“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

1. Introduction For s_{1}, …, s_{k} in ℤ with s_{1} ≥ 2,

ζ(s_{1}, …, s_{k}) = ∑_{n_{1}>···>n_{k}≥1} n_{1}^{−s_{1}} ⋯ n_{k}^{−s_{k}}.

k = 1 integer values of Riemann zeta function ζ(s). Euler: ζ(s)π^{−s} ∈ ℚ for s even ≥ 2. \underline{Fact}: No known other algebraic relations between values of Riemann zeta function at positive integers. \underline{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)) ≠ 0 ?

\underline{F. Lindemann} (1882): π is transcendental.

\underline{R. Apéry} (1978): ζ(3) is irrational.

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

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

ζ(3), ζ(5), …, ζ(2n + 1)

has dimension

≥ \frac{1 − ϵ}{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.

2. Sketch of Proof of Rivoal’s Theorem

\underline{Goal}: Given a sufficiently large odd integer a, construct a sequence of linear forms in (a + 1)/2 variables, with integer coefficients, such that the numbers

ℓ_{n} = p_{0n} + ∑_{i=1}^{(a−1)/2} p_{in}ζ(2i + 1)

satisfy, for n → ∞,

|ℓ_{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 + \frac{log α}{log β} ≃ \frac{log a}{1 + log 2}

(Nesterenko’s Criterion).

Explicit construction of the linear forms Previous works of R. Apéry, F. Beukers, E. Nikishin, K. Ball, D. Vasilyev, … \underline{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} \frac{(t − rn + 1)_{rn}(t + n + 2)_{rn}}{(t + 1)_{n+1}^{a}},

S_{n} = ∑_{k=0}^{∞} R_{n}(k), ℓ_{n} = d_{2n}^{a}S_{2n}.

Write the partial fraction expansion

R_{n}(t) = ∑_{i=1}^{a} ∑_{j=0}^{n} \frac{c_{ijn}}{(t + j + 1)^{i}}

with

c_{ijn} = \frac{1}{(a − i)!} (\frac{d}{dt})^{a−i} (R_{n}(t)(t + j + 1)^{a})_{|t=−j−1}.

Set p_{in} = d_{2n}q_{i,2n} where

q_{0,n} = − ∑_{i=1}^{a} ∑_{j=1}^{n} c_{ijn} ∑_{k=0}^{j−1} \frac{1}{(k + 1)^{i}}

and

q_{in} = ∑_{j=0}^{n} c_{ijn} (1 ≤ i ≤ a).

Estimate for |p_{in}|:

c_{ijn} = \frac{1}{2πi} ∫_{|t+j+1|=1/2} R_{n}(t)(t + j + 1)^{i−1}dt.

Estimate for |ℓ_{n}|:

S_{n} = \frac{((2r + 1)n + 1)!}{n!^{2r+1}} · I_{n},

I_{n} = ∫_{[0,1]^{a+1}} F (\underline{x}) · \frac{dx_{1}dx_{2} … dx_{a+1}}{(1 − x_{1}x_{2} ⋯ x_{a+1})^{2}} ,

F (x_{1}, x_{2}, …, x_{a+1}) = (\frac{∏_{i=1}^{a+1} x_{i}^{r}(1 − x_{i})}{(1 − x_{1}x_{2} ⋯ x_{a+1})^{2r+1}} )^{n} .

3. Shuffle Product for Series \underline{Reflexion Formula}:

ζ(s)ζ(s′) = ∑_{n≥1} n^{−s} · ∑_{n′≥1}(n′)^{−s′}

= ∑_{n>n′≥1} n^{−s}(n′)^{−s′} + ∑_{n′>n≥1} n^{−s} (n′)^{−s′}

+ ∑_{n≥1} n^{−s−s′}

= ζ(s, s′) + ζ(s′, s) + ζ(s + s′).

\underline{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) = ∑_{m>n≥1}(mn)^{−2} = \frac{π^{4}}{120}·

\underline{Other example}:

ζ(2)ζ(3) = ζ(2, 3) + ζ(3, 2) + ζ(5).

Shuffle relations arising from the series representation.

ζ(\underline{s})ζ(\underline{s}′) = ∑_{\underline{σ}} ζ(\underline{σ}),

where \underline{σ} = (σ_{1}, …, σ_{h}) ranges over the tuples obtained as follows:

\begin{matrix}\underline{s} → & ( & s_{1} & 0 & s_{2} & ⋯ & s_{k} & ) \\ \underline{s}′ → & ( & 0 & s′_{1} & s′_{2} & ⋯ & 0 & ) \\ \underline{σ} = & ( & s_{1} & s′_{1} & s_{2} + s′_{2} & ⋯ & s_{k} & ) \end{matrix}

Hence max{k, k′} ≤ h ≤ k + k′.

