Well-poised hypergeometric service for diophantine problems of zeta values

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

W

ADIM

Z

UDILIN

Well-poised hypergeometric service for diophantine

problems of zeta values

Journal de Théorie des Nombres de Bordeaux, tome

15, n

o

_{2 (2003),}

p. 593-626

<http://www.numdam.org/item?id=JTNB_2003__15_2_593_0>

© Université Bordeaux 1, 2003, tous droits réservés.

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

Well-poised hypergeometric

service

for

_{diophantine problems}

of

zeta

values

par WADIM ZUDILIN

RESUME. On montre comment les _concepts

_classiques

de séries

et

_{intégrales hypergéométriques}

bien

_{équilibrées}

devient crucial dans l’étude des

_propriétés

_{arithmétiques}

des valeurs de la fonction zêta de Riemann. Par ces _arguments, on obtient

(1)

un _groupe de

_permutations

_pour les formes linéaires en 1 et

03B6(4) =

03C04/90

donnant une

_majoration

conditionnelle de la mesure d’irrationalité de

_{03B6(4) ; (2)}

une récurrence d’ordre deux _pour

_03B6(4)

semblable à celles introduites _{par Apéry pour}

_03B6(2)

et

_03B6(3),

ainsi que des

récurrences d’ordre réduit pour les formes linéaires en des valeurs de la fonction zêta aux entiers

_{impairs ;}

₍₃₎

un _{gros groupe de}

permutations

pour une famille

_{d’intégrales multiples généralisant}

les

_intégrales

dites de Beukers pour

03B6(2)

et

_03B6(3).

ABSTRACT. It is

_explained

how the classical concept of

_well-poised

hypergeometric

series and

_integrals

becomes crucial in

_studying

arithmetic

_properties

of the values of Riemann’s zeta function.

_By

these

_well-poised

means we obtain:

(1)

a

_permutation

_group for linear forms in 1 and

_{03B6(4) =}

_03C04/90

_yielding

a conditional _upper bound for the

_{irrationality}

measure of

03B6(4);

(2)

a second-order

Apéry-like

recursion for

_03B6(4)

and some low-order recursions for

linear forms in odd zeta

_values;

₍₃₎

a rich

_permutation

_group for

a

family

of certain

_{Euler-type multiple integrals}

that

generalize

so-called Beukers’

_integrals

for

_03B6(2)

and

_03B6(3).

1. Introduction

In this

_work,

we deal with the values of Riemann’s zeta function

(zeta

values)

at

_{integral points s}

=

_{2, 3, 4, ....}

Lindemann’s

proof

of the transcendence

of ~r as well as Euler’s formula for even zeta

_values,

summarized

_by

the

inclusions ~ (2n)

E

Q7r2n

for n

_{= 1, 2, ... ,}

_yield

the

_{irrationality}

_(and

tran-scendence)

of

_{~’(2), ~’(4), ~(6), ....}

The

_story

for odd zeta values is not so

complete,

we know

_only

that:

~

~’ (3)

is irrational

(R.

_Apery

[Ap],

_1978);

~

infinitely

_many of the

numbers ((3), ~’ (5), ~ (7), ...

are irrational

(T.

Ri-voal

_{~Ril], [BR],}

_2000);

. at least one of the four numbers

((5), ((7), ((9), ((11)

is

irrational1

-(this

author

_{[Zu3], [Zu4],}

_2001).

The last two results are due to a certain

_well-poised

_{hypergeometric’}

con-struction,

and a similar

_approach

can be

_put

forward for

proving Ap6ry’s

theorem

_(see

_[Ri3]

and

_[Zu5]

for

_details).

After remarkable

_Apery’s

_proof

_[Ap]

of the

_{irrationality}

of

_{both ((2)}

and

~ (3),

there have

_appeared

several other

_explanations

of

_why

it is so; we are

not able to indicate here the

_complete

list of such

_publications

and mention the most known

_approaches:

~

orthogonal polynomials

[Bel], [Hat]

and

Pade-type

approximations

[Be2], [Sol], [So3];

~

multiple Euler-type

_integrals

[Bel], [Hat], [RV2];

~

_{hypergeometric-type}

_series

_{[Gu], [Nel];}

~ modular

_{interpretation}

[Be3].

G. Rhin and C. Viola have

_developed

a new

_{group-structure}

arithmetic

method to obtain nice estimates for

_{irrationality}

measures of

~’(2)

and

((3)

(see

[RV1],

[RV2],

[Vi]).

The

_permutation

_{groups in}

_{[RV1], [RV2]}

for

multi-ple integrals

can be translated into certain

hypergeometric

series and

inte-grals,

and this translation

_[Zu4]

leads one to classical

permutation

_groups

(due

to F. J. W.

_Whipple

and W. N.

_Bailey)

for

_{very-well-poised}

hypergeo-metric series.

The aim of this _paper is to demonstrate

_potentials

of the

_well-poised

hypergeometric

service

_(series

and

_integrals)

in

_{solving quite}

different

prob-lems

_concerning

zeta values. Here we concentrate on the

following

features:

~

hypergeometric permutation

_{groups for}

((4) (Sections 3-5)

and for linear forms in

_odd/even

zeta values

_(Section

_8);

~ a conditional estimate for the

irrationality

measure

of ~ (4)

via the

group-structure

arithmetic method

_{(Section 6);}

. an

_Ap6ry-like

difference

_equation

and a continued fraction

for ((4)

(Section 2)

and similar difference

_equations

for linear forms in odd

zeta values

_{(Section 7);}

’The first record of this _type, at least one of the nine numbers _{((5), C(7), C(21)} is irrational,

is due to T. Rivoal _[Ri2].

