International Conference in honour of Srinivasa Ramanujan Centre for Advanced Study in Mathematics

“NumberTheory & Discrete Mathematics”

International Conference in honour of Srinivasa Ramanujan Centre for Advanced Study in Mathematics

Panjab University, Chandigarh (October 2/6, 2000.)

Multiple Zeta Values

and

Euler-Zagier Numbers

by

Michel WALDSCHMIDT

http : //www.math.jussieu.fr/ _∼ miw/articles/ps/MZV.ps

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1. Introduction

For s ₁ , . . . , s _k in Z with s ₁ ≥ 2, ζ (s ₁ , . . . , s _k ) = X

n

> ··· >n

≥ 1

n ⁻ ₁ ^s

· · · n ⁻ _k ^s

.

k = 1 integer values of Riemann zeta function ζ (s).

Euler: ζ (s)π ^{− s} ∈ Q for s even ≥ 2.

Fact: No known other algebraic relations between values of Riemann zeta function at positive integers.

Expected: there is no further relation : Are the numbers

π, ζ (3), ζ (5), . . . , ζ (2n + 1), . . . algebraically independent?

Means:

For n ≥ 0 and P ∈ Q [X ₀ , X ₁ , . . . , X _n ] \ { 0 } , P ¡

π, ζ (3), ζ (5), . . . , ζ (2n + 1) ¢

6

= 0 ?

3

F. Lindemann (1882): π is transcendental.

R. Ap´ery (1978): ζ (3) is irrational.

T. Rivoal (2000): infinitely many irrational numbers among ζ (3), ζ (5), . . . , ζ (2n + 1), . . .

Theorem (T. Rivoal). Let ² > 0. For any suffi- ciently large n, the Q -vector space spanned by the n numbers

ζ (3), ζ (5), . . . , ζ (2n + 1) has dimension

≥ 1 − ²

1 + log 2 · log n.

The proof also yields:

There exists on odd integer j with 5 ≤ j ≤ 169 such that the three numbers

1, ζ (3), ζ (j )

are linearly independent over Q .

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2. Sketch of Proof of Rivoal’s Theorem

Goal: Given a sufficiently large odd integer a, con- struct a sequence of linear forms in (a + 1)/2 vari- ables, with integer coefficients, such that the numbers

` _n = p _0n +

(a − X 1)/2

i=1

p _in ζ (2i + 1) satisfy, for n → ∞ ,

| ` _n | = α ^{− n+o(n)} and

| p _in | ≤ β ^n+o(n) with

α ' a ^2a and β ' (2e) ^2a . It will follow that the (a + 1)/2 numbers

1, ζ (3), ζ (5), . . . , ζ (a)

span a Q -vector space of dimension at least 1 + log α

log β ' log a

1 + log 2

(Nesterenko’s Criterion).

5

Explicit construction of the linear forms Previous works of R. Ap´ery, F. Beukers,

E. Nikishin, K. Ball, D. Vasilyev, . . .

Pochammer symbol: (m) ₀ = 1 and, for k ≥ 1, (m) _k = m(m + 1) . . . (m + k − 1).

Set r = £

a(log a) ^{− 2} ¤

. Define

d _m = l.c.m. of { 1, 2, . . . , m } ,

R _n (t) = n! ^{a − 2r} (t − rn + 1) _rn (t + n + 2) _rn (t + 1) ^a _n+1 , S _n =

X ∞ k=0

R _n (k), ` _n = d ^a _2n S _2n . Write the partial fraction expansion

R _n (t) =

X a

i=1

X n

j =0

c _ijn (t + j + 1) ⁱ with

c _ijn = 1 (a − i)!

µ d dt

¶ a − i ¡

R _n (t)(t + j + 1) ^a ¢

| t= − j − 1 .

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Set p _in = d _2n q _i,2n where q _0,n = −

X a

i=1

X n

j =1

c _ijn

j − 1

X

k=0

1 (k + 1) ⁱ and

q _in =

X n

j=0

c _ijn (1 ≤ i ≤ a).

Estimate for | p _in | : c _ijn = 1

2πi Z

| t+j +1 | =1/2

R _n (t)(t + j + 1) ^{i − 1} dt.

