Integral identities and constructions of approximations to zeta-values

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

Y

URI

V. N

ESTERENKO

Integral identities and constructions of

approximations to zeta-values

Journal de Théorie des Nombres de Bordeaux, tome

15, n

o

_{2 (2003),}

p. 535-550

<http://www.numdam.org/item?id=JTNB_2003__15_2_535_0>

© Université Bordeaux 1, 2003, tous droits réservés.

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

Integral

identities and constructions

of

_{approximations}

to

zeta-values

par YURI V. NESTERENKO

RÉSUMÉ. Nous

_présentons

une construction

_générale

de combi-naisons linéaires à coefficients rationnels en les valeurs de la

fonc-tion zêta de Riemann aux entiers. Ces formes linéaires sont

ex-primées

en termes

_{d’intégrales complexes,}

dites de

_Barnes,

ce

_qui

permet de les estimer. Nous montrons

_quelques

identités reliant

ces

_intégrales

à des

_intgrales

_multiples

sur le cube unité réel.

ABSTRACT. Some

_general

construction of linear forms with

ra-tional coefficients in values of Riemann zeta-function at

_integer

points

is

_presented.

These linear forms are

_expressed

in terms of

complex integrals

of Barnes _type that allows to estimate them.

Some

_identity

_connecting

these

_integrals

and

_multiple

_integrals

on

the real unit cube is

_proved.

1. Introduction

Apery’s proof

of

_{irrationality}

of

₍₍₃₎

uses an

_elementary

and rather

com-plicated

construction of rational

_{approximations}

_un/vn

_{E Q}

to this number based on a recurrence relation. In

[1]

the

_{following integral interpretation}

of these _{sequences un, vn} was

proposed

Another

presentation

for the same _sequences can be found in

[4]

Here

_r(s)

is Euler

_{gamma-function}

and the contour C is the vertical

_straight

line that

_begins

_{at -1/2 -}

ioo and ends

_{at -1/2}

+ ioo.

In

_particular

the aim of this article is to _prove the coincidence of the

integrals

( 1 )

and

_(2).

Besides we _prove a more

_{general integral}

_identity

Manuscrit recu le 5 juin 2002.

(Theorem 2)

connected to the construction of functional linear forms in

polylogarithmic

functions

In this way since we have the

_equality

_{Lk(1) _ «(k)}

one can construct small linear forms in zeta-values at

_integer

_points,

which are more

_general

than

right-hand

sides of

₍₁₎

and

_(2).

2. General construction of linear forms in

polylogarithms.

Let

_{1, b~ &#x3E;}

_{1, j}

=

1,...,

_m, be

integers.

Define

where q

is a rational number that will be defined later.

Denote

and

Then

The last

_equality

defines the

_{constant -y}

=

b)

_&#x3E; 0.

In what follows an

_important

role

_belongs

to the function

The

_parameter

₍₅₎

is useful for a

_description

of the convergence domain of

(5).

We will assume

that 8 &#x3E; 1.

_2,

then the series

₍₅₎

converges in the circle

lzl

1. In the case 6 =

1,

this series

diverges

at the

point z

= 1.

In the

_sequel

the notation

_Aj

will be used for

_{special segments}

of the real line. Define

For the

_length

of

_Aj

we will use notation

For _any

_{integer i}

define

and denote

The rational function

_R(s)

can be

presented

as a sum of

simple

fractions

where P is the union of sets E

S2.

Note that the

_equality

_d(1)

= 0 is

possible.

For coefficients we have the

expression

where d =

d( i).

Farther

denote q

=

maXtEp

d(e).

Due to

₍₇₎

one can find the

following

equalities

This confirms that the sum

(5)

is a linear form in 1 and

polylogarithms

with

_polynomial

in

_1/z

coefficients. It is clear that

_analogous

result can

be

_proved

if we

_put

as coefficients of the series _any derivative of

R(s)

and

shift the lower limit of summation on _any admissible

integer

number. The

following

Proposition

defines our

general

construction.