2We refer the reader to _[Ba], Section 2.5, or to formula ₍₆₉₎ for a formal _definition, to _[An]

.

Euler-type multiple

_integrals

_{represented very-well-poised}

hypergeo-metric series

_and,

as a _consequence, linear forms in

odd/even

zeta

values

_(Section

_8).

All these features can be considered as a

_part

of the

_general

_{hypergeometric}

construction

_{proposed recently by}

Yu. Nesterenko

_{[Ne2], [Ne3].}

Hypergeometric

sums and

integrals

of Sections 3-6 are

prompted by

Bailey’s integral

transform

_(Proposition

2

_below),

and it is a

_pity

that the

permutation

group for

~(4) (containing

51840

elements!)

leads to an

esti-mate for the

_{irrationality}

measure of

_~(4)

under a certain

(denominator)

conjecture only.

We indicate this

_conjecture

_(supported

_by

our numerical

calculations)

in Section 6. The

_particular

case of the construction is

pre-sented in Section

_2;

this case can be

regarded

as a

toy-model

of that

follows,

and its main

_advantage

is a certain nice recursion satisfied

_by

linear forms

in 1 and

_~(4).

Section 7 is devoted to difference

_equations

for

_higher

zeta

_values;

such recursions make

_possible

to

_predict

a true arithmetic

(i.e., denominators)

of linear forms in zeta values.

The

_subject

of Section 8 is motivated

_{by multiple integrals}

that were

conjecturally Q-linear

forms in

odd/even

zeta values

depending

on

parity

of k

(see

[VaD]).

D.

Vasilyev

[VaD]

required

several clever but

cumbersome tricks to _prove the

_conjecture

for k = 4 and k = 5.

However,

one can see. no obvious

generalization

of

Vasilyev’s

scheme

and,

in

[Zu4],

we

have made another

_{conjecture, yielding}

the old one, about the coincidence

of the

_{multiple integrals}

with some

very-well-poised hypergeometric

series.

We now _prove the

_conjecture

of

[Zu4]

in more

_{general settings}

and

explain

how this result leads to a

permutation

_{group for} a

_family

of

multiple

inte-grals.

Acknowledgements.

I am

grateful

to F. Amoroso and F. Pellarin for

their kind invitation to contribute to this volume of Actes des 12èmes

ren-contres

_{arithmétiques}

de Caen

_(June

_29-30,

_2001).

I am

kindly

thankful’

to T. Rivoal for his comments and useful discussions on the

_subject

and to

G. Rhin for

_pointing

out the reference

_[Co],

where the recurrence for

((4)

was first discovered

by

means of

Apery’s

original

method.

Special

gratitude

is due to E. Mamchits for his valuable

_help

in

_computing

the _group 6 of Section 5 for linear forms in

_1,

_C(4).

2. Difference

_equation

for

₍₍₄₎

In his

_proof

of the

_{irrationality}

of

_((3),

_Ap6ry

consider the sequences un

and vn

of rationals

_satisfying

the difference

_equation

A

_prior£,

the recursion

₍₁₎

_implies

the obvious inclusions

n!3vn

E

_Z,

but a miracle

happens

and one can check

(at

least

experimentally)

the

inclusions

for each n =

_{1, 2, ... ;}

here and

_later,

by

Dn

we denote the least common

multiple

of the numbers

_{1, 2, ... ,}

n

(and

_Do

= 1 for

completeness),

thanks

to the

_prime

number theorem

The _sequence

is also a solution of the difference

_equation

(1),

and it

exponentially

tends

to 0 as n -3 00

(even

after

multiplying

it

by

Dn).

A similar

approach

has

been used for

_proving

the

_{irrationality}

of

_{((2) (see}

_{[Ap], [Po]),}

and several other

_Ap6ry-like

difference

_equations

have been discovered later

_(see,

_e.g.,

[Be4]).

Surprisingly,

a second-order recursion exists for

C(4)

and we are now able to

_present

and _prove it

_by

_{hypergeometric}

means.

Remark.

_During

_preparation

of this

_article,

we have known that the differ-ence

equation

for

C(4),

in

slightly

different

normalization,

had been stated

independently by

V. Sorokin

_[So4]

_by

means of certain

_{explicit Pade-type}

approximations.

Later we have learned that the same but

again

differ-ently

normalized recursion had been

_already

known

_[Co]

in 1981 thanks

to H. Cohen and G. Rhin

_(and

_{Ap6ry’s original}

’acc6l6ration de la

conver-gence’

method).

We underline that our

approach

presented

below differs

from that of

_[Co]

and

_[So4].

We also mention that no second-order recursion

for

_{C(5) and/or}

_higher

zeta values is known. Consider the difference

_equation

with the initial data

for its two

_independent

solutions Un and vn.

Theorem 1. For each n =

0,1, 2, ... ,

the numbers _Un and _vn are

positive

rationals

_satisfying

the inclusions

and there holds the limit relation

Application

of Poincar6’s theorem then

_yields

the

_asymptotic

relations

and

_{(see [Zul],}

_Proposition

₂₎

since the characteristic

_polynomial

a2 -

270A - 27 of the

_equation

₍₃₎

has

zeros

135±78B/3 =

(3f2~)3.