Estimate for | ` _n | : S _n =

¡ (2r + 1)n + 1 ¢

!

n! ^2r+1 · I _n , I _n =

Z

[0,1]

F (x) · dx ₁ dx ₂ . . . dx _a+1 (1 − x ₁ x ₂ · · · x _a+1 ) ² , F (x ₁ , x ₂ , . . . , x _a+1 ) =

à Q a+1

i=1 x ^r _i (1 − x _i ) (1 − x ₁ x ₂ · · · x _a+1 ) ^2r+1

! n

.

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3. Shuffle Product for Series Reflexion Formula:

ζ (s)ζ (s ⁰ ) = X

n ≥ 1

n ^{− s} · X

n

≥ 1

(n ⁰ ) ^{− s}

= X

n>n

≥ 1

n ^{− s} (n ⁰ ) ^{− s}

+ X

n

>n ≥ 1

n ^{− s} (n ⁰ ) ^{− s}

+ X

n ≥ 1

n ^{− s − s}

= ζ (s, s ⁰ ) + ζ (s ⁰ , s) + ζ (s + s ⁰ ).

Example:

ζ (s) ² = 2ζ (s, s) + ζ(2s).

For s = 2: ζ (2) = π ² /6, ζ (4) = π ⁴ /90, ζ (2, 2) = X

m>n ≥ 1

(mn) ^{− 2} = π ⁴ 120 · Other example:

ζ (2)ζ (3) = ζ (2, 3) + ζ (3, 2) + ζ (5).

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Shuffle relations arising from the series representation.

ζ (s)ζ (s ⁰ ) = X

σ

ζ (σ),

where σ = (σ ₁ , . . . , σ _h ) ranges over the tuples ob- tained as follows:

s → ( s ₁ 0 s ₂ · · · s _k )

s ⁰ → ( 0 s ⁰ ₁ s ⁰ ₂ · · · 0 ) σ = ( s ₁ s ⁰ ₁ s ₂ + s ⁰ ₂ · · · s _k ) Hence max { k, k ⁰ } ≤ h ≤ k + k ⁰ .

Example: k = k ⁰ = 1, s = s, s ⁰ = s ⁰ , then

s → (s 0) (0 s) s

s ⁰ → (0 s) (s ⁰ 0) s ⁰

σ = (s s ⁰ ) (s ⁰ s) s + s ⁰

so that

{ σ ₁ , σ ₂ , σ ₃ } = { (s, s ⁰ ), (s ⁰ , s), s + s ⁰ } .

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Other Description:

Alphabet with two letters X = { x ₀ , x ₁ } . Words: X ^∗ = { x ^a ₀

x ^b ₁

· · · x ^a ₀

x ^b ₁

} . Non-commutative polynomials: Qh X i .

For s ≥ 1 set y _s = x ^s ₀ ^{− 1} x ₁ .

For s = (s ₁ , . . . , s _k ) with s _i ≥ 1, set x _s = y _s

· · · y _s

= x ^s ₀

^{− 1} x ₁ x ^s ₀

^{− 1} x ₁ · · · x ^s ₀

^{− 1} x ₁ .

The number k of factors x ₁ is the depth of the word x _s and of the tuple s.

The number p = s ₁ + · · · + s _k of letters is the weight . The set of such x _s ’s is X ^∗ x ₁ togeether with the null word e (corresponds to ∅ with k = 0).

Convergent words: x ₀ X ^∗ x ₁ S

{ e } .

Set ζ (w) = ζ (s) for w = x _s with ζ ( ∅ ) = ζ (e) = 1.

Convergent polynomials: Qh X i ^conv ⊂ Qh X i .

Extend ζ by linearity to Qh X i conv .

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Law ∗ on Qh X i ^conv :

e ∗ w = w for w ∈ X ^∗ x ₁

and, for s ≥ 1 and t ≥ 1, w and w ⁰ in X ^∗ x ₁ , (y _s w ) ∗ (y _t w ⁰ ) =

y _s (w ∗ y _t w ⁰ ) + y _t (y _s w ∗ w ⁰ ) + y _s+t (w ∗ w ⁰ ).

Then

x _s ∗ x _s

= X

σ

x _σ .

Proposition. For w and w ⁰ in x ₀ X ^∗ x ₁ , ζ (w)ζ (w ⁰ ) = ζ (w ∗ w ⁰ ).

Connection with quasi-symmetric functions Commutative infinite alphabet: t = { t ₁ , t ₂ , . . . } Formal power series: Q [[t]].

