Birational transformations and values of the Riemann zeta-function

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

C

ARLO

V

IOLA

Birational transformations and values of the

Riemann zeta-function

Journal de Théorie des Nombres de Bordeaux, tome

15, n

o

_{2 (2003),}

p. 561-592

<http://www.numdam.org/item?id=JTNB_2003__15_2_561_0>

© Université Bordeaux 1, 2003, tous droits réservés.

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

Birational transformations and values

of the Riemann zeta-function

par CARLO VIOLA

RÉSUMÉ. Dans sa _preuve du théorème

_d’Apéry

sur l’irrationalité

de

_03B6(3),

Beukers

_[B]

a introduit des

_intégrales

doubles et

_triples

de fonctions rationnelles donnant de bonnes suites

_{d’approximations}

rationnelles de

_03B6(2)

et

_03B6(3).

La méthode de Beukers a

été,

par la

suite,

améliorée par Dvornicich et

_Viola,

_par

_Hata,

et par Rhin

et Viola. Nous

_présentons

ici un survol de nos résultats récents

([RV2]

et

_[RV3])

sur les mesures d’irrationalité de

03B6(2)

et

_03B6(3)

obtenus _par de nouvelles méthodes

_algébriques

mettant en

_jeu

les

actions de transformations birationnelles et de _groupes de per-mutations sur des

_intégrales

doubles et

_triples

du _type de celles introduites par Beukers. Dans les deux dernières

parties,

nous

donnons une méthode constructive pour obtenir les

transforma-tions birationnelles

appropriées

pour les

intégrales triples

à par-tir des transformations

correspondantes

pour les

intégrales

dou-bles. Cette méthode est

_{également appliquée}

pour obtenir l’action de transformations birationnelles sur des

intégrales quadruples

du

type de celles introduites _par

_Vasilyev.

ABSTRACT. In his

_proof

of

_Apery’s

theorem on the

irrational-ity

of

_03B6(3),

Beukers

_[B]

introduced double and

_triple

_integrals

of suitable rational functions

_{yielding good}

sequences of rational

ap-proximations

to

_03B6(2)

and

_03B6(3).

Beukers’ method was

subsequently

improved by

Dvornicich and

_{Viola, by Hata,}

and

_by

Rhin and Viola. We

_give

here a _survey of our recent results

_([RV2]

and

[RV3])

on the

_{irrationality}

measures of

03B6(2)

and

03B6(3)

based upon

a new

algebraic

method

involving

birational transformations and

permutation

groups

acting

on double and

triple

integrals

of

Beukers’ _type. In the last two sections we

_give

a constructive

method to obtain the relevant birational transformations for

triple

integrals

from the

_analogous

transformations for double

_integrals,

and we also

_apply

such a method to _get birational transformations

1. The

_{irrationality}

measures of

((2)

and

((3)

Let

_{1£( a)}

denote the least

_{irrationality}

measure of an irrational number

a,

i.e.,

the least

exponent

A such that for any e &#x3E; 0 there exists a constant qo =

qa (e)

_&#x3E; 0 for which

for all

_integers

p and q with q &#x3E; _qo. We recall that an

irrationality

measure

of a number a is

usually

obtained

through

the construction of a convenient

sequence

(rn~sn)

of rational

approximations

to a,

by

applying

the

following

well known

PROPOSITION Let a E and let

_{(rn~, (sin)}

be sequences

of

integers

satis-f ying

and

for

some

_positive

numbers R and S. Then

a ft

Q,

and

The

_{irrationality}

measures of

~(2) _

7r2/6

and of

~(3),

where (

denotes the Riemann

_{zeta-function,}

were

_extensively

studied in _{recent years.} Since

Apery’s

paper

[A],

several

improvements

in the search for _sequences of

good

rational

_{approximations}

to

₍₍₂₎

and to

₍₍₃₎

were obtained.

_Thus,

the

and

Soon after

_Ap6ry’s

results

_[A],

Beukers

_[B]

found

_through

a different method the same _sequences of rational

_{approximations}

to

₍₍₂₎

and

₍₍₃₎

previously

obtained

_by

_Ap6ry.

Beukers’ method was based on the arith-metical

_study

of the

_integrals

for

_~(2),

and

for

_~(3),

and of their

_asymptotic

behaviours oo. The

integrals

(1.3)

and

_(1.4)

_yield

the same

_inequalities

li(((2))

11.85078... and

/-t(((3))

13.41782... found

_by

_Ap6ry.

We remark that all the succes-sive

_improvements

_(1.1)

and

_(1.2)

on

_Ap6ry’s

_{irrationality}

measures were obtained

_through

the

_study

of arithmetic and

_{analytic properties}

of suitable variants of Beukers’

_integrals

_(1.3)

and

_(1.4).

2. Beukers’ and Hata’s methods

Beukers’ method

_[B]

_employs

two different

representations

for each of the

_integrals

_(1.3)

or

(1.4).

Consider

(1.3)

for

_{simplicity. By}

n-fold

partial

integration

one

_gets

whence

we obtain

Hence the

_integral

_(1.3)

can be also

_expressed

in the form

where

_{Pn (x, y)}

E

_{Z [x, y]}

is a suitable

_polynomial

(instead

of a rational function as in

(1.3))

satisfying degx

Pn

=

degy

Pn

= _n.

Using

the series

expansion

one sees that

_(2.2)

_equals

_an +

_bnS(2)

with

_bn

E Z and

_dnan

E

Z,

where

dn

=

l.c.m.{1, ... , n}.

Thus the

right

side of

_{(2.1), being}

of

_type

_(2.2),

shows that the

_integral

has the

_required

arithmetic form an +

_bn((2),

whereas the left side of

_(2.1)

is suitable to obtain

_asymptotic

estimates

_{of an + bn( (2)}

and

Ibn’

as n -+ oo.