Proposition

1. Let a, p be

integers satisfying inequalities

a

aj, j

E

S2,

and p

&#x3E; 1. Then

_for

_{any z}

_from

the _convergence domain

_of

the series

the

_{following identity}

holds

Proof. By

(7)

one can find

and

This proves

Proposition

1.

Note that in the case 6 &#x3E; 2 we have

A1 (1)

=

0,

since

Another reason for this

_equality

is the

_divergence

of

L1

(z)

and the converge-nce of

Lk (z),

k &#x3E;

_2,

at the

_point

z = 1.

Consider some more

_interesting

_partial

choices of

_parameters

_ai,

b~.

The

following

Proposition

describes a construction of simultaneous Pade

ap-proximations

for

_{polylogarithms}

at

_{neighbourhood}

of

_infinity.

Proposition

2. Let r be

_integers,

1 r _m, and

_parameters

_{aj, bj}

following

conditions

For _{any ~,}

_{1 ~}

_r,

_define

a

function

by

the

equality

These

_fqcnctions

have the

_{representation}

In

_particular

and

Note that

_polynomials

do not

_depend

on the index _0. In the case r = 1 and J = 1 the above construction

gives

the Pade

approximation

of the first kind and for r = _{m -} 1 and J = 1 it

gives

the

approximations

of the second kind.

Proof.

In the conditions we have

This is a reason

why

the

equalities

(13)-(15)

follow from

(10)

and

(11).

The

_inequalities

₍₁₆₎

follow from

_(14).

-1)(v)

= 0

-

The

_inequalities

₍₁₇₎

are

valid,

since due to

₍₁₂₎

we have = 0

Proposition

3. Assume that the

_parameters

_aj,

_bj

_satisfy

the conditions

for

some c. Then

for

polynomials

Ak(x),

defined

in

Proposition

1 the

fol-lowing equalities

hold

The condition

₍₁₈₎

in this context was

proposed by

K. Ball and used in

first

_by

T. Rivoal

_[6].

Proof.

Since all

_parameters

are

_integers

and due to

_identity

=

7r , we find

Therefore the function

_R(s)

has the

_property

Since

₍₁₈₎

we derive that for I E P the inclusion c - I E P and

_equality

d(c - 1)

=

d(l)

hold. Moreover when I _runs

through

the _set

P,

also the

number c - I runs

_through

this set. Hence

₍₁₉₎

_implies

that

Compared

together

the last

_equality

and

_(7),

we find

’

Hence

Then and this proves

Proposition

3. 0

In

_particular

the last

_{Proposition implies}

that in case under consideration

with 8 &#x3E; 2 and even

It+ 6

the number

G(1)

is a linear combination of 1 and

values of Riemann zeta-function at odd

_points

with rational coefficients.

But for

_{odd it}

+ 6 the number

_G(1)

is a

_polynomial

in

’1f2

with rational

coefficients.

3.

_{Integral representations.}

For

_applications

of above construction of linear forms one need _upper

bounds for the absolute value of these forms. Here we will find

analytic

representations

for functions see

(9),

which allow in some cases to

find estimates and even

asymptotics

for their values.

3.1.

_{Hypergeometric}

functions. The

_generalized

_{hypergeometric}

func-tion with

_parameters

_{al, ... , a"a,}

_{b2, ... , bum}

E ~ is

_{given by}

the series

where the

_symbol

_(a)o

=

1,

(a)"

=

a(a

+

1) ... (a + v - 1), v &#x3E;

1. Here

one assumes that _ai,

_b~

are distinct from

negative integers.

The series

(20)

absolutely

converges for any

Izl

1 and for

izl

= 1 if _an additional condition

is

_satisfied,

see

(3~.

In this subsection we consider a

_partial

case of the construction

(9),

corresponding

to the

_choice

=

_1,

_a =

Lemma 1. Let

_bl

= 1 al .... a"1,

b2,...,bm

be

_{positive integers}

with

a =

_{al :5}

aj, j

E

S2,

and

_{R(s), Gl (z)}

be

_{functions defined by}

₍₃₎

and

_(9).

