Renormalization of Multitangent Functions and Applications. Combinatorics of Mathematical Renormalization: a special day.

Renormalization of Multitangent Functions and Applications.

Combinatorics of Mathematical Renormalization: a special day.

Olivier Bouillot, Paris XI University.

IHES , February 27

, 2012 .

Plan

1 Reminders on Multizeta Values.

2 Introduction to Multitangent Functions.

3 Multitangent Functions Renormalization.

4 Relations Between Multizeta Values Obtained via the Study of Multitangent Functions, for Small Weights.

5 Some Conjectures.

Reminders on Multizeta Values: Definitions.

Let : S

= {s ∈ seq( N

) ; s

≥ 2} .

Multizeta Values Definition.

Let s ∈ S

.

We put: Ze

= Ze

= X

1≤n_r<···<n₁<+∞

1 n

· · · n

.

Today, multizetas values have deep connections with many other mathematical topics:

1

Number theory.

2

Quantum groups, knot theory or mathematical physics.

3

Resurgence theory and holomorphics invariants.

4

Feynman diagrams.

5

P

− {0 ; 1 ; ∞} (with Grothendieck-Ihara’s program) .

6

Absolute Galois group of Q .

Reminders on Multizeta Values: Integral Representation.

Multizeta values can be represented by two different ways:

by a sum or by an iterated integral.

Example: ∀n ∈ N

, 1 n

=

Z

Z

v

dv du

u .

+∞

X

n=1

1 n

=

Z

Z

+∞

X

n=1

v

dv

! du u =

Z

Z

dv 1 − v

du u .

More generally, we let: Wa

= Z

0<t₁<···<t_r<1

ω

· · · ω

,

where ω

= dz

z and ω

= dz

1 − z .

Then: Ze

= Wa

.

Reminders on Multizeta Values: the Two Multiplication Tables 1/2 .

Shuffle product or symmetrality. Quasi-shuffle product or symmetrelity.

Consider the alphabet X = {x

; x

} . Consider the alphabet Y = {y

; i ∈ N } . Fact : S

' x

X

x

: Fact : S

' {ω = y

ω e ∈ Y

; i ≥ 2} :

s 7−→ x

= x

x

· · · x

x

. s 7−→ y

= y

· · · y

. We define ζ : x

X

x

−→ C by We define ζ : x

X

x

−→ C by

ζ(x

) = Ze

, and we extend it by linearity ζ(y

) = Z e

, and we extend it by linearity to Q ε ⊕ x

Q <X> x

. to Q ε ⊕ {y

; i ≥ 2} Q <Y > .

Shuffle product: Stuffle product:

We define recursively by: We define ? recursively by:







ε ω = ω ε = ω .

(x

ω

) (x

ω

) = x

ω

(x

ω

) + x

(x

ω

) ω

.



 



 



ε ? ω = ω ? ε = ω .

(y

ω

) ? (y

ω

) = y

ω

? (y

ω

) + y

(y

ω

) ? ω

+ y

(ω

? ω

) . Fact : For (ω

; ω

) ∈ x

Q <X> x

, Fact : For (ω

; ω

) ∈ {y

; i ≥ 2}Q <Y >

ζ(ω

)ζ(ω

) = ζ(ω

ω

) . ζ (ω

)ζ(ω

) = ζ(ω

? ω

) .

Reminders on Multizeta Values: the Two Multiplication Tables 2 / 2 .

Shuffle product or symmetrality. Quasi-shuffle product or symmetrelity.

Example: Example:

Ze

Ze

= Ze

+ Ze

+ Ze

. Ze

Ze

= Ze

+ 3Ze

+ 6Ze

.

So:

Ze

= 2Ze

+ 6Ze

.

Reminders on Multizeta Values: Renormalization.

Lemma :

For all θ ∈ C , there exists a unique symmetrel extension of Ze

to seq( N

) , such that Ze

= θ .

Remark : We can choose θ = 0 for simplicity, but there exists some simple passage formulas if we chose θ 6= 0 .