Thus,

we can consider

vn/Un

as

_convergents

of a continued fraction for

((4)

and

making

the

equivalent

transform of the

fraction

_([JT],

Theorems 2.2 and

_2.6)

we obtain

Theorem 2. There holds the

_{following continued-fraction expansion:}

where the

_polynomial

_b(n)

is

_defined

in

_(4).

Unfortunately,

the linear forms

do not tend to 0 as n -~

00.3

A motivation of a

_{hypergeometric}

construction considered below leans on the two series

- - - "I2

(Gutnik’s

form of

_Apery’s

sequence

[Gu],

[Nel]),

and

(Ball’s sequence),

and on the coincidence of these series

proved

by

T. Rivoal

[Ri2],

[Ri3]

with a

help

of the difference

equation

(1).

These

_arguments

make

_possible

to

_give

a new

_{’elementary’}

_proof

of the

irrationality

of

((3)

(see

[Zu5]

for

_details).

Consider the rational function

and the

_{corresponding}

series

In some _sense, the series

(11)

is a mixed

generalization

of both

(8)

and

(9).

Lemma 1. There holds the

_equality

Proof.

The

_polynomials

are

integral-valued

and,

as it is well

_known,

where

_P"(t)

is _any of the

_polynomials

_(13).

The rational function

has also ’nice’ arithmetic

_{properties. Namely,}

Hence, for j

=

_{1, 2, ...}

_we obtain

-Therefore the inclusions

_{(14), (16), (17)}

and the Leibniz rule for

differenti-ating

a

product

imply

that the numbers

satisfy

the inclusions

Now, writing

down the

_{partial-fraction}

_expansion

of the rational func-tion

_(10),

we obtain that the

quantity

has the desired form

₍₁₂₎

with

we deduce that

Un,

Dn U;~’,

E Z as

required.

0

Now,

with a

help

of

Zeilberger’s algorithm

of creative

telescoping

Chapter

6)

we

_get

the rational function

(certificate)

~S’n(t)

:=

sn(t)Rn(t),

where

satisfying

the

_following

_property.

Lemma 2. For each

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

there holds the

_identity

where the

_polynomial

_b(n)

is

_given

in

_(4).

where is

_given

in

_(23).

0 Lemma 3. The

_quantity

₍₁₁₎

_satisfies

the

_difference

_equation

₍₃₎

_{for n =}

1,2,....

Proof.

Since

_Rn(t)

=

O(t-3)

and

Sn(t)

=

O(t-2)

_as t _{-3 oo} for n &#x3E;

_1,

differentiating

identity

(24)

and

_summing

the result over t =

_{1, 2, ..}

_we

arrive at the

_equality

(n

+

_{b(n)Fn -}

3n 3(3n -

_{1) (3n}

+ =

Sn (1)

-It remains to note

_that,

_1,

both functions and

_{Sn(t) _}

sn(t)Rn(t)

have second-order zero at t = 1. Thus

Sn(1)

= 0 for _~c =

1, 2, ...

and we obtain the desired recurrence

(3)

for the

_quantity

(11).

0

Lemma 4. The

_{coefficients Un,}

_Un,

_{Un, Un",}

_Vn

in the

_{representation}

₍₁₂₎

satisfy

the

_difference

_equation

₍₃₎

_for

n =

1, 2, ....

Proof.

Write the

_{partial-fraction expansion}

₍₂₀₎

in the form

where the formulae

₍₁₈₎

remain valid for all J~ E Z

_{a,nd j = 1, 2, 3, 4.}

Mul-tiply

both sides of

₍₂₄₎

_by

_(t

+

k)4,

take

_{(4 -}

_j)th

derivative of the

result,

substitute t = -k and _{sum over} all k _E

Z;

this

procedure

yields

that,

for

each j

=

1, 2, 3, 4,

the numbers

(21)

written as

satisfy

the difference

_equation

_(3).

_Finally,

the sequence

also satisfies the recursion

_(3).

0

Since

’

- .. ,

-in accordance with

_{(21), (22)}

we obtain

hence as a _consequence of Lemma 4 we arrive at the

following

result.

The sequences ~c" :=

Un/6

and vn

:=

Vn/6

satisfy

the difference

equa-tion

₍₃₎

and initial conditions

_(5);

the fact

_IFni

2013~ 0 as n - _oo, which

_yields

the limit relation

_(7),

will be

_proved

in Section 4. This

_completes

our

proof

of Theorem 1.

The conclusion

₍₆₎

of Theorem 1 is far from

_{being precise;}

in

_fact,

(ex-perimentally)

there hold the inclusions

and,

moreover, there exists the sequence of

positive integers

0,1, 2, ... ,

such that

This sequence can be determined as follows: if _vp is the order of

_prime

_p in

(3n)!/n!3,

then

here and below and

_{x}

:= x -

LxJ

denote

respectively

the

integral

and

fractional

_parts

of a real number x. For

primes

_p &#x3E; we obtain the

explicit

(simple)

formula

hence

where :=

r’(x)/r(x).

Thus,

we obtain that the linear forms

do not tend to 0 as n ~ oo.