\underline{Example}: k = k′ = 1, \underline{s} = s, \underline{s}′ = s′, then

\begin{matrix}s → & (s & 0) & (0 & s) & s \\ s′ → & (0 & s) & (s′ & 0) & s′ \\ σ = & (s & s′) & (s′ & s) & s + s′ \end{matrix}

so that

\{\underline{σ}_{1}, \underline{σ}_{2}, \underline{σ}_{3}\} = \{(s, s′), (s′, s), s + s′\}.

\underline{Other Description}: Alphabet with two letters X = {x_{0}, x_{1}}.

Words:  X^{∗} = {x_{0}^{a_{1}}x_{1}^{b_{1}} ⋯ x_{0}^{a_{h}}x_{1}^{b_{h}}}.

Non-commutative polynomials: ℚ⟨X⟩. For s ≥ 1 set y_{s} = x_{0}^{s−1}x_{1}. For \underline{s} = (s_{1}, …, s_{k}) with s_{i} ≥ 1, set

\begin{aligned}x\underline{s} & = y_{s_{1}} ⋯ y_{s_{k}} \\ & = x_{0}^{s_{1}−1}x_{1}x_{0}^{s_{2}−1}x_{1} ⋯ x_{0}^{s_{k}−1}x_{1}. \end{aligned}

The number k of factors x_{1} is the depth of the word x\underline{s} and of the tuple \underline{s}. The number p = s_{1} +⋯+s_{k} of letters is the weight. The set of such x\underline{s}’s is X^{∗}x_{1} togeether with the null word e (corresponds to ∅ with k = 0).

Convergent words: x_{0}X^{∗}x_{1} ⋃{e}. Set ζ(w) = ζ(\underline{s}) for w = x\underline{s} with ζ(∅) = ζ(e) = 1.

Convergent polynomials: ℚ⟨X⟩_{conv} ⊂ ℚ⟨X⟩. Extend ζ by linearity to ℚ⟨X⟩_{conv}.

Law ∗ on ℚ⟨X⟩_{conv}:

e ∗ w = w for w ∈ X^{∗}x_{1}

and, for s ≥ 1 and t ≥ 1, w and w′ in X^{∗}x_{1},

(y_{s}w) ∗ (y_{t}w′) =

y_{s}(w ∗ y_{t}w′) + y_{t}(y_{s}w ∗ w′) + y_{s+t}(w ∗ w′).

Then

x\underline{s} ∗ x\underline{s}′ = ∑_{\underline{σ}} x\underline{σ}.

Proposition. For w and w′ in x_{0}X^{∗}x_{1},

ζ(w)ζ(w′) = ζ(w ∗ w′).

Connection with quasi-symmetric functions Commutative infinite alphabet: \underline{t} = \{t_{1}, t_{2}, …\} Formal power series: ℚ[[\underline{t}]]. To w = x\underline{s} ∈ X^{∗}x_{1} associate

F_{w}(\underline{t}) = ∑_{n_{1}>···>n_{k}≥1} t_{n_{1}}^{s_{1}} ⋯ t_{n_{k}}^{s_{k}}.

Then for w and w′ in X^{∗}x_{1} we have

F_{w}(\underline{t})F_{w′}(\underline{t}) = F_{w∗w′}(\underline{t}).

For w ∈ x_{0}X^{∗}x_{1}, ζ(w) is the value of F_{w}(\underline{t}) with t_{n} = 1/n, (n ≥ 1).

4. Shuffle Product for Integrals Let \underline{s} = (s_{1}, …, s_{k}) with s_{i} ≥ 1; set p = s_{1} +⋯+s_{k}. Define ϵ_{i} ∈ {0, 1} for 1 ≤ i ≤ p by

x\underline{s} = x_{ϵ_{1}} ⋯ x_{ϵ_{p}}.

For instance for \underline{s} = (2, 3) with p = 5:

x_{(2,3)} = x_{0}x_{1}x_{0}^{2}x_{1}, (ϵ_{1}, …, ϵ_{5}) = (0, 1, 0, 0, 1). Define differential forms:

ω_{0}(t) = \frac{dt}{t} et ω_{1}(t) = \frac{dt}{1 − t}·

Let Δ_{p} be the simplex in ℝ^{p}:

Δ_{p} = \{\underline{t} ∈ ℝ^{p} ; 1 > t_{1} > ⋯ > t_{p} > 0\}. Proposition. For s_{1} ≥ 2,

ζ(\underline{s}) = ∫_{Δ_{p}} ω_{ϵ_{1}}(t_{1}) ⋯ ω_{ϵ_{p}}(t_{p}).