Then the

_following

_identity

holds

Proof.

Since

_R(x)

=

0,

1 -

x -1,

_we derive

"

Due to

_r(v+a)

=

r(a).

_0,

the above

_equalities

prove the

identity.

0 In

_particular

for

Lemma 1

_implies

The

_{generalized hypergeometric}

function has several

_integral

represen-tations,

see

[3], [7],

and in this _way it has

analytic

continuation in the whole

complex plain

with some

_cuttings.

_Although

some of the formulae in this

subsection are

classical,

we have included their

proofs

for the convenience

of the reader. In two

_following

lemmas we assume that the

_parameters

aj, bj

are

complex

numbers.

The first one is a multidimentional

generalization

of Euler

representation

for Gaussian

_{hypergeometric}

function.

Lemma 2. Let

_{bl -}

1 and

_{b2, ...,}

_b,.,,,,

be

_complex

numbers

satisfying

.

The

_{integral J1}

_(z)

is

_{defined by}

In these conditions the

_following

assertions hold.

l. The

_integral

_{Jl (z)}

_absolutely

_converges

_for

_any z E

_C,

_{arg(1 -}

_{z) I}

x,.

2. For _{any z,}

_lzl

_1,

we have

3.

_{If 5}

&#x3E;

_1,

see

(21),

we have

(23)

for

_any

complex

_z,

lzl

1.

Proof.

1. The first assertion is

_trivial,

since in the conditions the function 1 - zx2 ~ ~ ~ x"1 is distinct from 0 on the cube

[0;

2. To _prove the second one, see

[7],

subsection

4.1,

we

apply

the

identity

to the function

_{(1 -}

_~2’’’ Since

_{I z}

_1,

the

_{corresponding}

series

uniformelly

converges on the cube

[0;

Hence

and this _proves the

_identity.

3.

_Primarily

let us _prove for _any real

a, 0 a rrz - 1,

the

following

identity

where both the series and the

_integral

converge.

Convergence

of the series is the consequence of

2(m -1) - (m - 1) -

a =

m - a - 1 &#x3E;

0,

see

(7~,

subsection 2.2.

Let us _prove that the

integral

in

(24)

_{converges. Let .~ be real number}

such that 0 À 1. Then we derive

The first

_equality

is obtained

by

the transformation of variables ~Z =

Àt2,

xj =

Since _{the convergence of the series}

₍₂₄₎

one can

_compute

the limit for

a -3 1. This _proves that the

_integral

in

₍₂₄₎

_{converges and the}

_equality

(24)

itself.

Since

_{1 - xj}

1 - _{x2 ~ ~ ~ x",} on the cube

[0,1]~"~,

we obtain

This _{proves the convergence of the}

_integral

_Jl(1).

For _{any z,}

_lzl

_1,

we have

11 -

_{zx2 ~ ~ ~} 1- ~2 ~ - ~ Xm. Therefore the

integral

Jl(z)

uniformely

converges on the set

_lzl

_1,

and is continuous

function on this set. The sum of

(20)

is continuous function too. Hence the

_equality

₍₂₃₎

is valid for _every z with

lzl

1. 0

Lemma 2 demonstrates that the function from Lemma 1 can be

expressed by

Euler’s

_multiple

_integral

_only

in the case when the set

_Si

consists of one element.

_Following

_{representation}

of

_generalized

hypergeo-metric function as a Mellin-Barnes

_type

integral

is valid in less restrictive

conditions on

_parameters,

see

[3],

subsection 5.3.1.

Lemma 3. Let

_bl

= 1 and _{al, ... , a;",,}

b2, ... ,

bm be complex numbers,

a~ ~

0, -1, -2, ... ,

and let the

_{integral I(z)}

be

_{defined by}

the

_equality

there the

_{path of integration}

L is

_{coming from}

-ioo to ioo and

_separates

the

_poles

_of

_r(s

+

_{1 j}

m

from

the

poles of

r(-s).

Besides

Under these conditions the

_following

assertions hold.