By

combining

such estimates with the

asymptotic

estimate

dn

=

exp(n+o(n))

given by

the Prime Number

Theorem,

and

applying

the

Proposition

in Section

_1,

one obtains the

irrationality

measure

A

_slightly

more

complicated

_argument,

again

based on

_{repeated partial}

integration, yields

the

_analogue

of the formula

_(2.1)

for the

_integral

_(1.4),

As

above,

the

_right

side of this formula is

_easily

seen to be

_an

+

_26~(3)

with

_bn

E 7~ and

_dnan

E

?~,

while the left side of

_(2.3)

_yields

the

_asymptotic

estimates of

_a~

+

_2&#x26;~(3)

and Thus one

_gets

the

irrationality

measure

In

_[H2]

and

_[H3],

Hata

_improved

the above

_{irrationality}

measures

through

the

_study

of variants of the

_integrals

_(1.3)

and

_(1.4),

where the

_exponents

of the factors x, 1 - x, y, 1 - y,

_{1 - xy appearing}

in the rational function in

_(1.3),

or of the factors _~, 1 - _{x, y, 1 - y, z,} 1 - z, 1 -

(1 -

xy)z

in

(1.4),

are not all

_equal.

For

_instance,

in

_[H2]

Hata considered the

_integral

and transformed

_(2.4),

_by

12n-fold

_partial

_integration

with

_respect

to _x, into an

_integral

of

_type

_(2.2)

with

_deg.

_Pn

= 18~t and

degy

Pn

=

16n,

thus

showing

that

_(2.4)

_equals

an +

_bn~(2)

with an

E Q

and

bn

E Z. More

gen-erally,

k-fold and

_{(12n -}

_k)-fold

_partial

_integrations

with

_respect

to x and

y

respectively,

for any k such that 0 k

12n,

transform

(2.4)

into an

integral

of

_type

_(2.2)

divided

_by

_{~1 ~"~ ,}

where now

y)

=

Fk (x)Gk (y)

for suitable

_{Legendre-type polynomials}

_Fk

and

_Gk.

On

_choosing

k

appro-priately,

Hata

_got

a

good

control of the denominator of the rational

_part

an of the

integral

(2.4)

through

the

p-adic

valuation of the above

~1~"~

and of the binomial coefficients

_occurring

as coefficients of

_Fk

and

Gk.

Thus Hata

_proved

the

_{irrationality}

measure

p(((2) )

6.3489. In his addendum to the _paper

_[H2],

he

_subsequently

_applied

to the

_integral

_(2.4)

a

_change

of

variables introduced in

_[RV1]

_showing

that

_(2.4)

_equals

By

combining

(2.4)

and

_(2.5),

Hata

_proved

that

_it(((2))

5.687.

~

3. _{Permutation groups for} double

_integrals

The

_equality

of the

_integrals

_(2.4)

and

_(2.5)

is a

_special

instance of a far more

_general

phenomenon

discovered

by

Rhin and

Viola,

and described in

_[RV2]

for double

_integrals,

and in

_[RV3]

for

_{triple integrals}

of Beukers’

type.

This

_{phenomenon depends}

upon the actions of suitable birational

In this section we outline the main results of

[RV2].

Let

where

_{h, z, j,}

are _any

_{non-negative integers}

(this

condition

_clearly

ensures that

_z,

_{j, k,}

t )

is

finite).

Let T :

(x,

y)

H

_{(X, Y)}

be the birational transformation defined

_by

the

_equations

-It is

_easily

seen that T has

_period

5 and _maps the _open unit _square

(o,1 ) 2

onto itself.

_Moreover,

we have

and

so that both the rational function

_(3.3)

and the measure

(3.4)

are invariant under the action of T.

If we

apply

the birational transformation T to

_{I(h, i, j, k, l),}

_i.e.,

if we make in the

_integral

_(3.1)

the

_change

of variables

and then

_{replace X,}

Y with _{x, y}

_{respectively, by}

virtue of

_(3.4)

we obtain the

_integral

_Thus,

it is natural to associate with the action of T on

_{i, j,}

_k,

l )

the

_cyclic

_permutation

Also,

if we

apply

to the transformation

i.e.,

if we

_interchange

the variables _{x, y} in

(3.1),

we

_get

the

_integral

I(k, j, i, h, l).

Hence with the action of Q2 on we associate

the

_permutation

°

~2 = (h k)(i 7)~

We use the

_subscript

2 for _Q2 and _u2, and later on for _p2, and

_~2

_(see

related with double

_integrals.

In Section

_4,

the

_analogues

for

_{triple integrals}

will be denoted

_by

a similar notation with the

_subscript

3. The

_permutation

_group

generated by

T and _a2 is

_{clearly isomorphic}

to the _{dihedral group}

_T5

of order

_10,

and the value of

_I(h,

_i,

_j,

_k,

_{l )}

is invariant under the action of the

group T on

t~, i, j,

k,

l. Note that

-is the

_integral

_(2.4),

and

is

_(2.5),

so that the

_equality

of

(2.4)

and

_(2.5)

is a

special

case of the formula

obtained

_by

_applying

to

_h,

_{i, j,}

_k,

I the

_permutation

_{-ra2 E T}

_(here

and in the

_sequel,

a

_product

of

_permutations

is meant to be the

_permutation

obtained

_by

_applying

first

_~C3

and then

_a).

Besides the

_integers

we consider the five

auxiliary integers

the last of which is the

_exponent

of the factor

_{1- xy}

in the rational function

appearing

in the

_integral

_(3.1).

_Also,

it is natural to extend the actions of the above

_permutations

T and _u2 on _any linear combination of

_h,

_i,

_j,

_k,

1

by linearity.

Thus

_{r(j+k-h) = -r (j) + -r (k) -,r (h)}

=

k + I - i,

_etc. Hence

T and _U2 act on the ten

_integers

_(3.5)

and

_(3.6)

as follows:

and

Let

_{do =}

1

_and,

as

above,

dn =

l.c. m. ~ 1, ... , n~

for _any

integer

n &#x3E; 1. We also need some further notation: we denote

by

_max,

mar, mad’, ...

the successive maxima in a finite _sequence of real numbers.

Precisely,

if

A _

(ai,..., an)

is

any finite sequence of real numbers

( with n &#x3E;

3)

and .

i 1, ... , zn

is a

_reordering

of

_{1, ... , n}

such that

For the

_integral

_(3.1),

let S be the sequence of the

integers

(3.6):

and let

We

_incidentally

remark that our

_assumption

_{h, i, j,}

&#x3E; 0

_{easily implies}

that at most two of the

_integers

_(3.6)

can be

_{strictly negative}

(in

which

_case,

the two

_negative

_integers

must be

_cyclically

consecutive in

_(3.6)).

