehmer p = 1 and

the "symmetric"

form of the

conjecture

of Schinzel

and Zassenhaus p = oo

(see

also

[1]

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

d(a)

&#x3E; The

equality

holds for the

polynomial

2. We

conjecture

that the answer to the above

question

is

affirmative,

so that

Coo =

log

2. In this

paper, we take up the

interpolation

determinant

again

(see

[3],[5],[9],[10],

[19])

and fill the gap between

inequalities

(3)

and

(4)

(Theorem 2).

One can also consider the mean value of

conjugates

of an

algebraic

integer

(5)

The lower bound for

ml (a)

where a is a

totally

positive integer

was consi-dered

by

I. Schur

[14],

C.L.

Siegel

[15],

C.J.

Smyth

[17].

In

1988,

M.

Langevin

[7]

solved Favard’s

problem proving

that

t(a)

:= maxi,j

lai-aj

I

&#x3E; 2 - c for an

algebraic integer

of a

sufhciently large degree.

The author

[6]

proved

that

t2 (a)

&#x3E;

4 e -

c. The

problem

of

finding

an upper bound

for

:=1/

aj )

is known as a

separation

problem.

In this

article,

we

apply

the lower bound for

M2 (a)

to estimate

rrLp (a)

from below

(Theorem 3).

2. STATEMENT OF THE RESULTS.

The notations are the

following.

Let

G(x)

be a real valued function in

[0; 1]

such that

G(O) =

1,

G(l) =

0. Let also the derivative of

G(x)

be continuous and

negative

in the interval

(0; 1).

Put

Put also for

brevity

Let a be a

reciprocal algebraic integer,

i.e. d =

2m,

m E N,

a2m =

a2m-1 =

1/2(X2, ... , am+1 - l/am

where

Ia2l &#x3E; ... &#x3E; laml ;2:

1.

Suppose

also that a is not a root of

unity.

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;

duo(£).

The constant

do (e)

and the constants

d1

(e),d2 (e),d3 (e),

d4,

d5 (P)

used below are effective.

Taking

G(x) = (1 -

X)2,

we

get

I =

1/3,

J =

1/5,

L =

(6)

Corollary

1. For every &#x3E; 0 there exists

d1 (c)

such that

whenever d &#x3E;

d1

(e).

This

inequality

obviously implies

Louboutin’s result. On the other

hand,

taking

G(x)

= 1 - we have I = 1 -

2/7r,

J =

3/2 -

4/-7r,

L =

7r2/ 8.

Hence

We can

replace

in the

inequality

above

log

laj

by

laj

1-1,

and so Theorem 1

yields

the

following Corollary.

Corollary

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

d2(~)

such that

for

d &#x3E;

d2(e)

we have

where

Corollary

2

implies

the

inequality

(2),

since T~ = 1. The

following

theorem fills the gap between

(3)

and

(4).

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

d3(E)

such that

for

d &#x3E;

d3 (e)

we have

-where the constant

bp

is

given

by

We are not

solving

the

problem

of

computing

the maximum in

(9)

for a

fixed p from the interval

(1; oo).

However,

notice that if

G(x) _ (1 -

x)u7

and p = 2 then

by

(5)-(7)

and

(9)

we

get

b2

&#x3E; 6.2679.

(7)

Theorem 3.

If

a is an

algebraic

integer

which is not a root

of

unity,

then

for

every p &#x3E; 0 there exists

d5(p)

such that

for

d &#x3E;

d5(p)

we have

In

particular,

Proof of

Theorem 1. Let

f (x)

be a continuous

non-negative

function in i

=

1

[0; 1]

such

that j f(z)dz

=

1,

and let

G(x)

= f f (y)dy.

Put

o x

Define

Consider the determinant

where the matrix consists of N -

~ko

+

l~1

+ ... +

ks) d

columns,

u, =

0,1, ... ,

l~r -1, j -

1, 2, ... ,

d. Here pr is the r-th

prime

number

(po =

l, pl - 2,p2 =

3,...).

Recall that a is

reciprocal

and a2m

-1/al, ... , ~m+1 - l/am.

Then see

([3],

[5], [9], [10],

[19])

the determinant D is

given by

where the first

product

is taken over

i, j =1, 2, ... ,

m and 0 u v s

(if

u = v, then i

j ~ .

The second

product

is taken over

all i, j

=

l, 2, ... ,

m;

u, v =

0, l, 2, ... ,

s. Let us denote these

(8)

We first consider

Pl.

We have:

Next,

we have for the

product

P2

Combining

these results we find

Now from each term

afu -

in the first

product

we take

This is the

key

point

of our

argument.

Write the determinant D as

(9)

Denote yl =

y2 -

~3/~2?... ~m-1 =

yrrt =

Then D can be

expressed

in the form

where

p ( y1, ... ,

ym )

is a

polynomial

in y2 , ... , The power sj is

given

by

Using

the maximum modulus

principle

and the

inequalities

1,

j - 1,2,...,m,

we have

where

Iyrl

=

Iygl

]

= ... = = 1. Now

by

Hadamard’s

inequality

we find

(see

[5])

On the other hand

(see

[9]),

(10)

Similarly,

Since

and

(11)

For a

sufhciently large

d we have

This

inequality implies

(8).

D

Proof of

Theorem 2. If a is not

reciprocal,

then

by

Smyth’s

result

[16]

2 log 0,

and the theorem follows from

Mp(a) 2::

M1(a).

Let

a be

reciprocal.

Then

by

(8)

and

by

Holder’s

inequality

we have

where

1/p

+

llq

= 1.

Note first that for a

reciprocal

a

For d

tending

to

infinity

we have

Hence

(12)

Proof of

Theorem 3. We have

If a is

reciprocal,

then the inner sum

equals

for

even j

and zero

for

odd j.

Hence

Utilizing Corollary

3 we have

if d is

large enough

and the statement of Theorem 3 follows.

Suppose

now that a is not

reciprocal.

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

conjugates

of a

lying strictly

outside the unit circle. Put

(13)

Then

We shall show now that the last

expression

is

greater

than

where 0 =1.32471....

Indeed,

if

then

Therefore,

the function is

increasing

in the interval

(1; oo)

and

by

Smyth’s

theorem

Put for

brevity

p = zd and r =

yd.

We are

going

to prove that

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

Indeed,

g(0)

= 0 and

Therefore,

with our

hypotheses

(14)

REFERENCES

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

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

(1971), 355-369.

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

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

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

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

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

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

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

461-479.

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

(1983), 707-708.

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

[11] 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.

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

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

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

J. 12 (1965), 81-85.

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

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

(1945), 302-312.

[16] 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.

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

663-681.

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

[19] 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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