### 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

### The

mean### values of

### logarithms

### of

### algebraic

### integers

par ART016ARAS DUBICKAS

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

### conjugués

d’unentier

_{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>

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 > 1.

1. INTRODUCTION.

Let a be an

_{algebraic }

number of _{degree }

d > 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

for

_{the "symmetric }

deviation" of _{conjugates }

from the unit circle. Denote
### for p > 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) > }

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) }

> 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) }

> 1 + _{1/52d }

_{log(6d). }

In _{1978,}

C.L. Stewart _{[18] }

_{proved }

that _{M(a) }

> 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 theseresults

_{showing }

that for each £ > _{0, }

there exists an effective ### d(é)

such thatfor d >

_{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 > 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

> 1_{+ ~y/d }

whenever a is not a
we have

where d > 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,

thenLouboutin’s result can be written as follows

### Taking

p =oo, we can write the

_{inequality }

### (2)

in the### following

formThe function p --~

### Mp(a)

is_{nondecreasing. }

Hence the _{inequality}

cp where 1 p oo and _{cp }> 0 lies between the

_{conjecture}

of Lehmer _{p }= 1 and

### the "symmetric"

form of the### conjecture

of Schinzeland 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 truethat

_{d(a) }

> 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

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

> 2 - c for an### algebraic integer

of a### sufhciently large degree.

The author### [6]

### proved

that_{t2 (a) }

> _{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 > ... > 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 > 0 there exists such that we have

whenever d >

_{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 =

### Corollary

1. For_{every }> 0 there exists

_{d1 (c) }

such that
whenever d >

### 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.

HenceWe can

_{replace }

in the _{inequality }

above ### log

### laj

### by

### laj

### 1-1,

and so Theorem 1### yields

the_{following Corollary.}

### Corollary

2._{For every - }> 0 there exists

_{d2(~) }

such that _{for }

d > _{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 > 0. Then there is

_{d3(E) }

such that _{for}

d > _{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 }

> 6.2679.
Theorem 3.

_{If }

a is an _{algebraic }

_{integer }

which is not a root _{of }

_{unity, }

then
### for

every p > 0 there exists_{d5(p) }

such that _{for }

d > _{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.

Puto 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

We first consider

_{Pl. }

We have:
### Next,

we have for the### product

### P2

### Combining

these results we findNow from each term

_{afu - }

in the first _{product }

we take
This is the

_{key }

_{point }

of our _{argument. }

Write the determinant D as
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 havewhere

_{Iyrl }

= _{Iygl }

_{] }

= ... = = 1. Now ### by

Hadamard’s### inequality

we find### (see

### [5])

On the other hand

_{(see }

_{[9]),}

### Similarly,

Since

and

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 havewhere

_{1/p }

+ _{llq }

= 1.
Note first that for a

### reciprocal

aFor d

_{tending }

to _{infinity }

we have
Hence

### Proof of

Theorem 3. We haveIf a is

### reciprocal,

then the inner sum### equals

for### even j

and zerofor

_{odd j. }

Hence
### Utilizing Corollary

3 we haveif d is

_{large enough }

and the statement of Theorem 3 follows.
### Suppose

now that a is not### reciprocal.

Ifthen

for d >

_{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. PutThen

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

theoremPut for

_{brevity }

_{p }= zd and

_{r }=

_{yd. }

We _{are }

### going

to prove thatfor z > 0 and 0 y 1.

### Indeed,

### g(0)

= 0 and### Therefore,

with our### hypotheses

REFERENCES

[1] D. Bertrand, Duality on tori and multiplicative dependence relations. J. Austral. Math.
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[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.
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[7] M. Langevin, Solution des problèmes de Favard. Ann. Inst. Fourier 38 _{(1988), }no. _{2, 1-10.}

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461-479.

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[10] E.M. Matveev, A connection between Mahler measure and the discriminant _{of algebraic}
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[11] M. Meyer, Le problème de Lehmer: méthode de Dobrowolski et lemme de _{Siegel "à }la
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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.

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[14] I. Schur, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit
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(1945), 302-312.

[16] C.J. _{Smyth, }On the product of the conjugates outside the unit circle _{of an algebraic integer.}
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663-681.

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