1. For _any.z E C such that

_I

_7r,

_{z 0 0,}

the

_integral

_I(z)

absolutely

converges.

2.

_Suppose

z is a

_positive

real number

arg(-z)

= ±7r and 6 _&#x3E;

1,

_see

(21);

then the

_{integrnl I(z)}

_converges

_absolutely.

3. For any z, ~

arg(-z))

I

7r such that both the

integral

I(z)

and the series

₍₂₀₎

_converge the

_{following equality}

holds

Due to

and

see

[81,

we find

where _ci here and later are

_positive

constants

_depending

_only

on _z,.aj,

_bj

and the

_path

L. Since

then

This

_{inequality implies}

the _{absolute convergence of the}

_integral

from

₍₃₂₎

for

_{1 arg( -z)1}

~ and for

_{1 arg( -z)1 =}

x, if we

additionaly

postulate J

&#x3E; 1.

To _prove the last assertion we _suppose

Izl

1 and

1 arg( -z)1

7r. Due

to

_uniqueness

theorem the

_identity

will be true for other

_points

_{of converge}

at

_lzl

1.

Denote T =

N+1/2,

where N is

sufficiently large integer

number. Since

the

_inequality

₍₂₈₎

is satisfied on the contour C

composed

of

_segments

connecting

the

_{points iT, T + iT, T - iT, -iT,}

the

_integral

_fc

_{BII (s )ds}

tends

to zero when N - +00. Then the

_integral

_1(z)

can be evaluated as the sum of the residues of

~(~)

at

_{points (}

=

0,1, 2, ...

taken with the

sign

minus

Since

_r(a

-E-

_n)

=

r(a) . (a)n

the last

identity

proves the assertion. 0

Corollary

1. Let

_bl

= 1 and _{al, ... ,}

b2, ...,

bm

be

complex

numbers

Then

_for

_any z E C under condition

I arg(-z) I

1r the

_{following identity}

holds .

where the

_{path of integration}

L is the same as in Lemma 3.

Suppose

that J &#x3E; 1 and z is a real number 0 z

1,

arg(-z)

=

:l:1r;

then the above

_identity

is

_satisfied.

Proof.

For

_Izi

1 the

_identity

is a _consequence of Lemmas 2 and 3. For

remaining z

it is valid due to

_uniqueness

of the

analytic

continuation. 0

3.2.

_{Hypergeometric integrals.}

Here we will find

_integral

representa-tions for sums defined in

_Proposition

1. These

_{representations}

are

useful in

_applications

for

_computation

of

_asimptotics

for constructed linear forms.

Let r &#x3E; 2 be

_integer

and be

_complex

number. Denote

where L is a

path

in

complex plane

the same as in Lemma 3. It is _easy

to check with the

_inequalities

like

_(28),

that the

_integral

₍₂₉₎

_{converges for}

l3lul

r. The

following

assertion _express the function in terms of

integrals

(29)

and is

_analogous

in this

_respect

to Lemma 3.

Theorem 1. In conditions

_{of Proposition}

1 there exist constants

de-pending only

on _r, such that

for

_{any z} =

e1riu,

l3lul

2,

the

following

inequalities

hold

All the

_integralls

are evaluated on

stright

line Rs =

1/2-a

_comming

bottom-up.

Relations of this kind were used at first

_by

L. A.

_Gutnik,

_[2]

in

_opposite

direction,

and T.

_{Hessami-Pilerhood,}

_[5].

Lemma 4. For _any

_integer

r &#x3E; 2 there exist rational numbers

_da

such that the

_following

_{asymptotic equality}

holds

Proof.

We _prove this assertion

_by

induction on r. For r = 2 it is _true with

do

= 1. Assume that this assertion is true for _some r &#x3E; 1.

Differentiate

₍₃₀₎

and

_apply

formulas

we derive

Last

_equality

_proves the assertion needed. 0

Another

_proof

of Lemma 4 one can find in

[9,

Lemma

2.2.].