Therefore

0. One can show that

where M and N are defined

_by

(3.10).

In

[RV2],

Theorem

_2.2,

this result is established in a

_slightly

_stronger

form, i.e.,

with a more

_precise

definition for N. In

_{fact, depending}

on the numerical values for

h, i, j, k, l,

the N defined in

_[RV2]

can be either

max’s,

or

max"s

max’S.

_However,

the definition

_(3.10)

for N turns out to be

_appropriate

in

_practice,

since the N defined in

_[RV2]

_equals

max’s

for all the

_"good"

numerical choices of

h,

i,

j,

(i.e.,

those

_eventually

_{yielding good irrationality}

measures of

~ (2) ) .

The

_proof

of

_(3.11)

_given

in

_[RV2]

is

_independent

of a

_{representation}

of

_type

_(2.2)

for the

_integral

_(3.1),

and relies on the invariance of the value of under the actions of the

_permutations

T and _0"2, and on a method of descent based on suitable linear

_{decompositions}

of the rational function

_appearing

in

_(3.1).

_Moreover,

the same method of descent shows that the

_integer

b in

_(3.11)

can be

expressed

as a double contour

integral,

in the form

It is worth

_remarking

that if i

_{-I- j - l}

&#x3E;

minlh,

_{i, j,}

kl,

the

_integral

_(3.1)

cannot be transformed

_{by partial}

_integration

into an

integral

of

_type

(2.2)

to which Hata’s method

applies.

Thus it is essential to

_dispense

with the

partial integration

method and with

_{Legendre-type}

_polynomials,

and to

apply directly

to

_(3.1)

the method of descent alluded to above.

As is obvious from

_(3.7)

and

_(3.8),

the

_permutation

_{group T} =

(T, 0*2)

is intransitive over the set of the ten

_integers

_(3.5)

and

_(3.6).

We will now

enlarge

our

_permutation

_group, under the further

assumption

that the

integers

(3.6)

are all

_{non-negative, by}

_introducing

a new

_permutation

_cp2

(defined

below in

_(3.19)),

which mixes _up

_(3.5)

with

_(3.6),

so that the

_larger

permutation

group

~2

=

(CP2,

_r,

~2)

thus obtained is transitive over the set

naturally

into

_play

the factorials of the

_integers

_(3.5)

and

_(3.6),

and the

p-adic

valuation of such factorials

yields

strong

arithmetic information on the denominator of the a

E Q

in

(3.11).

The

permutation

_{CP2 is related} to Euler’s

_integral

_{representation}

of Gauss’s

_{hypergeometric}

function,

and to the invariance of this function under the

_interchange

of the two

_parameters

appearing

in the numerator of the

_{hypergeometric}

series.

We recall that Gauss’s

_{hypergeometric}

function is defined as follows:

where a,

{3 and -t

are

_complex

_parameters,

y :A 0,

_-1,

-2, ... ,

and y

is a

complex

variable

_satisfying

_Iyl

1. The Pochhammer

_symbols

in

_(3.13)

are defined

by

and

_similarly

for

_(Q)n

and

_(7)n.

Euler’s

_integral

_{representation,}

valid for is

where 1’ denotes the Euler

_{gamma-function,}

and

_gives

the

_analytic

contin-uation of outside the unit disc

_’y’

1.

_{Since, by}

_(3.13),

if Re -t &#x3E;

maxfre

a, and

minire

a, &#x3E; 0 we

_get

from

(3.14)

(3.15)

We henceforth assume the

_{non-negative integers}

(3.5)

to be such that

(3.6)

are also

non-negative.

Then we _may take in

(3.15)

whence

If we divide

(3.16)

_by

we

_get

Let _p2 be the

_integral

transformation

_changing

into

and let

_{W2 be}

the

_{corresponding}

_{permutation, mapping}

the

_integers

_(3.5)

respectively

to i

_{+ j -}

_{I, I}

+ h -

_j,

j, k, 1,

and extended to any linear

combination of the

_integers

_(3.5)

_by

_linearity.

_{Thus CP2 is the}

_permutation

acting

on the ten

_integers

_(3.5)

and

_(3.6)

as follows:

By

(3.17),

the value of the

_quotient

_(3.18)

is

_clearly

invariant under the action on the

_integers

(3.5)

and

(3.6)

of the

_permutation

_group

generated by

and

Cr2-We want to determine the structure of the

_permutation

_group

_~2

_acting

on

(3.5)

and

(3.6),

and in

_particular

its order For this _purpose, we show that

_~2

can be viewed as a

permutation

_group

acting

on five

_integers

only,

i.e.,

on the sums

To _prove

_this,

it

_clearly

suffices to show that and ~2

permute

the

integers

(3.20),

and that if a

_permutation

_{e E}

~2

acts

_identically

on the

integers

(3.20),

it acts

_identically

also on the

integers

(3.5),

and therefore also on

(3.6).

It is

_plain

that the actions of the above

permutations

on the

integers

(3.20)

are as follows:

Moreover, if e

E

~2

acts

_identically

on the

_integers

(3.20)

we

_get,

by

and

_similarly

and so on.

Hence e

acts

_identically

on the

_integers

(3.5).

Since the

symmet-ric group

65

of the 5!

_permutations

of five elements

_(i.e.,

of the

_integers

(3.20))

is

_{generated by}

a

cyclic

permutation

of the five elements and a

transposition,

from

_(3.21)

we conclude that

°

is

_isomorphic

to

_65.

Therefore

₁₉₂₁

= 120.

We now consider

again

the actions

(3.7), (3.8)

and

(3.19)

of the above

permutations

on the ten

_integers

_(3.5)

and

_(3.6).

If we

apply

to

(3.18)

_any

permutation e

E

_4P2,

we

_get

the transformation formula

Thus we associate

_{with Q}

the

_quotient

resulting

from the transformation formula

_(3.23)

for

_I(h,

_{i, j, k,}

_d).