Sketch of proof:

• By symmetr´ elity, for sequences s ∈ seq( N

) beginning with 1s, we can recursively remove the 1s to the right.

For example: Ze

= Ze

Ze

− Ze

− Ze

.

• So, if MZV

= Vect

(Ze

)

, we have:

∀s ∈ seq( N

) , Ze

∈ MZV

h Ze

i .

Reminders on Multizeta Values: Third Type of Relations.

Lemma : Let y

= x

x

, for i ∈ N

.

Then, y

y

− y

? y

is a linear combination of words of x

X

x

and is in the kernel of ζ .

Example : x

x

x

− x

? x

x

= x

x

− x

x

, so Ze

= Ze

.

Reminders on Multizeta Values: Diophantine Conjecture.

Diophantine Conjecture:

The kernel of ζ is exactly described by the three types of relations that we have seen:

• shuffle relations ←→ relations of symmetrality.

• stuffle relations ←→ relations of symmetrelity.

• regularization relations

Henceforth, we shall be interested only in these three types of relations between

multizeta values.

Introduction to Multitangent Functions: Definition.

Definition:

Let S

= {s ∈ seq( N

) ; s

≥ 2 et s

≥ 2} . For all sequence s ∈ S

, we consider:

Te

: C − Z −→ C

z 7−→ X

−∞<n_r<···<n₁<+∞

1

(n

+ z )

· · · (n

+ z)

.

Remarks : 1. Multitangent functions are a generalization of Eisenstein series (r = 1) .

2. Multitangent functions appear naturally in problems of holo-

morphic dynamics.

Introduction to Multitangent Functions: First Properties.

Property :

1

Differential property.

Let s = (s

; · · · s

) ∈ S

.

The function Te

is holomorphic on C − Z ; it is a uniformly convergent series on any compact subset of C − Z and satisfies:

∂Te

∂z = −

r

X

i=1

s

Te

.

2

Parity property.

∀z ∈ C − Z , ∀s ∈ S

, Te

(−z) = (−1)

Te

(z ) .

3

Symmetrelity.

Te

is symmetrel.

Introduction to Multitangent Functions: Reduction into Monotangent Functions, First Version.

Remark : A monotangent function is a multitangent function with length 1 . Let: MZV = Vect

(Ze

)

d

.

m(s) = max(s

; · · · ; s

), for all s ∈ seq( N

) .

Property: Reduction of Convergent Multitangent Functions into Monotangent Functions.

∀s ∈ S

, ∃(z

; · · · ; z

) ∈ MZV

, Te

=

m(s)

X

k=2

z

Te

.

Sketch of proof:

1. Partial fraction expansion of 1

(n

+ X )

· · · (n

+ X )

. 2. Using an analytic argument:

∀z ∈ C − R , |Te

(z)| ≤ 4r 2

|=m z|

e

1 − e

.

Introduction to Multitangent Functions: Examples of Reduction into Monotangent Functions.

Weight 6

Weight 4 Te

= 8

5 ζ(2)

Te

− 2ζ(3)Te

+ ζ(2)Te

. Te

= 2ζ(2)Te

. Te

= − 12

5 ζ(2)

Te

. Te

= 8

5 ζ(2)

Te

+ 2ζ(3)Te

+ ζ(2)Te

. Te

= −3 ζ(3)Te

+ ζ(2)Te

. Te

= 8

5 ζ(2)

Te

. Te

= 3 ζ(3)Te

+ ζ(2)Te

. Te

= − 2

5 ζ(2)

Te

+ ζ(3)Te

.

Te

= 0 . Te

= − 2

5 ζ(2)

Te

− ζ(3)Te

.

Weight 5 Te

= 4

5 ζ(2)

Te

.

Multitangent Functions Renormalization: the Property.

Property: Multitangent Functions Renormalization.

There exists an extension of multitangent functions to seq( N

) such that:

1. Te

is always symmetrel.

2. ∀z ∈ C − Z , Te

(z) = π tan(πz) .

This extension automatically satisfies: the differential property.

the parity property.