To w = x _s ∈ X ^∗ x ₁ associate F _w (t) = X

n

> ··· >n

≥ 1

t ^s _n

· · · t ^s _n

. Then for w and w ⁰ in X ^∗ x ₁ we have

F _w (t)F _w

(t) = F _{w ∗ w}

(t).

For w ∈ x ₀ X ^∗ x ₁ , ζ (w) is the value of F _w (t) with

t _n = 1/n, (n ≥ 1).

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4. Shuffle Product for Integrals

Let s = (s ₁ , . . . , s _k ) with s _i ≥ 1; set p = s ₁ + · · · + s _k . Define ² _i ∈ { 0, 1 } for 1 ≤ i ≤ p by

x _s = x _²

· · · x _²

.

For instance for s = (2, 3) with p = 5:

x _(2,3) = x ₀ x ₁ x ² ₀ x ₁ , (² ₁ , . . . , ² ₅ ) = (0, 1, 0, 0, 1).

Define differential forms:

ω ₀ (t) = dt

t et ω ₁ (t) = dt 1 − t · Let ∆ _p be the simplex in R ^p :

∆ _p = { t ∈ R ^p ; 1 > t ₁ > · · · > t _p > 0 } . Proposition. For s ₁ ≥ 2,

ζ (s) = Z

∆

ω _²

(t ₁ ) · · · ω _²

(t _p ).

Example:

ζ (2, 3) = Z

∆

dt ₁

t ₁ · dt ₂

1 − t ₂ · dt ₃

t ₃ · dt ₄

t ₄ · dt ₅ 1 − t ₅ · Proof. Expand 1/(1 − t) = P

s ≥ 0 t ^s .

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Shuffle on X ^∗ :

e tt w = w tt e = w,

and, for i and j in { 0, 1 } , u and v in X ^∗ ,

(x _i u) tt (x _j v) = x _i (u tt x _j v) + x _j (x _i u tt v )

Example. Computation of y _{2 tt} y ₃ = x ₀ x _{1 tt} x ² ₀ x ₁ : get x ₀ x ₁ x ² ₀ x ₁ once, x ² ₀ x ₁ x ₀ x ₁ three times and x ³ ₀ x ² ₁ six times. Hence

y _{2 tt} y ₃ = y ₂ y ₃ + 3y ₃ y ₂ + 6y ₄ y ₁ . Corollary.

ζ (w)ζ (w ⁰ ) = ζ (w tt w ⁰ ) for w and w ⁰ in x ₀ X ^∗ x ₁ .

Proof. The Cartesian product ∆ _p × ∆ _p

is the union

of (p + p ⁰ )!/p!p ⁰ ! simplices.

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Example. From

y _{2 tt} y ₃ = y ₂ y ₃ + 3y ₃ y ₂ + 6y ₄ y ₁ we deduce

ζ (2)ζ(3) = ζ (2, 3) + 3ζ (3, 2) + 6ζ(4, 1).

On the other hand the shuffle relation for series gives ζ (2)ζ (3) = ζ (2, 3) + ζ (3, 2) + ζ (5),

hence

ζ (5) = 2ζ (3, 2) + 6ζ(4, 1).

There are further relations.

Example:

x ₁ tt x ₀ x ₁ = x ₁ x ₀ x ₁ + 2x ₀ x ² ₁ and

x ₁ ∗ x ₀ x ₁ = x ₁ x ₀ x ₁ + x ₀ x ² ₁ + x ² ₀ x ₁ , hence

x _1tt x ₀ x ₁ − x ₁ ∗ x ₀ x ₁ = x ₀ x ² ₁ − x ² ₀ x ₁ . Fact: ζ (x ₀ x ² ₁ ) = ζ (x ² ₀ x ₁ ).

Euler : ζ (2, 1) = ζ (3).

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Proposition. For w and w ⁰ in x ₀ X ^∗ x ₁ , ζ (w)ζ (w ⁰ ) = ζ (w ∗ w ⁰ ),

ζ (w)ζ (w ⁰ ) = ζ (w tt w ⁰ ) and

ζ(x _{1 tt} w − x ₁ ∗ w) = 0.