If

_{e, e’}

E

~2

lie in the same left coset of T in

_~2,

_i.e.,

_{if e}

=

e’e"

with

e"

_E

_T,

the

integers

e(h), e(i), e(j), e(k), Q(1)

obviously

coincide with

_e!(h),

_e,(i),

e’(j),

e~(k), e’(1),

up to a

_permutation.

Hence the

quotient

(3.24)

for

e

equals

the

analogous

quotient

for

e’.

Thus with each left coset of T

in

_~2

we _{may associate the}

_{corresponding quotient}

(3.24),

where e

is _any

representative

of the coset considered.

_Also,

for _{any e} E

_P2

we

_simplify

the

quotient

(3.24)

by removing

the factorials of the

_{integers appearing}

both in the numerator and in the denominator.

_If,

after this

_{simplification,}

the

resulting quotient

of factorials has v factorials in the numerator and v in the

_denominator,

we _say

that e

is a

permutation

of level v

(or

that the left

coset

_LoT

is of level

_v).

In other

_{words, e}

is of level v if the intersection of the set

_fLo(h),

_{e(i), e(j), e(k),}

with the set of the

_integers

_(3.6)

contains v elements.

Since

_I’P21

= 120 and

ITI

=

10,

there are 12 left cosets of T in

_~2.

In

Moreover,

the 5

_quotients

of factorials associated with the 5 left cosets of level 2 are all

_distinct,

and

_similarly

the 5

quotients

of factorials associated with the 5 left cosets of level 3 are all distinct.

For the rest of this section we define

where

T

is the _sequence of the ten

_integers

_(3.5)

and

_(3.6).

Then,

a

fortiori,

(3.11)

holds with M and N

_{given by}

_(3.25).

_Also,

for _{any e} E

_~2,

_e(j)

+

e(k) - e(h)

=

e( j + k - h)

is _one of the

integers

(3.5)

or

(3.6),

and

_similarly

for

_e(k)

+

_{g(l) - e(i),}

etc. Hence we

_{have, by}

(3.11),

where M and N are defined

_by

_(3.25),

and where _a. and

_bg

_depend

_only

on the left coset

_eT.

For fixed

_{h, i, j, 1~, l ,}

we now abbreviate

with

_{a,, E Q}

and

_bn

E Z. Pick _{any g} E

_P2

of level 2: for

_instance,

take

9 =

C(J2.

By

(3.16),

the transformation formula for

In

corresponding

to the

permutation

C(J2 is

where

with

_a.

_{E ~}

_{and V.}

E Z.

_By

_{(3.27), (3.28)}

and

_(3.29),

and

_by

the

irra-tionality

of

_~(2),

we obtain

On

_multiplying

_(3.30)

_{by dMndrrn,}

with M and N

_{given by}

_(3.25),

we

_get

where

_{An -}

and are

_{integers, by}

(3.11)

and

(3.26).

For a

prime

_p, let

denote the fractional

_part

of

_Using

the

_p-adic

valuation of the factorials

appearing

in

_(3.31),

it is _easy to see

([RV2],

pp.44-45)

that _any

prime

divides

_An

=

The above discussion

_applies

to each transformation formula for

_In

cor-responding

to a left coset of T in

~2

of level 2. The 5 left cosets of level 2

yield

the 5

_inequalities

for w obtained

by

applying

the powers of the per-mutation T to

(i

+ j -1, l

+ h -

j, h,

i)

in

(3.32).

Let S? be the set of real numbers W E

_{~0,1 )}

_satisfying

at least one of such 5

inequalities.

We con-clude that _any

_prime

_p &#x3E;

Mn,

for which E

f2,

divides the

_integer

An =

We

_incidentally

remark that Y2 is

_clearly

the union of

finitely

many intervals with rational

endpoints

aq, ,Bq

E

_(0,1).

A similar

_{analysis applies}

to the 5 transformation formulae for

In

corre-sponding

to the left cosets of level

3,

and

_yields

a subset

S2’

C S2 such that

p2

divides

_An

=

dMndNnan

for

any

prime

p &#x3E;

Mn

satisfying in/p}

E S2’.

Finally,

the transformation formula

_{corresponding}

to the left coset of level 5

_gives

no further

prime

factors

of An

besides the above

([RV2],

_pp.

46-49) .

Let

From the above discussion we

_get

An.

Also,

it is _{easy to} show that

dn

and

_Hence,

if we define

we have Z and

Therefore, by

(3.27),

and the

_asymptotic

estimates for

_Dn,

_In

and oo in

(3.34)

yield

an

irrationality

measure of

~(2),

by applying

the

Proposition

in Section 1.

Since

_dMndNn

=

exp((M+N)n+o(n))

by

the Prime Number

Theorem,

from

_(3.33)

we

_get

where

_1/J(x)

=

r’(x)/1’(x)

is the

logarithmic

derivative of the _Euler gamma-function. Therefore

Note that if the

_integral

_In

_given

_by

_(3.27)

is Beukers’

_integral

_(1.3),

_i.e.,

if

h = i = j = k = l,

the five

_inequalities

of

type

(3.32)

become

_2(hw~

_2(hw)

and therefore are

false,

whence S2 = _0. Thus

vanishes for the

_integral

_(1.3).

A

_judicious

choice of the

_parameters

in

(3.27),

i.e.,

one where

_h,

_i,

_j,

are not all

_equal

but their differences are not too

_large,

forces to lose a certain amount on the

_asymptotics

for

I~,

and

Ibnl

in

_comparison

with the

_integral

_{( 1.3),}

but allows to

_gain

much more on the divisor

_Dn

of

_{dMndNn, by}

virtue of the arithmetical correction

_(3.36)

in the

_asymptotic

formula

_(3.35).

Let

....

If we assume the

_integers

(3.5)

and

(3.6)

to be all

_{strictly positive,}

it is _easy to see that the function

f (x,

y)

has

exactly

two

_{stationary points}

_(xo,

_yo)

and

_{(xl, yl)}

_satisfying

_{x(l - x)y(l - y) 0}

_0,

with 0 _{xo, yo} 1 and

xl,yl

0,

xlyl &#x3E; 1.