3.

_Well-poised

_{hypergeometric}

construction Consider the set of

_{eight positive}

_integral

_parameters

and

_assign

to h the rational function

In the last

_{representation}

we

_pick

out the rational functions

of the form

_{(15), (13),}

_having

some nice arithmetic

_properties

_([Zu4],

Sec-tion

_7).

’

It is _easy to

_verify

_that,

due to

_(26),

for the rational function

₍₂₈₎

the difference of numerator and denominator

_degrees

is

_equal

to

_3,

hence

The series

Lemma 6. The

_quantity

_{F(h) is}

a linear

form

in 1 and

((4)

with rational

coefficients.

Proof.

Order the

_parameters

_{hl, ... ,}

_h6

as

_{hi . ~ ~ hs}

and consider the

partial-fraction expansion

of the rational function

_(28):

where

Then we obtain

with

and the

_{well-poised origin}

of the series

_{(30) (namely,}

the

_property

_R(t

-ho) =

-R(t),

hence

_Ajk

=

by

(32),

cf.

[Zu4],

Section

8,

with r = 2 and

q = 6)

yields A2 = A4 = 0,

while the residue _sum theorem

accompanied

with

₍₂₉₎

_{implies Al}

= 0

(cf.

[Nel],

Lemma

1).

0

Remark. The

_question

of denominators of the rational numbers

_A3

and

_Ao

that _appear as the coefficients in

F(h)

can be solved

_{by application}

of

Nesterenko’s denominator theorem

_[Ne3]

_(announced

by

Yu. Nesterenko in his Caen’s

_talk).

_Namely,

consider the set

where

mi &#x3E; " - &#x3E;

m5 are the five successive maxima of the set N.

Unfortunately,

we have not succeeded in

_using

the inclusion

₍₃₃₎

for arithmetic

_{applications; actually,}

our

_experimental

calculations show that

the

_stronger

inclusion for the linear forms

_F(h),

indicated at the

_beginning

of Section

_6,

holds.

Using

standard

_arguments,

the

_property

₍₂₉₎

and the fact that

_R(t)

has second-order zeros at

_integers

t = 1 -

h_i},

I

one deduces the

_following

_{hypergeometric-integral representation}

of the

se-ries

_(30).

Lemma 7

_{(cf. [Nel],}

Lemma

_2).

There holds the

_equality

with

_{any tl}

_{1 - hi}

_tl

The series

₍₃₀₎

as well as the

corresponding hypergeometric

integral

(34)

are known in the

_theory

of

_{hypergeometric}

functions and

integrals

as

very-well-poised objects,

i.e.,

one can

split

their

_top

and bottom

_parameters

in

pairs

such that

and the second

_parameter

has the

_special

form 1

Remark..

As

it is

_easily

_seen, the _sequence

_Fn

of Section 2

_corresponds

_(after

a suitable shift of the summation

_parameter

t)

to the choice

of the

_parameters

h. Hence the

_equalities

_{Un -}

_U~’

= 0 in the

representation

(12)

can be deduced from Lemma 6.

4.

_Asymptotics

We take the new set of

_positive

_parameters

and for each n =

0,1, 2, ...

relate them _to the old

_parameters

by

the

for-mulae

Then Lemma 6

_yields

that the

_quantities

_Fn

= _:=

F(h)

are linear

forms in 1 and

₍₍₄₎

with rational

_{coefficients,}

say

and the

_goal

of this section is to determine the

_asymptotic

behaviour of these linear forms as well as their coefficients _un

and vn

as n - oo.

To the set

₍₃₆₎

_assign

the

_polynomial

and the function

defined in the cut

_{T-plane C}

_B

_’1-I}]

U where

"1i ~ ’12 ~ ... ~ "16

denotes the ordered version of the _{set 771, ’TJ2, ... , ’16.} The first condition in

₍₃₇₎

_implies

that

₍₃₉₎

is a

fifth-degree polynomial;

moreover, the

symmetry

under substitution _{"10 - T} and the second condition in

₍₃₇₎

_yield

that this

_polynomial

has zeros

The last four zeros can be

easily

determined

by

solving

a certain

_biquadratic

(in

terms of

_{r~o/2 - T)}

_equation.

Set and

Proposition

1. The

_follourlng

limit relations hold:

Proof.

The

_proof

is based on

_application

of the

_saddle-point

method to

the

_{integral representation}

of Lemma 7 for the

_quantities

_F"

and a similar

integral

representation

(see

formula

_{(48) below)}

for the coefficients U,,; the

fact that both limits in

₍₄₂₎

are

equal

follows

immediately

from the limit

relation

’I.,

-since

_-Co

0

_Cl

under the conditions

_(37).

Without loss of

_generality,

we will restrict ourselves to the ’most sym-metric’ case

(35),

_i.e.,

that

_corresponds

to the linear forms in

_{1, C(4)}

constructed in Section 2. In the case

(43),

the zeros

(40)

of the

_{corresponding polynomial}

(39)

are as follows:

By

Lemma

_7,

with _any ti E

_1R,

-n

_ti

0.