The removal to the right algorithm does not apply:

Te

(z) = Te

(z)

| {z }

known by hypothesis

× T e

(z)

| {z }

convergent multitangent

function

− Te

(z)

| {z }

problematics

=unknown

− T e

(z )

| {z }

convergent multitangent

function

.

Multitangent Functions Renormalization: Notations.

Let S

= {s ∈ seq( N

) ; s

≥ 2} . S

= {s ∈ seq( N

) ; s

≥ 2} .

S

= {s ∈ seq( N

) ; s

≥ 2 et s

≥ 2} .

Let us consider the symmetrel moulds He

, He

and Ce

, defined by:

∀s ∈ S

, He

(z ) = X

0<nr<···<n₁<+∞

1

(n

+ z)

· · · (n

+ z)

and He

(z) = 1 .

∀s ∈ S

, He

(z) = X

−∞<n_r<···<n₁<0

1

(n

+ z)

· · · (n

+ z )

and He

(z) = 1 .

∀s ∈ seq( N

) , Ce

(z) =



 

 

 

 

1 si s = ∅ . 1

z

si l (s) = 1 .

0 si l (s) > 1 .

Multitangent Functions Renormalization: Trifactorisation 1 / 2 .

Property:

∀s ∈ S

, Te

= X

(s1 ;s2 ;s3 )∈S?

d×seq(N∗)×S? f s1·s2·s3 =s

He

Ce

He

.

We write this in a simpler way: Te

= He

× Ce

× He

.

Notation:

In a general manner, we write (A

× B

)

when we consider the sum:

X

s¹·s²=s

A

B

.

Multitangent Functions Renormalization: Trifactorisation 2 / 2 .

Sketch of proof:

Recall that Te

(z) = X

−∞<n_r<···<n₁<+∞

1

(n

+ z)

· · · (n

+ z)

.

We deal with the position of 0 in the decreasing sequence n

< · · · < n

: n

< · · · < n

< 0 < n

< · · · < n

.

or

n

< · · · < n

< n

= 0 < n

< · · · < n

.

The terms to the left of 0 give He

(z) . The term n

= 0 gives Ce

(z) .

The terms to the right of 0 give He

(z) or He

(z) . We obtain all the terms He

Ce

He

where s

· s

· s

= s .

Multitangent Functions Renormalization: Consequences of trifactorisation 1 /3 .

Lemma:

Let (Φ

; Φ

) ∈ H( C − Z )

.

He

et He

have a unique symmetrel extension to seq( N

) such that







He

= Φ

. He

= Φ

.

Sketch of proof:

Identical to the renormalisation property of the multizeta values, by the removal to the right of the 1s algorithm.

Multitangent Functions Renormalization: Consequences of trifactorisation 2 /3 .

Example:

Te

(z ) = He

(z ) + He

(z ) 1

z + He

(z)

| {z }

divergent term

+ 1

z

He

(z)

| {z }

divergent term

+ He

(z)

| {z }

divergent term

= He

(z ) + He

(z ) 1

z + He

(z)

| {z }

divergent term

+ 1

z

He

(z)

| {z }

divergent term

+He

(z) He

(z)

| {z }

term divergent

− He

(z )

| {z }

convergent term

−He

(z) .

Multitangent Functions Renormalization: Consequences of trifactorisation 3 /3 .

Property: Multitangent Functions Renormalization.

There exists an extension of multitangent functions to seq( N

) such that:

1. Te

is always symmetrel.

2. ∀z ∈ C − Z , Te

(z) = π tan(πz) .

This extension automatically satisfies: the differential property.

the parity property.

Sketch of proof:

The extension is given by:



 

 

 

 

 

 

 

 

 

 

He

(z) = X

n≥1

1 n + z − 1

n

.

He

(z) = X

n≥1

1 n − 1

n − z

.

∀s ∈ seq( N

) , Te

= (He

× Ce

× He

)

.

Multitangent Functions Renormalization: Generating Functions 1/5.

Problem: Divergent multitangent functions are not written in an internal way...