_Plainly

and the double contour

_{integral representation}

_(3.12)

for

_bn

_{easily yields}

If we denote

from

_{(3.34), (3.35), (3.37), (3.38)}

and the

_Proposition

in Section 1 we

_get

the

_{irrationality}

measure

The choice

whence we obtain

4. _{Permutation groups for}

_triple

_integrals

The

_theory

outlined in Section 3 has its

_analogue

for

_{triple integrals}

of Beukers’

_type,

and this has been worked out in

_[RV3].

In this section we

briefly

summarize the results of

_[RV3].

_Again

we introduce a

permutation

group

arising

on the one hand from the action on the

_triple

_integrals

of a three-dimensional birational _{transformation 3}

_(defined

_by

_{(4.2) below),}

analogous

to the two-dimensional transformation T

given by

(3.2),

and on the other hand from the

_{hypergeometric}

_integral

transformation

_given

by

the relation

_(3.15).

_Thus,

the

_{hypergeometric}

_part

of our method for

_triple

integrals

still relies on the formula

(3.15)

_{involving simple integrals,}

while a _process to derive the three-dimensional transformation 3 from the two-dimensional transformation T is

definitely

less

obvious,

and is described in

Section 5.

_Moreover,

as we have seen in Section

_3,

the rational function

appearing

in the double

_integral

contains 5

_factors,

the

trans-formation T has

period

_5,

and the

permutation

_group

~2 -

(Sp2, T,

is

isomorphic

to the

_symmetric

_group

₆₅

of

_permutations

of 5 elements. The situation for

_triple

_integrals

that we treat in this section is more

compli-cated,

since no number has the role

_{played by}

5 for double

_integrals.

In

fact,

the rational function in the

_triple

_integral

_(4.1)

considered in this sec-tion contains 7

_factors,

_but,

in order to ensure that the

integral

has the

required

arithmetic

_{expression, only}

6

_exponents

of such factors are taken to be

_independent,

_by

virtue of the linear condition

_(4.6)

below.

_Also,

the

transformation 3 has

period 8,

and the

_permutation

_group

_arising

from

. t9 and from the

hypergeometric integral

transformation can be

naturally

embedded in the

_alternating

_group

_2ll0

of the even

permutations

of 10 elements.

We assume

_{h, j, k, l,}

_q,

_{r, s &#x3E;}

0 and h k + r, since these

inequalities

are

_easily

seen to _{be necessary and} sufficient for the

_integral

_{(4.1 )}

to be

finite. Let {} :

_{(x, y, z)}

~

_(X,

_Y,

_Z)

be the birational transformation defined

by

It is _easy to check that 3 has

_period

8 and _maps the _open unit cube

_(0,1)3

onto itself.

_Moreover,

under the action of 3 we have

and

If we

_apply

the transformation t9 to the above

_integral,

_i.e.,

if we make in

_(4.1)

the

_change

of variables

and then

_{replace X, Y,}

Z with _x, y, z

respectively, by

(4.3)

the

integral

(4.1 )

becomes

Thus it is natural to define m = k + r -

h,

whence m &#x3E; 0 and

and to assume the linear condition

which eliminates from

_(4.4)

the undesired extra factor 1 -

_{(1 - x)z.}

_Also,

written in the form

This

_integral

is obtained from

_(4.1)

_by

_applying

to the

_parameters

the

_cyclic

permutation

(h j

k 1 r

s).

_Therefore,

if for _any

_{non-negative integers}

h, j, k,

I,

m, q, r, s

satisfying

(4.5)

and

(4.6)

we define

where m is a "hidden"

_parameter,

we see that the transformation 3

given

by

(4.2)

changes

the

_integral

_(4.7)

into

_{I ( j,}

_{k, l,}

_{m, q, r, s,}

_h).

Thus we associate with the action of V on the

cyclic

permutation

If we

_apply

to the

_integral

_(4.7)

the transformation

i.e.,

if we

interchange

the variables _{x, y} in

(4.7),

by

(4.5)

we

_get

the

integral

I(k, j,

h,

s, r, q, m,

t ).

Hence with the action of ~3 on

I(h, j7

k, l,

_{m, q, r,}

s)

we associate the

_permutation

The

_permutation

_group

generated by t9

and a3 is

clearly isomorphic

to the dihedral group

T8

of order

_16,

and the value of

_{l(h,j,k,l,m,q,r,s)}

is invariant under the action

of e on h, j, k, 1,

m, q, r, s.

we consider the

_eight

auxiliary

_integers

where the double

expression

for each of the

_integers

_(4.9)

is obtained

_by

applying

(4.5)

or

(4.6).

Also,

we extend the actions of the

_{permutations t9}

and _er3 on _any linear combination of the

_integers

(4.8)

_by

linearity,

and we can do this because

so that 0 and _{Q3 preserve} the relations

(4.5)

and

(4.6).

In

particular,

1? and a3 act on the sixteen

_integers

(4.8)

and

(4.9)

as follows:

and

Let

be the _sequence of the

_integers

_(4.9),

and let

where we use

_again

the notation

(3.9).

It is _{easy to} see that M &#x3E; N &#x3E;

(~ &#x3E;

0. In

_[RV3],

Theorem

_2.1,

we _prove that

As with

_(3.11),

the

_proof

of

_(4.11)

is based on the invariance of the value of the

_integral

_(4.7)

under the actions and _a3, and on a suitable method of descent. We also have the

_analogue

of

_(3.12),

since the

_integer

b in

_(4.11)

is

_given

_by’

where C =

{x

E ~ :

Ixl

=

y},

C.

=

ly

E C :

ly - llxl

=

~2}

and

C

The linear relations

_(4.5)

and

_(4.6)

are essential for the

validity

of

(4.11),

₇ since

_{if j}

_{+ q}

&#x3E; 1 + _s, in

_general

the

_integral

_(4.1)

is a linear combination of

1,

’(2)

and

₍₃₎

with rational coefficients

_(see

_[RV3],

Remark

_2.2).