Proof of

Theorem 1. With the coefficients

da

introduced in Lemma 4 we

find

where

Lemma 4

_implies

that the function

_U(s)

has 1 as

period,

besides in a

neighbourhood

af _every

_{integer point}

v the

_assymptotic

_equality

holds

The

_integral

in

_right-hand

side of

₍₃₁₎

_equals

to the sum of the residues at

integer points v

&#x3E;

_{1/2 -}

a. Therefore

The last

_{equality implies}

.._1 i

This allows to enforce the inductive

_argument

on r. 0

The

_following

Theorem connects

_multiple

_integrals

of

_special

kind and

complex

integrals

(29).

Theorem 2. Let _{m, r,} 1 r _m, be

_zntegers,

c be

_negate

real number

and _{al, ... am}

_{b2, ...,}

_complex

numbers

_satisfying

Then

_for

any

complex

number z, ~

arg zi

~ the

_{following identity}

holds

where both

_integrals

_converge. Here

and the

_{path of integration}

L is the

_stright

line

_RC

= _c,

coming from

-ioo

to ioo.

Proof. Obviously,

the

_integral

on the left-hand side of

(32)

_converges.

Let us _prove the absolute convergence of the

integral

in

right-hand

side

for

_largzl

7rr.

Denote the function under the

_right

_integral

of

₍₃₂₎

as

~(~).

Then we

have

where

Using

(26), (27)

we obtain

Since on the

path

of

_integration

we

_get

This

_{inequality implies}

_{absolute convergence of the}

_integral

in

_right-hand

side of

₍₃₂₎

for

_{I arg zi}

rx.

According

to Lemma 2 the left-hand side of

₍₃₂₎

can be

expressed

in the

form

Apply

to the inner

_integral

of the

_right-hand

side of

₍₃₄₎

_Corollary

1

with the

_path

L

_given

in conditions of the

_Proposition.

Then we derive

Now one can

change

the order of

_integration

and to make the external

integration

on

_(:

On the

_path

L the

_following

_equalities

hold

Therefore

and we obtain

(32).

For

_{any (6L}

we have

This

_implies

that the

_integral

_converges and does not

_depend

on the variable

(.

In addition the

_integral

absolutely

converges due to Lemma 3. This

grounds

the

possibility

to

change

the order of

_integration

in

_(35).

0

Note that for

_{integer parameters}

_aj,bj

the

_integral

in

_right-hand

side of

₍₃₂₎

_owing

to the relation

_x)

=

4

can be written in the

form

_(29).

And

_conversely

the

_integral

₍₂₉₎

can be

expressed

as

_multiple

with

in the case of convergence.

References

[1] F. _{BEUKERS, A} note on the _{irrationality of 03B6(2)} and 03B6(3). Bull. London Math. Soc. 11 _(1979),

268-272.

[2] L. A. _GUTNIK, The _{irrationality of} certain quantities involving 03B6(3). Acta Arith. 42 _(1983), 255-264.

[3] YU. L. _LUKE, Mathematical functions and their approximations. Academic Press, New York,

1975.

[4] YU. _NESTERENKO, A _few remarks on _03B6(3). Math. Notes 59 _(1996), 625-636.

[5] T. _{HESSAMI- PILERHOOD,} Linear _{independence of} vectors with _{polylogarithmic} coordinates. Vestnik Moscow _University Ser.1 _{(1999), no6,} 54-56.

[6] T. _RIVOAL, La _fonction Zêta de Riemann prend une _infinité de valeurs irrationnelles aux

entiers impairs. C. R. Acad. Sci. Paris 331 _(2000), 267-270.

[7] L. J. SLATER, Generalized _{hypergeometric functions. Cambridge} Univ. Press, 1966.

[8] E. T. WHITTAKER, G. N. WATSON, A course _of modern _{analysis. Cambdidge University}

Press, 1927.

[9] V. V. ZUDILIN, On _{irrationality of} values of Riemann zeta function. Izvestia of Russian Acad. Sci. 66, 2002, 1-55.

Yuri V. NESTERENKO Moscow State _University

Faculty of Mechanics and Mathematics

Vorob’evy gory 119899 Moscow Russia