We would like to express divergent multitangent functions with convergent multitangent functions.

Let us consider colored multizeta values and colored multitangent functions:

For

ε s

∈ seq ( Q / Z × N

) and e

= e

, for k ∈ [[ 1 ; n ]] , we denote:

Ze

ε₁,· · ·, ε_r s₁,· · ·,s_r

= X

1≤n_r<···<n₁

e

· · · e

n

· · · n

.

He

ε1,· · ·, εr s₁,· · ·,s_r

+

(z) = X

0<nr<···<n₁<+∞

e

· · · e

(n

+ z)

· · · (n

+ z)

.

Te

ε₁, ,· · ·, ε_r s₁, ,· · ·,s_r

(z) = X

−∞<n_r<···<n₁<+∞

e

· · · e

(n

+ z)

· · · (n

+ z)

.

Multitangent Functions Renormalization: Generating Functions 2/5.

For a symmetrel mould Me

, we can consider its generating functions, which we denote Mig

:



 

 

 

 

Mig

= 1 . Mig

ε₁,· · ·, ε_r v₁,· · ·,v_r

= X

s₁,···,s_r≥1

M e

ε₁,· · ·, εr s₁,· · ·,s_r

v

· · · v

.

Zig

, Zig

, Hig

, Hig

and Tig

.

Multitangent Functions Renormalization: Generating Functions 3/5.

Let µ

= 1

r

! · · · r

! where the sequence n = (n

; · · · ; n

) ∈ seq( N

) attains r

times its highest value, r

times its second highest value, and so on.

Lemma: The Generating Function Zig

(J. Ecalle) . For all k ∈ N

, let doZig

and coZig

be defined by:

doZig

ε₁,· · ·,ε_r v₁,· · ·,v_r

k

=



 



 



X

1≤n_r<···<n₁<k

e

· · · e

(n

− v

) · · · (n

− v

) , if r 6= 0 .

1 , if r = 0 .

coZig

ε₁,· · ·,ε_r v₁,· · ·,v_r

k

=



 



 



(−1)

X

1≤n_r≤···≤n₁<k

µ

n

· · · n

, if ε 6= 0 .

0 , if u = 0 .

Then, the mould Zig

admits a mould factorization:

Zig

= lim

k−→+∞

(coZig

× doZ ig

) .

Multitangent Functions Renormalization: Generating Functions 4/5.

Theorem: The Generating Functions T ig

.

Let: 

 

 

 

 

Qig

(z) = 0 . Qig

ε₁ v₁

(z ) = −Te

ε₁ 1

(v

− z) . Qig

ε₁,· · ·, εr v₁,· · ·,v_r

(z ) = 0 , si r ≥ 2 .



 

 

δ

= 0 . δ

ε₁,· · ·,ε_r v1,· · ·,vr

= ( (i π)

r! 1

(ε

) · · · 1

(ε

) , if r is even.

0 , if r is odd.

Then:

T ig

(z) = δ

+ Zig

× Qig

(z) × Zig

.

Multitangent Functions Renormalization: Generating Functions 5/5.

Sketch of proof:

1. T ig

(z ) = Hig

(z)

| {z }

=Zig^•(z)

×Cig

(z) × Hig

(z)

| {z }

=Zig₋^•(z)

.

2. We use the expression of the generating function Zig

: T ig

(z) = lim

N−→+∞

coZig

× Tig

(z) × coZig

,

where: Tig

ε₁,· · ·,u_r v₁,· · ·,v_r

N

(z) =

doZig

(z) × Cig

(z) × doZig

(z)

ε₁,· · ·,u_r v₁,· · ·,vr

= X

−N<n_r<···<n₁<N

e

· · · e

(n

− V

+ z) · · · (n

− V

+ z) . 3. We use a partial fraction expansion of Tig

.

4. We compute coZig

× Tig

(z) × coZig

to conclude.

Multitangent Functions Renormalization: Reduction into Monotangent Functions, second version.

Property: Reduction into Monotangent Functions.