We now assume the

_{non-negative integers}

(4.8)

to be such that

_(4.9)

are also

_{non-negative.}

If in the formula

_(3.15)

we

change y

into

-yz/(1 - x),

and take

we

_easily

obtain

Multiplying

by

and

_integrating

in 0 y

1,

0 z

1,

we

_get

Therefore

Let p3 be the

integral

transformation

changing

into

and let tp3 be the

corresponding permutation, mapping

the

_integers

_(4.8)

respectively

to

_q’,

_{j, k,}

_r~,

_{m, r, q, s,} and extended to _any linear combination

lI take this _opportunity to correct a _misprint in _{[RV3] occurring} in the _proof of the above integral representation (4.12) for b. On _p.2?? of _[RV3], in the line between the formulae _(3.2) and _(3.3), the _inequality UIU2U3 &#x3E; 1 should be replaced by (gi g2 - 1)g3 &#x3E; 1.

of the

_integers

_(4.8)

_by

_linearity.

In

_fact,

in

_analogy

with

_(4.10)

we have

so that W3 also _preserves

(4.5)

and

_(4.6).

_{Hence W3} acts on

(4.8)

and

(4.9)

as follows:

We can also

apply

the

_{hypergeometric}

transformation with

_respect

to z.

If in

_(3.15)

we

_change

x into z

_and

into 1 - _xy, and take

we have

Multiplying

by

xh(1 -

and

_integrating,

we

_get

whence

We denote

_{by X}

this

_{hypergeometric}

_integral

_{transformation,}

and

_{by X}

the

corresponding

permutation

acting

on

(4.8)

and

(4.9).

_Again,

_{x preserves}

(4.5)

and

_(4.6),

since

Therefore

The value of the

_quotient

_(4.13)

is

_clearly

invariant under the action on the

integers

(4.8)

and

_(4.9)

of the

_permutation

_group

generated by

and

0’3-The structure of

_~3

can be

_analysed

_by

an

_argument

similar in

_principle

to the discussion on the structure of

~2

made in Section

_3,

_although

more

complicated.

~3

can be viewed as a

_permutation

_group

_acting

on ten

which,

for

_brevity,

we denote

by

_{ul , ... , ulo}

respectively.

In

fact,

the actions of CP3,

X, t9

and _a3 on such

_integers

are the

_following:

I , , ,

and if a

permutation g

E

~3

acts

_identically

on the set

we

_get,

_by

_linearity,

and

_similarly

_2e(j)

=

e(2j)

=

O(U2) - e(u5)

+

g(U9) - e(u4)

+

g(U7) =

2i

(where

we use

_{u9 = j + q = l + s),}

etc.

_{Hence e}

acts

_identically

on the

integers

(4.8),

and therefore also on

(4.9).

Since

(4.14)

are even

permutations

of

_U,

we have an

_embedding

of the group

_~3

in the

_alternating

group

fllio

of all the even

_permutations

of U.

The _group

_{(fP3’ ~9)}

is

_clearly

transitive over

_U,

and hence so is

_~3.

More-over,

43

is

imprimitive

over

U,

with blocks of

imprimitivity given by

the elements of the

_partition

of the set U. In

_fact,

the

_permutations

_x*,

i9* and

₍₇₃

of P defined

_by

are

clearly

induced

by

_~p3, and _~3

respectively,

whence the

mapping

Sp3 t-4 _{x H}

_{x*, ~ ~}

_t9*,

extends to a

homomorphism

.3 ~

65

of the group

~3

into the

_symmetric

_group

_C~5

of all the

_permutations

of ~. Such a

_homomorphism

is

_easily

seen to be

surjective.

Hence,

if we denote its kernel

_by

K,

we have an exact _{sequence of}

_{multiplicative}

_groups:

The structures of the two

_$anking

_{groups in}

_(4.15)

_yield

information on the structure of

_~3,

and in

_particular

determine the order It is easy

to show that K is

_isomorphic

to _{the additive group}

_(Z/2Z)4

_(see

_[RV3],

p.283).

Thus we have

IKI

=

24,

1651

=

5!,

whence

Also,

a further

_argument

shows that

in

_analogy

with

_(3.22).

The rest of this discussion is similar to the one in Section 3. With _any

permutation g

E

_~3

we associate the

_quotient

and if lie in the same left coset of 8 in

_~3,

the

_quotient

_(4.16)

_{for g}

equals

the

_analogous

_quotient

for

_J.

_Also,

we _say

_{that e}

is a

_permutation

of level _v, or that the left coset

_~00

is of level _v,

_if,

after

_simplifying

_(4.16),

we have v factorials in the numerator and in the denominator.

Since

₁₉₃₁

= 1920

16,

there _are 120 left cosets of 8 in

~3,

yielding

120 distinct

_quotients

of

_factorials,

which can be classified as fol-lows

_{(see [RV3], pp. 286-287) :}

We now define

where

is the sequence of the

_integers

_(4.8)

and

_(4.9).

By

(4.11)

we

have,

for any

a

where ag and

bl1

depend only

on the left coset

e4.

For fixed

_{h, j, k, t ,}

_{m, q, r, s,} let

with and

_bn

E Z. From

_(4.11)

we

_get

_dMndNndQnan

E Z. As in

Section

_3,

each transformation formula for

_In

_{corresponding}

to a left coset

of ein

_~3

of level &#x3E; 0

_yields

information on the

p-adic

valuation of the

integer

A.n

=

dMndNndQnan,

thus

_allowing

to eliminate divisors of

A.n

of the

types

p,

p’

or

p3

for suitable

_primes

_g~.

_However,

the

_resulting

arithmetic discussion

clearly depends

on the level of the left coset associated with the transformation formula considered. In Section 5 of

_[RV3]

we show that the choice

satisfying

h + m = k ₊ r

_required,

with M = k =

19,

N

_{= q’}

=

18, Q = j =

17,

yields

_a suitable _set of 15

(among

the

30)

left cosets of level

_4,

such that the

_{corresponding}

15 transformation formulae

for

_In

suff ce for the

_study

of the

_p-adic

valuation of

An .