_Using

the

_asymptotic

formula

for z E C with Re z = _{const &#x3E;}

_0,

taking

_ti = _-nTo and

changing

variables

t =

as n -+ _00, where

Since

and To is a zero of the

_polynomial

(39)

(which

is

(T-3)2(T-1)6-T2(T-2)s

in the restricted

_case),

we conclude that

f’(TO)

= 0 and _To is the

unique

maximum of the function Re

_{f (T)}

on the contour. Thus the

_integral

₍₄₄₎

is determined

_by

the contribution of the

_saddle-point

To

(see

[Br],

Section

5.7):

hence

This _proves the limit relation

_(41).

In the

_{neighbourhood}

of t

_{= -k,}

where k =

n + 1, ... , 2n+ 1,

the function

R(t)

has the

_expansion

by

(31).

On the other

hand,

and if ,C is a closed clockwise contour

_surrounding

_{points t}

= -n -

_{1, ... ,}

-2n -

_1,

then

Taking

the

_rectangle

with vertices

_{±it2 ±}

_N,

for some fixed real

_t2

&#x3E; 0 and any N &#x3E; 2n +

_1,

as the contour ,C and

_using

the estimates

on the lateral sides of the

_rectangle,

from

(47)

we deduce that

where the constant in

_O(N-2)

_depends

on

_t2

_only.

_Tending

N - oo and

making

the substitution t H

_{-t - ho}

= _{-t -}

(3n

₊

2)

in the first

integral,

we obtain

(cf. [Zu2],

Lemma

_3.1).

_Finally,

take

t2

= _-nsi =

-nImTl,

change

the

variable t = _-nT and

apply

the

asymptotic

formula

Since

for r E C with Im T = _{S 1 &#x3E;}

0,

we obtain

By

(45)

and the definition of the

_point

71

(that

is the zero of the

polyno-mial

_(39)),

hence

_{/’(71) -}

_4xiTi

=

_0,

_we conclude that ₇ = ₇₁ is the

_unique

maximum of the function on the line Im T = _sl.

Therefore,

the

_saddle-point

method _says that the

_asymptotics

of the

_integral

in

₍₄₉₎

is determined

_by

the contribution of the

_point

T = _Tl that

yields

the desired

limit relation

The

_proof

of

_Proposition

1 is

_complete.

0

Remark. The limit relation

₍₄₆₎

_yields

that -~ 0 as n - _oo, and this is the fact that we have

promised

_{to prove} for Theorem 1

(see

the

paragraph

after Lemma

_5).

To be

_honest,

the

_fact,

that the

_asymptotics

of the linear forms and their coefficients in the case

(35)

is determined

by

the zeros

(3

t

_2v’3)3

of a

quadratic

polynomial

with

integral

coefficients,

_gave us the

idea to look for a second-order difference

equation.

5.

_Group

structure for

₍₍₄₎

This section can be viewed as a continuation

of

the

_story

in

[Zu4],

Sec-tions

_4-6,

where we

_explain

the Rhin-Viola _group structures for

~(2)

and

Proposition

2

_(Bailey’s

_integral

_transform

_[Ba],

_Section

_6.8,

_formula

_(1)).

There holds the

_identity

where k = 1 + 2a - _{c -} d -

e, and the

parameters

are connected

_{b y}

the

relation

By

Lemma 7 the transform

₍₅₀₎

_{rearranges the}

_{parameters h}

as follows:

Consider the set of 27

_{complementary}

_parameters

_e,

and set

Then

_Bailey’s

transform can be written as follows:

where 6 from

₍₅₁₎

is the

_following

second-order

_permutation

of the

param-eters

_(52):

Further,

the h-trivial _group

_(i.e.,

_{the group of}

_permutations

of the

pa-rameters

_{hl, h2, ... ,}

_h6)

is

_generated

_by

second-order

permutations

of

_hk,

1 k

_5,

and

_h6.

The action of these five

_permutations

on the set

₍₅₂₎

is

as follows:

and the

_quantity

(due

to the definition

₍₂₈₎₎

is stable under the action of

_(56).

_Setting

(58)

6 =

_{6 (e) :}

1"

(e01 ,

_{eo2, eo4, e06 ,}

_e02,

_e03,

_e05,

_{e12, e15, e24 ,}

e36 }

and

_combining

the above

_stability

results we arrive at the

_following

fact. Lemma 8. The

_quantity

is stable under the action

_of

the _group

Moreover,

the

_{quantities h-l}

and

are also 0-stable.

Proof.

Routine calculations show the

_stability

of

_H(e)/II(e)

under the

ac-tion of

_b,

_{~2, ~3, ~14, ~5}

with a

_help

of

(55)

and

(57).

Hence

H(e)/II(e)

is

stable under the action of the

e-permutation

group

generated

by

these six

permutations

(54), (56).

The

_stability

of

_{h_1}

under the action of

₍₅₆₎

is

_obvious,

and b does not

change

the

_parameter

_{h_1}

_by

_(51).

_Finally,

that

_yields

the

_stability

of

_E(e)

under the action of 6. The

_proof

is

With the

_help

of a C++ _program we have discovered that the

_{group 6}

con-sists of 51840

_elements,

hence the left factor includes

_51840/6!

= 72

left

_cosets;

here

_6e

is _{identified with the h-trivial group}

_{b2 , b3,}

_4,

_5).

It is

_interesting

to mention that the _group

₆₀

_{acting trivially}

on the set

₍₅₈₎

consists of

_just

4 elements: go =

id,

Rerraark. In the most

_symmetric

case

(35)

all

complementary

_parameters

(52)

are

_equal

to n that means that _any

_permutation

from 0 does not

change

the

_quantity

_F(la).