∀s ∈ seq( N

) , ∃(z

; · · · ; z

) ∈ MZV

, Te

= δ

+

m(s)

X

k=1

z

Te

,

where δ

=





 (i π)

r! , if s = 1

and if r is even.

0 , else.

Important remark: z

= 0 ⇐⇒ s 6= 1

or

s = 1

.

r is even .

Multitangent Functions Renormalization: Examples of Reduction into Monotangent Functions.

Weight 2

Weight 4

Te

= −3 ζ(2) . Te

= −ζ(2)Te

. Te

= −ζ(2)Te

.

Te

= 0 . Te

= − 1

2 ζ(2)Te

.

Te

= 0 . Te

= 0 .

Te

= −ζ(2)Te

. Te

= − 1

2 ζ(2)Te

.

Weight 3 Te

= 3

2 ζ(2)

.

Relations Between Multizeta Values Obtained via the Study of Multitangent Functions: a Commutative Diagram.

MTGF ⊗ MTGF

multiplication by symmetrelity

//

reduction⊗reduction

MTGF

reduction

MTGF ⊗ MTGF

multiplication by symmetrelity

MTGF

// MTGF

Property:

The symmetrelity of Te

and the precedent commutative diagram show the symmetrelity of Ze

.

Remark: We obtain other relations between multizeta values, for example

some of regularization.

Relations Between Multizeta Values Obtained via the Study of Multitangent Functions: No T ¹ in relation of reduction.

Property:

There is no composant Te

in the reduction of Te

into monotangent if:

s 6= 1

or

s = 1

. r is even .

Remark: We obtain some symmetrality relations between multizeta values

and some regularization relations.

Relations Between Multizeta Values Obtained via the Study of Multitangent Functions: For weight 3 et 4.

Multitangent functions give us the following relations between multizetas values:

By the commutative diagram:

6Ze

+ 8Z e

= 5 Ze

. 2Ze

+ Z e

= Ze

.

By the absence of composant Te

: Ze

= Ze

.

2Ze

+ 4Ze

= Ze

.

Consequences: • For the weight 3, we find all the regularization relations.

• For the weight 4, we find all the symmetrelity and symmetrality

relations. We find sufficiently many relations to deduce those of

regularization.

Relations Between Multizeta Values Obtained via the Study of Multitangent Functions: For weight 5.

Convergent multitangent functions give us the following relations between multizeta values:

By the commutative diagram:

Ze

+ Ze

+ Ze

= Ze

Ze

. Ze

+ 2Ze

+ Ze

+ Ze

= Ze

Ze

. Ze

+ 2Ze

+ 2Ze

+ 3Ze

+ 7Ze

= Ze

Ze

+ Ze

. 4Z e

+ 6Ze

+ 15Ze

= 10Ze

Ze

. By the absence of composant Te

:

Ze

+ 3Ze

+ 6Ze

= Ze

Z e

. Consequences: • The last one is not independant of the others.

• For the weight 5, we obtain all the symmetrelity relations, but the following symmetrality relation is missing:

3Ze

+ 6Ze

+ Ze

= Ze

Ze

.

Relations Between Multizeta Values Obtained via the Study of Multitangent Functions: General Statement.

Conjecture:

• With convergent multitangent functions, we find :

1

All the symmetrelity relations.

2

A few of symmetrality relations.

3

A few of regularization relations.

• Using the divergent multitangent functions, we find:

1

All the symmetrelity relations.

2

A few more of symmetrality relations.

3

A few more of regularization relations.

Some Conjectures: Projection on Multitangent Function Space.

Let : MZV = Vect

(Ze

)

d

and MZV

= Vect

(Ze

)

. MTGF = Vect

(Te

)

. and MTGF

= Vect

(Te

)

. Here, N

= N − {0 ; 1} .

Conjecture: Projection on Multitangent Function Space.

∀(s

; s

) ∈ S

× S

, Ze

T e

∈ MTGF

.

Remark: It is sufficient to show that: ∀s ∈ S

, Ze

T e

∈ MTGF

.

This has been verified for all the sequences s ∈ S

of weight less

than 12 .