Using

such 15 transformation

_{formulae, by}

a treatment similar to the one described in Section 3 we define a divisor

_Dn

of

_dMndNndQn

such that

_Dnan

E ?L and

By

(4.17)

we have

and the

_asymptotic

estimates for

_Dn,

_In

and

_Ibnl

in

_(4.20)

_yield

an

ir-rationality

measure of

((3),

again by

applying

the

Proposition

in Section

1.

Let . - -

If the

_integers

_(4.8)

and

_(4.9)

are all

_{strictly positive,}

has

exactly

two

_{stationary points}

and

_{(xl, yl,zi)}

_satisfying

_{x(1 x)y(l}

-z)

0,

with 0 xo, yo, zo 1 and xl, yl, Zl

0,

yn &#x3E;

1

zj

(1-

XIY1)-I.

Then,

in

analogy

with

(3.37)

and

(3.38),

for the

integral

(4.17)

we have

and

With the values

_(4.18)

we

_get

Thus,

using

(4.19),

(4.21), (4.22), (4.23), (4.24)

in

_(4.20),

and

_applying

the

Proposition

in Section

_1,

we find that the choice

_(4.18)

_yields

the

irrational-ity

measure

5. Invariants

As we have seen in Section

_4,

the main new tool in our treatment of

_triple

integrals,

in

_comparison

with the

_theory

for double

_integrals

outlined in

Section

_3,

is

_{represented by}

the three-dimensional birational transformation 3 defined

_by

_(4.2).

In this section we

_give

a constructive method to derive 3

from the two-dimensional birational transformation T defined

_by

(3.2).

Our method

_employs

suitable rational functions and measures invariant under the action of a two-dimensional involution ? related with T.

Denote the

_permutation

ra2

by

T.

_First,

we remark that T = _Tcr2

-(h

k)

is a

_product

of

_{transpositions,}

and hence has

_period

2.

Ac-cordingly,

the

_{corresponding}

birational transformation T = _{(12" :}

(x7 Y)

_~

(X, Y)

given by

is an

involution, i.e.,

has also

period

2. Since T = =

TT2?

for the

permutation

group T we

_obviously

have

Thus the involutions ? and o~2,

together

with the

hypergeometric integral

transformation V2, suffice to obtain the results of Section 3.

Using

an

_{involution ’TJ}

(in

_any

dimension)

has the

advantage

that one can

easily

construct several functions invariant under the action

_{of q.}

For if the

transformation q :

(xl , ... , zn)

F4

_{(X 1, ... , Xn )}

defined

by

the

_equations

Therefore,

if

_{F(u, v)}

_{is any}

_symmetric

function of two variables and

G (t 1, ... ,

is any function of n

_variables,

we

_get

and we conclude that

is invariant under the action of t7. For

instance,

taking

F(u, v)

= _{uv, n} = 2

and

_{G(tl, t2)}

=

_t2,

from the second

_equation

in

(5.1)

_{we see} that the rational

function

is invariant under the action of the involution T.

We want to

_get

three-dimensional birational

_{transformations, acting}

on the Beukers

_type

_integral

_(4.1),

from the two-dimensional birational

trans-formations

_previously

used for the

_integral

_(3.1).

_Therefore,

it is nat-ural to seek a rational function

Z (x, y, z)

such that the transformation

(x, y, z)

e

_{(X, Y, Z)}

obtained

_by

_associating

the involution

_(5.1)

with the

equation

Z =

Z(x,

_y,

z):

maps the open unit cube

(0,1)3

onto itself and satisfies

Thus we demand a solution

Z(x,

_y,

z)

of the differential

equation

(5.3),

with X and Y

_{given by}

_(5.1),

such that

’

Since X and Y are

independent

of _z, the

jacobian

determinant factorizes:

where we denote

the differential of

_Z(x,

_y,

_z)

with

respect

to z

only.

Since the measure

dx dy/(I -

xy)

is invariant

_(up

to the

_sign)

under the action of

_(5.1),

if we

multiply

and divide

_by

1- XY the left side of

_(5.3)

and

_{by 1- xy}

the

_right

side, by

virtue of

_(5.5)

we

_get

the differential

_equation

whence

with C =

C(x, y)

independent

of _z. Therefore

The condition

_(5.4)

becomes

O

i.e.,

for 0 _{x, y, z} 1. In

_particular,

_{taking z}

= 0 and z =

_1,

_we

require

whence,

choosing

either

_sign

for the

exponent,

Since

_(5.1)

is an

_involution,

if we had

(5.6)

we should also have the same

inequality

with

_{(x, y)}

and

_{(X, Y)}

_{interchanged,}

whence XY = _xy. But

this is

_false,

since the function _~y is not invariant under the action of T. Hence the differential

_equation

_(5.3),

with X and Y

_given

_by

_(5.1),

has no solutions

_satisfying

_(5.4).

However,

since the

_jacobian

determinant

is invariant

_(up

to the

_sign)

under the

interchange

of x with 1 - _x, or of X with 1-

_X,

_etc.,

or under _any

permutation

of _y,

zl

or of

ZI,

we can

_apply

_{the method} described above to solve the differential

_equation

obtained

_{by twisting}

_(5.3)

with _any of the above mentioned

_interchanges

or

permutations.

For

_instance,

we _may demand a solution

Z(x,

_y,

z)

satisfying

(5.4)

of the twisted differential

_equation

again

with X and Y

_{given by}

_(5.1).

Since

_(5.2)

is invariant under the action of

_(5.1),

the measure

is also invariant

_(up

to the

_sign).

Hence

_(5.7)

can be written in the form

whence

with C =

C(x, y).

Therefore

From

_(5.4)

we obtain

O

i.e.,

for 0

_{x, y, z}

1.

_{Taking z}

= 0 and z = 1 _we

require

These

_{inequalities yield}

1 1

if we choose the

_exponent

-I-1,

or

if we choose the

_{exponent -1.}

In either case we

_get

is also invariant under the action of

_(5.1).

Thus

and the conditions

_(5.9)

or

(5.10)

yield

or

respectively. By

(5.8)

and

_(5.11)

or

(5.12),

the differential

problem

(5.1)-(5.4)-(5.7)

has the solution

if we choose the -

sign

in

(5.7),

i.e. the

_exponent

+1 in

(5.8),

or the solution

if we choose the +

_sign

in

_(5.7).