This fact

_{explains why}

do we dub this case as

tmost

_{symmetric’.}

6. Denominators of linear forms

As we have mentioned in Remark to Lemma

6,

‘trivial’ arithmetic

(33)

of the linear forms

_H(e)

=

F(h)

does not lead us to a

qualitative

result

for

_((4).

We are able to estimate the

irrationality

measure of

~(4)

under the

following

condition,

which we have checked

numerically

for several values

of h

_satisfying

₍₂₆₎

and

_(27).

Denominator

_Conjecture.

There holds the

inclusion4

where

_{mi &#x3E; m2 &#x3E; m3 &#x3E;}

rn4 are the

four

successive maxima

of

the set e

in

₍₅₂₎

and

with

If this

_conjecture

is

_true,

then

_taking

_any element _g e 6 and

_writing

conclusion of Lemma 8 as

we deduce

_that,

for _any

_prime

_p &#x3E;

where

_9C

=

6 (ge)

and

ordp (u( (4) -

v)

:=

min{ordp u, ordp v}

for rational

numbers _u, v.

Finally,

setting

with

from

₍₅₉₎

we obtain the inclusion

Now,

to each =

0,1, 2, ... assign

the

_parameters

h in accordance with

₍₃₈₎

and set

so that the set of

_{complementary}

_parameters

e ~ n

corresponds

to the set h.

Then,

in the above

_notation,

we can write the inclusion

(60)

as

The

_asymptotic

behaviour of the linear forms

_H(en)

E

_Q((4)+Q

and their coefficients as n - oo is determined

by Proposition

1;

in

addition,

by

the _consequence

₍₂₎

of the

_prime

number

_theorem,

while the arithmetic lemma of

_{Chudnovsky-Rukhadze-Hata}

_(see,

_e.g.,

_[Zu2],

Lemma

_4.4)

_yields

where

is a

1-periodic

function.

Recalling

the notation of

_Proposition

1 and

_combining

its results with

saying

above,

as in

[RV2],

the

proof

of Theorem

_5.1,

we arrive at the fol

-lowing

statement.

Proposition

3. Under the denominator

_conjecture,

let

If Co

&#x3E;

C2,

then the

_{irrationality}

_exponent

_of

_C(4)

_satisfies

the estimate

Recall that the

_{irrationality}

of a real irrational

num-ber a is the least

possible

_exponent

such that for _any e &#x3E; 0 the

_inequality

has

_only

_finitely

many solutions in

integers

p, q with q &#x3E; 0.

With a

_help

of

_Proposition

3 we are able to state the

_following

conditional result.

Theorem .3. The

_{irrationalaty}

_exponent

_of

_C(4)

_satisfies

the estimate

provided

that the denominator

_conjecture

holds.

Proof. Taking q

=

(68,57;

22,23,24,25,26,27)

we obtain

and .

= 27 + 26 + 25 + 24 - 69.76893283... = 32.23106716....

The estimate

₍₆₁₎

can be

_compared

with the ’best known’ estimate

which follows from the

_general

result of Yu. Aleksentsev

_[Al]

on

approxi-mations of ~r

by

algebraic

numbers.5

7. Further difference

_equations

for zeta values A natural

_{very-well-poised generalization}

of Ball’s _sequence

_(9),

where n =

1, 2, ... , gives

rise for

searching

difference

equations

satisfied

by

both linear forms and their rational coefficients.

_{Applying Zeilberger’s}

algorithm

of creative

_telescoping

in the manner of Section 2 we deduce the

following

result for the linear forms

Theorem 4. The numbers _{un, 7 Wn vn} in the

_{representation}

₍₆₃₎

are

_positive

rationals

_satisfying

the third-order

_difference

_equation

with

5In fact, the result of _[Al] is _proved for approximations of 1r by algebraic numbers of sufficiently large degree.

where

The characteristic

_polynomial

A3 _ 188A~ -

2368A + 4 of the difference

equation

(64)

determines the

_asymptotic

behaviour of the linear forms

₍₆₃₎

and their coefficients as n - oo.

A similar

_(but

_quite

_cumbersome)

fourth-order recursion with character-istic

_{polynomial a4 -}

828A - 132246A

+ 260604A - 27 has been discovered

by

us for the linear forms

_F7,n

and their coefficients. These recursions allow us to

_verify

the inclusions

up to n =

1000,

although

_{we are} able _to

prove that

where

using

our arithmetic results

[Zu2],

Lemmas 4.2-4.4.

Another

_story

deals with the

_quantities

where

_{i4,, Wn,}

_vn

aare

_positive

rationals. We have discovered a

(quite

cum-bersome)

fourth-order difference

_equation

satisfied

_by

_i4,,

_wn,

_vn;

its char-acteristic

_polynomial

is

where

_On

is

_given

in

_(66),

while our calculations _{up to n} = 1000 with _a

help

of the recursion mentioned above show that

What is a trick that makes arithmetic as it is?

8.

_{Multiple-integral representation}

_

of

_{very-well-poised}

_{hypergeometric}

series

In

_[Zu4],

Section

_9,

we

_conjecture,

for

_integer

k &#x3E;

_2,

the coincidence of

the

_{very-well-poised hypergeometric}

series

₍₆₂₎

and the

_{multiple integral}

where

_Qo

:= 1 and

for k &#x3E; 1. The

_integrals

_J2,n

and

_J3,n

have been studied

_by

F.