Some Conjectures: Examples of Projections on Multitangent Function Space.

||s|| = 4 Z e

Te

= 1 2 Te

.

||s|| = 5 Z e

Te

= 1

6 Te

− 1

6 Te

. Ze

Te

= 1

6 Te

− 1 6 Te

.

||s|| = 6

Z e

Te

= − 1

6 Te

. Ze

Te

= − 1 24 Te

. Z e

Te

= − 1

8 Te

. Ze

Te

= − 1 6 Te

.

||s|| = 7

Z e

Te

= − 1

30 Te

− 1

15 Te

+ 1

15 Te

+ 1 30 Te

. . .

.

Some Conjectures: Multizeta values and Multitangent Functions Cleansing 1/3 .

For multizeta values :

Theorem: (conjectured by J. Blumlein, proved by par J. Ecalle.) Let MZV = Vect

(Ze

)

d

.

MZV

= Vect

(Ze

)

N2)

.

Then, for all sequence s ∈ S

, Ze

can be expressed (explicitly) in terms of MZV

.

In other words:

MZV = MZV

.

Some Conjectures: Multizeta values and Multitangent Functions Cleansing 2/3 .

For multitangent functions:

Conjecture:

Let MTGF = Vect

(Te

)

. MTGF

= Vect

(Te

)

N2)

.

Then, for all sequence s ∈ S

, T e

can be expressed (explicitly) in terms of MTGF

.

In other words:

MTGF = MTGF

.

Some Conjectures: Multizeta values and Multitangent Functions Cleansing 3/3 .

p = 6

Te

= 1

6 Te

+ 1

4 Te

− 1 4 Te

. Te

= 1

6 Te

− 1

4 Te

+ 1 4 Te

. Te

= − 1

3 Te

.

p = 7

Te

= 1

6 Te

− 1

6 Te

− 1

3 Te

+ 7

48 Te

+ 23

48 Te

+ 1 3 Te

Te

= 1

5 Te

. Te

= 1

3 Te

+ 1

3 Te

+ 13

24 Te

+ 5

24 Te

− 1

3 Te

.

Some Conjectures: Absence of Relations between Multizeta Values and Multitangent Functions of Different Weights 1/2 .

For multizeta values:

Conjecture:

There is no relation between multizeta values of different weights:

X

k∈N^∗

Z

= M

k∈N^∗

Z

, o` u Z

=



 

 

 

 

 

 

Q , if k = 0 . {0} , if k = 1 . Vect

(Ze

)

||s||=kd

, if k ≥ 2 .

Some Conjectures: Absence of Relations between Multizeta Values and Multitangent Functions of Different Weights 2/2 .

For multitangent functions:

Conjecture:

There is no relation between multitangent functions of different weights:

X

k∈N^∗

T

= M

k∈N^∗

T

, o` u T

=



 

 

 

 

 

 

Q , si k = 0 . {0} , si k = 1 . Vect

(Te

)

df

||s||=k

, si k ≥ 2 .

Some Conjectures: What are the Implications Between the Various Conjectures?

Projection on multitangent function space

= ⇒

Multitangent Function cleansing

If the conjecture concerning the projection on multitangent functions space is true, then:

multizeta values cleansing

⇐⇒

multitangent functions cleansing

X

k∈N^∗

Z

= M

k∈N^∗

Z

= ⇒ X

k∈N^∗

T

= M

k∈N^∗

T

.



 



 

 X

k∈N^∗

T

= M

k∈N^∗

T

projection on MTGF

= ⇒ X

k∈N^∗

Z

= M

k∈N^∗

Z

.

Conclusion:

1

Multitangent functions seem to be an interesting functional model for the study of multizetas values.

2

There is a deep link between multizeta values and multitangent functions, the reduction into monotangent functions ; another important link, but a conjectural one, is projection on multitangent function space.

3

Multitangent functions don’t allow us to find the full dimorphy of multizeta values: we find only 1 + 1

2 symmetries.

4

The missing relations seem to be retrievable, but in a more complicated

way...