If in the transformations

_(x,

y,

z)

H

(X, Y, Z)

obtained

_{by associating}

_(5.1)

with

_(5.13)

or with

(5.14)

we

inter-change

x with 1 - x, X with 1 -

X,

_y with _z, and Y with

Z,

we

_get

the involutions

_(x,

_y,

_z)

~

_{(X, Y, Z)}

_given

_by

or

_by

which _map

_(0,1)3

onto itself

_{and, by}

virtue of

_(5.7),

_satisfy

respectively,

as

_required.

_Denoting

the involution

_(5.15)

_{by ~9,}

the

trans-formation 3

_given

_by

_(4.2)

is obtained as 3 =

ia3, i.e., by interchanging x

and y

in

_(5.15).

We remark that the success of the twist method described above relies on the invariance of

dxdy/(1 - xy) (up

to the

_{sign), (1 - x)y/(1 - xy)}

and

_{(1 - y)/(1 - xy)}

under the action of the involution ? defined

by

(5.1).

6.

_Yasilyev’s

_integral

If we

_{interchange x}

with 1-x in

(1.3)

and

(1.4),

we

_get

the denominators

1 -

_{(1 - x)y}

and 1 -

_{(1 - (1 -}

_x)y)z.

_Therefore,

the

_integral

where the

_polynomial

_{Qk(xl, ... , Xk)}

is

_recursively

defined

_by

is a natural

_{generalization}

of Beukers’

_integrals

(1.3)

and

(1.4).

The above

_integral

_In(k)

was studied

by Vasilyev

[Va],

who

proved

that

and

with

_An,

_Bn,

_Cn,

_{A~,, B~, Cn}

E

_Z,

and that the linear forms

_(6.1)

and

_(6.2)

tend

_{exponentially}

to zero oo.

Moreover,

Vasilyev

conjectured

that,

for

_{any h}

&#x3E;

_2,

with E Z

_{( j}

=

0,1,

... ,

7 h).

In this section we

apply again

our twist

method,

introduced in Section

5,

to obtain four-dimensional birational transformations which can be used to

_investigate

the arithmetical

_properties

of a suitable

generalization

of the

integral

In (4),

where the

_exponents

of the factors 1- _{zi , ... , ~4, 1- x4,}

04(~1,~2,~3,~4)

are not all

equal.

Accordingly,

we seek birational transformations

(x,

_{y, z,}

t)

H

(X, Y,

_Z,

_T)

mapping

the _open unit

_hypercube

_(0,1)~

onto itself and

_satisfying

We start from the involution

_{Q3~92 : (x,}

_y,

_z)

H

_{(X, Y, Z)}

_given

_by

the

equations

which _maps

_(0,1)3

onto itself and preserves

(up

to the

sign)

the measure

Also,

it is

_easily

seen that under the action of

(6.3)

we have the invariants

and

We seek a solution _z,

t)

of the twisted differential

equation

with

_X,

Y and Z

_{given by}

_(6.3),

such that

By

(6.4),

the measure

is invariant

_(up

to the

_sign)

under the action of

_(6.3).

_{Thus, following}

the method of Section

_5,

_(6.6)

can be written as

Therefore .

with C =

C(x,

y,

z).

From

(6.7)

we

_get

o

i.e.,

for 0 _x, y,

z, t

1.

_Taking

t = 0 and t = 1 _we

require

Choosing

either

_sign

for the

_exponent,

we

_easily

obtain the condition

By

(6.5),

this rational function is invariant under the action of

_(6.3).

Hence

by

the method of Section 5 we

_get

C = x/(1 - (1 - xy)z)

and

if we choose the

_exponent

_+1,

or C =

x(1 -

(1 - y)z)

and

if we choose the

exponent -1.

If in the transformations

(x, y, z, t)

H

(X, Y, Z, T)

obtained

_{by associating}

_(6.3)

with

_(6.8)

or with

(6.9)

we

inter-change x

with

_{1- x,}

X with 1-

_X,

and

_apply

the

_permutations

_{( y z t)}

and

(Y

Z

_T),

_by

virtue of

_(6.6)

we conclude that the involutions

(~, y, z, t)

H

or

by

map

(0,1)4

onto itself and

satisfy

or

respectively.

References

[A] R. APÉRY, Irrationalité de _03B6(2) et _03B6(3). Astérisque 61 _(1979), 11-13.

[B] F. BEUKERS, A note on the irrationality of 03B6(2) and _03B6(3). Bull. London Math. Soc. 11

(1979), 268-272.

[DV] R. DVORNICICH, C. _VIOLA, Some remarks on Beukers’ _integrals, in: Colloq. Math. Soc. János _{Bolyai 51, Budapest (1987),} 637-657.

[H1] M. _{HATA, Legendre type polynomials} and _{irrationality} measures. J. _{reine angew. Math.} 407 _(1990), 99-125.

[H2] M. _{HATA, A} note on Beukers’ _integral. J. Austral. Math. Soc. (A) 58 (1995), 143-153.

[H3] M. _{HATA, A} new _{irrationality} measure _for 03B6(3). Acta Arith. 92 (2000), 47-57.

[RV1] G. RHIN, C. VIOLA, On the irrationality measure _{of 03B6(2).} Ann. Inst. Fourier 43 (1993),

85-109.

[RV2] G. RHIN, C. VIOLA, On a _{permutation group} related to _03B6(2). Acta Arith. 77 _(1996), 23-56.

[RV3] G. RHIN, C. VIOLA, The group structure _{for 03B6(3).} Acta Arith. 97 _(2001), 269-293.

[Va] D. V. _{VASILYEV, Approximations of} zero _by linear _forms in values _of the Riemann

zeta-function. Dokl. Belarus Acad. Sci. 45 no. 5 _(2001), 36-40 _(Russian).

[Vi] C. VIOLA, On _Siegel’s method in diophantine approximation to transcendental numbers. Rend. Sem. Mat. Univ. Pol. Torino 53 _(1995), 455-469.

Carlo VIOLA Dipartimento di Matematica Universita di Pisa Via Buonarroti 2 56127 Pisa Italy