Beuk-ers

[Bel]

in the connection with

_Ap6ry’s

_proof

of the

_{irrationality}

of

₍₍₂₎

and

_((3).

In

_(Zu4~,

we _{prove the coincidence of}

F3~n

and with the

help

of

Bailey’s

identity

([Ba],

Section

_6.3,

formula

₍₂₎₎

and Nesterenko’s

_integral

theorem

_([Ne2],

Theorem

_2),

and use similar

_arguments

for

showing

that

F2,n

=

J2,n.

For

general

integer

k &#x3E;

2,

the

integrals

(67)

are introduced

_by

O. Vasilenko

_[VaO]

who states several results for

_Jk,o.

The cases k =

_4,5

and an

_{arbitrary integer n}

in

(67)

are

_{developed by}

D.

_Vasilyev

[VaD];

in

particular,

he

_conjectures

the inclusions

(cf. (65)),

and proves them if k = 5.

There is a

regular

_way to obtain difference

equations

for the

quanti-ties

_(67);

it is a

_part

of the

_general

_{WZ theory developed by}

H. Wilf and

D.

_Zeilberger

_[WZ].

_However,

difference

equations

for and

_JS,n

_by

these

means are out of calculative abilities of our

_computer,

so we cannot use a

troutine matter’ to

_verify

the

_identity

_Fk~n

= even when J~ =

4, 5.

The aim of this section is to deduce the desired coincidence of

₍₆₂₎

and

₍₆₇₎

from a

_{general analytic}

result on a

multiple-integral

_{representation}

of

_{very-well-poised hypergeometric series.}

6As

it is mentioned _by G. E. Andrews in _[An], Section 16, "an _{entire survey paper} could be written just on _integrals connected with _well-poised series" . The _following theorem would extend this survey a little bit.

Consider two

_objects:

_{very-well-poised}

_{hypergeometric}

series

and

_multiple

_integrals

Theorem 5. For each k &#x3E;

_1,

there holds the

_identity

provided

that

Remark. Condition

₍₇₂₎

is

_required

for the absolute convergence of the series

₍₆₉₎

in the unit circle

_(and,

in

_particular,

at the

_point

_(-1)~+i),

while condition

₍₇₃₎

ensures the convergence of the

corresponding multiple

integral

(70).

The restriction

₍₇₄₎

can be removed

by

the

theory

of

analytic

continuation if we write

r(hj

+

for j

=

_1,

k ₊ 2 _as Pochhammer’s

symbol

when

_summing

in

_(69).

In the case of

integral

_parameters

_h,

the

quantities

(69)

are known to be

Q-Iineax

forms in

_even/odd

zeta values

_depending

on

_parity

of k &#x3E; 4

_(see

[Zu4],

Section

_9).

_Therefore,

if

_positive

_integral

_parameters

a and b

satisfy

then the

_quantities

₍₇₀₎

are

_Q-linear

forms in

even/odd

zeta values.

Spe-cialization + 1 and

_b?

= 2n ₊ 2

gives

one the desired coincidence

of

₍₆₂₎

and

_(67).

The

_{choice aj}

= _rn + 1 and

bj = (r

+

_1)n

+ 2 in

_{(70) (or,}

equivalently,

ho

=

(2r+1)~+2

and hj

= rn+1

for j

=

1, ... ,

k+2 in

(69))

with the

_integer

r &#x3E; 1

_depending

on a

_given

odd

_integer

k

_presents

almost

the same linear forms in odd zeta values as considered

_by

T. Rivoal in

[Ril]

for

_proving

his remarkable result on infiniteness of irrational numbers

in _

the set

_{((3), ((5), ((7), ....}

In

_addition,

we have to

_mention,

under

_hypothesis

_(75),

the obvious

stability

of the

_quantity

under the action of the

_(h-trivial)

_group

_6k

of order

_(k

+

_2)!

_containing

all

permutations

of the

_parameters

_{hi, ... ,}

This fact can be

applied

for

number-theoretic

_applications

as in

[RV1], [RV2]

and Sections

_5,

6 above. In

the cases k = 2 and lk = 3 the

change

of variables

(Xk- 17

Xk)

_r-

(1-

xk,1-xx-1)

in

₍₇₀₎

_produces

an additional transformation G of both

(70)

and

(69);

for k &#x3E; 4 this transformation is not

_yet

available since condition

₍₇₅₎

is broken. The groups

(02, b)

and

_{(Ø3, b)}

of orders 120 and 1920

_respectively

are known: see

[Ba],

Sections 3.6 and

_7.5,

for a

_{hypergeometric-series}

_origin

and

_[RV1],

_[RV2]

for a

multiple-integral

explanation.

G. Rhin and C. Viola

make a use of these groups to discover nice estimates for the

irrationality

measures of

₍₍₂₎

and

_((3).

_Finally,

we want to note _{that the group}

_6k

can

be

_{easily interpretated}

as the

_permutation

_group of the

_parameters

as in Section 5

(see

[Zu4],

Section

9,

for

details).

Lemma 9. Theorerrc 5 is true

_{if k}

= 1.

Proof.