5. Symbolic Multizeta

Define Ze(s) for each s = (s ₁ , . . . , s _k ), with k ≥ 0 and s _i ≥ 1. Next define Ze(w) for w in X ^∗ x ₁ by Ze(x _s ) = Ze(s). Convergent symbols: Ze(s) with s ₁ ≥ 2 or k = 0; these are the Ze(w) with w in x ₀ X ^∗ x ₁ together with Ze(e) = Ze( ∅ ).

Algebra of Convergent MZV:

MZV _conv is the commutative algebra over Q gener- ated by the convergent symbols Ze(s) with the rela- tions

Ze(w)Ze(w ⁰ ) = Ze(w ∗ w ⁰ ), Ze(w)Ze(w ⁰ ) = Ze(w tt w ⁰ ) and

Ze(x ₁ tt w − x ₁ ∗ w) = 0

for w and w ⁰ in x ₀ X ^∗ x ₁ .

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Main Diophantine Conjecture. The specializa- tion morphism from MZV _conv into C which maps Ze(s) onto ζ (s) is injective.

Algebras MZV ^∗ and MZV

: generators Ze(s) with s = (s ₁ , . . . , s _k ), k ≥ 0, s _j ≥ 1, and ∗ (resp. tt ) defined by

Ze(w) ∗ Ze(w ⁰ ) = Ze(w ∗ w ⁰ ) resp.

Ze(w) tt Ze(w ⁰ ) = Ze(w tt w ⁰ ) for w ∈ X ^∗ x ₁ .

Remark.

x ₁ ∗ x ₁ = 2x ² ₁ + x ₀ x ₁ and x ₁ tt x ₁ = 2x ² ₁ , hence

Ze(x ₁ ) ∗ Ze(x ₁ ) = 2Ze(x ² ₁ ) + Ze(x ₀ x ₁ ) while

Ze(x ₁ ) tt Ze(x ₁ ) = 2Ze(x ² ₁ )

and ζ (x ₀ x ₁ ) = ζ (2) 6 = 0.

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Conjecture of

Zagier, Drinfeld, Kontsevich and Goncharov.

For p ≥ 2 let d _p denote the dimension of the Z - module in MZV _conv spanned by the 2 ^{p − 2} elements Ze(s) for s = (s ₁ , . . . , s _k ) of length p and s ₁ ≥ 2.

Conjecture. We have

d ₁ = 0, d ₂ = d ₃ = d ₄ = 1 and

d _p = d _{p − 2} + d _{p − 3} for p ≥ 4.

For each p ≥ 1, define Z p as the Q -vector space spanned by the Ze(s) with s convergent of weight p;

set Z ⁰ = Q . Then the sum of Z ^p (p ≥ 0) is direct, and the conjecture means

X

p ≥ 0

q ^p dim _Q Z ^p = 1

1 − q ² − q ³ ·

Remark: d _p → ∞ by Rivoal’s result.

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6. Further Results

Ecalle: for weight ´ ≤ 10, independent generators are Ze(2), Ze(3), Ze(5), Ze(7), Ze(9),

Ze(6, 2), Ze(8, 2).

Polylogarithms

Classical: for s ≥ 1 and | z | < 1, Li _s (z) = X

n ≥ 1

z ⁿ n ^s .

Higher dimension: for s = (s ₁ , . . . , s _k ) with s _i ≥ 1, Li _s (z ) = X

n

> ··· >n

≥ 1

z ⁿ

n ^s ₁

· · · n ^s _k

Then

Li _u

_v (z ) = Li _u (z )Li _v (z ) If s ₁ ≥ 2, then Li _s (1) = ζ (s).

Work of Petitot (Lille): the functions Li _w , with w

in X ^∗ , are linearly independent over C .

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7. Related Topics

Further connections with:

Combinatoric (theory of quasisymmetric functions, Radford’s Theorem and Lyndon words)

Lie and Hopf algebras

Resurgent series (´ Ecalle’s theory)

Mixed Tate motives on Spec Z (Goncharov’s work) Monodromy of differential equations

Fundamental group of the projective line minus three points and Belyi’s Theorem

Absolute Galois group of Q

Group of Grothendieck-Teichm¨ uller Knots theory and Vassiliev invariants K -theory

Feynman diagrams and quantum field theory Quasi-triangular quasi-Hopf algebras

Drinfeld’s associator Φ _KZ (connexion of Knizhnik-

Zamolodchikov).