Thanks to a

limiting

case of

Dougall’s

theorem,

(see,

e.g.,

[Ba],

Section

4.4,

formula

(1)),

provided

that &#x3E;

_Re(hi

+

hand,

the

_integral

on the

right

of

(71)

has Euler

_type,

that is

provided

that 1 +

_{R.e ho}

&#x3E;

_Re(h1

+

_h2

+

_h3)

and

_{Re h2}

&#x3E; 0.

_{Therefore, .}

multiplying

equality

(76)

by

the

_{required product}

of

_{gamma-functions}

we

deduce

_identity

₍₇₁₎

if k = 1. 0

Renaark. If we _{arrange about}

Jo(ao)

to be

_1,

the claim of Theorem 5 remains valid if k = 0 thanks _to another

consequence of

Dougall’s

theorem

([Ba],

Section

_4.4,

formula

_(3)).

Lemma 10

_([Ne2],

Section

_3.2).

Let and

_to

E lf8 be numbers

satisfying

the conditions

R,e ao &#x3E; ta &#x3E; 0,

R,e a &#x3E; to &#x3E; 0,

and

_{Reb&#x3E; Reao+Rea.}

Then

_for

_any non-zero z E

_{C B (1, +00)}

the following

_identity

holds:

where

_{(-z)t =}

_{Izlteitarg(-z),}

_~

arg(-z)

1r

for

z

E C B [0,+(0)

and

arg(-z)

= f:1r

for

z E

_(0,1~.

The

_integrals

on the

right-hand

side

of

(77)

converges

absolutely.

In

addition, if

lzl

1,

both

integrals

in

(77)

can be

identified

with the

_absolutely

_convergent

Gauss

_{hypergeorreetric}

series

Set ék

= 0 for k even

_{and Ek}

= 1 or -1 for k odd.

Lemma 11. For each

_integer

k &#x3E;

_2,

there holds the relation

provided

that

_{Re ao}

&#x3E; to &#x3E;

_{0, Re ak &#x3E;}

to

&#x3E;

_{0, Re bk}

&#x3E;

_{Re ao}

-E-

_Ileak,

and the

_integral

on the

_left

_converges.

Proof.

We start with

_mentioning

that the first recursion in

₍₆₈₎

and induc-tive

_arguments

_yield

the

_inequality

By

the second recursion in

_(68),

_Qk

=

Qk-1 -

(1 - zxk)

for k &#x3E;

2,

where

For each

_{(xl, ... , xk-1)}

E

(o,1)k-1,

the number z is real with the

property

z 0 for J~ even and 0 z 1 for J~

_odd,

since in the last case we have

by

(78).

Therefore,

splitting

the

_integral

₍₇₀₎

over

[0,

1]k

=

[0,

1]k-1

X

_{[0, 1]}

and

_applying

Lemma 10 to the

_integral

we arrive at the desired relation. 0

Proo, f of

Theorem 5. The case k = 1 is considered in Lemma 9. Therefore

we will assume that k &#x3E;

_{2, identity}

₍₇₁₎

holds for k -

_1,

_and,

in

_addition,

that

The restrictions

₍₇₉₎

can be

_easily

removed from the final result

by

the

theory

of

_analytic

continuation.

where the real number _so &#x3E; 0 satisfies the conditions

and the absolute convergence of the last

Barnes-type

integral

follows from

_[Ne2],

Lemma 3.

_Shifting

the variable t + s in

_{(80) (with}

a

_help

of the

_equality

eEIc1rit .

eEIc-l1ris =

eA,-,7ri(t+s) -

_e6l"t),

_applying

Lemma

_11,

and

_{interchanging}

double

_integration

_(thanks

to _{the absolute convergence}

of the

_integrals)

we conclude that

where sl - so +

_to.

Since

_{R,e hk+2}

1 and

_{0, -1, -2, ... ,}

the last

Substituting

this final

_expression

in

₍₈₁₎

we obtain

If k is _even, we take = -1 in the first

integral

and _êk-l = 1 in the

second one. Therefore the both

_integrals

are

equal

to

that

_gives

the desired

_identity

_(71).

The

_proof

of Theorem 5 is

_complete.

D

Another

_family

of

_multiple

_integrals

is known due to works of V. Sorokin

_{[So2], [So3].}

Recently,

S. Zlobin

_[Zll],

[ZI2]

has

_proved

_(in

more

general

settings)

that the

integrals

(70)

can be

reduced to the form

₍₈₂₎

with z = 1.

Therefore,

Theorem 5

gives

one a

way to reduce the

integrals

S(l)

to the

very-well-poised

hypergeometric

series

₍₆₉₎

under certain conditions on the

_parameters

_aj,

_b~,

_c~, and _ri

satisfying

natural restrictions of convergence, the

integral

S(z)

is a

Qfz-’]-linear combination of modified

_{multiple polylogarithms}

Following

a

_spirit

of this

_section,

we would like to finish the _paper with the

_following

Problem. Find a

multiple integral

over

(0,1~5

that

_represents

the series

defined

in

_{(30) (or,}

_{equavalently,}

the

_integral

₍₃₄₎₎

_of

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Wadim ZUDILIN

Department of Mechanics and Mathematics Moscow Lomonosov State _University

Vorobiovy Gory, GSP-2 119992 Moscow Russia

ML: _{http://vain.0153d.ras.ru/index.html} E-mail : _{vadimQips .} ras . ru