\underline{Example}:

ζ(2, 3) = ∫_{Δ_{5}} \frac{dt_{1}}{t_{1}} · \frac{dt_{2}}{1 − t_{2}} · \frac{dt_{3}}{t_{3}} · \frac{dt_{4}}{t_{4}} · \frac{dt_{5}}{1 − t_{5}} ·

Proof. Expand 1/(1 − t) = ∑_{s≥0} t^{s}.

Shuffle on X^{∗}:

e⊔⊔w = w⊔⊔e = w,

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

(x_{i}u)⊔⊔(x_{j}v) = x_{i}(u⊔⊔x_{j}v) + x_{j}(x_{i}u⊔⊔v)

\underline{Example}. Computation of y_{2}⊔⊔y_{3} = x_{0}x_{1}⊔⊔x_{0}^{2}x_{1}: get x_{0}x_{1}x_{0}^{2}x_{1} once, x_{0}^{2}x_{1}x_{0}x_{1} three times and x_{0}^{3}x_{1}^{2} six times. Hence

y_{2}⊔⊔y_{3} = y_{2}y_{3} + 3y_{3}y_{2} + 6y_{4}y_{1}.

Corollary.

ζ(w)ζ(w′) = ζ(w⊔⊔w′)

for w and w′ in x_{0}X^{∗}x_{1}.

Proof. The Cartesian product Δ_{p} × Δ_{p′} is the union of (p + p′)!/p!p′! simplices.

\underline{Example}. From

y_{2}⊔⊔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).

\underline{There are further relations}. \underline{Example}:

x_{1}⊔⊔x_{0}x_{1} = x_{1}x_{0}x_{1} + 2x_{0}x_{1}^{2}

and

x_{1} ∗ x_{0}x_{1} = x_{1}x_{0}x_{1} + x_{0}x_{1}^{2} + x_{0}^{2}x_{1},

hence

x_{1}⊔⊔x_{0}x_{1} − x_{1} ∗ x_{0}x_{1} = x_{0}x_{1}^{2} − x_{0}^{2}x_{1}.

\underline{Fact}:  ζ(x_{0}x_{1}^{2}) = ζ(x_{0}^{2}x_{1}).
Euler :  ζ(2, 1) = ζ(3).

Proposition. For w and w′ in x_{0}X^{∗}x_{1},

ζ(w)ζ(w′) = ζ(w ∗ w′),
ζ(w)ζ(w′) = ζ(w⊔⊔w′)

and

ζ(x_{1}⊔⊔w − x_{1} ∗ w) = 0.

5. Symbolic Multizeta

Define Ze(\underline{s}) for each \underline{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\underline{s}) = Ze(\underline{s}). Convergent symbols: Ze(\underline{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 generated by the convergent symbols Ze(\underline{s}) with the relations

Ze(w)Ze(w′) = Ze(w ∗ w′),
Ze(w)Ze(w′) = Ze(w⊔⊔w′)

and

Ze(x_{1}⊔⊔w − x_{1} ∗ w) = 0

for w and w′ in x_{0}X^{∗}x_{1}.

Main Diophantine Conjecture. The specialization morphism from MZV_{conv} into ℂ which maps Ze(\underline{s}) onto ζ(\underline{s}) is injective.

Algebras MZV^{∗} and MZV^{⊔⊔}: generators Ze(\underline{s}) with \underline{s} = (s_{1}, …, s_{k}), k ≥ 0, s_{j} ≥ 1, and ∗ (resp. ⊔⊔) defined by

Ze(w) ∗ Ze(w′) = Ze(w ∗ w′)

resp.

Ze(w)⊔⊔Ze(w′) = Ze(w⊔⊔w′)

for w ∈ X^{∗}x_{1}.

\underline{Remark}.

x_{1} ∗ x_{1} = 2x_{1}^{2} + x_{0}x_{1} and x_{1}⊔⊔x_{1} = 2x_{1}^{2},

hence

Ze(x_{1}) ∗ Ze(x_{1}) = 2Ze(x_{1}^{2}) + Ze(x_{0}x_{1})

while

Ze(x_{1})⊔⊔Ze(x_{1}) = 2Ze(x_{1}^{2})

and ζ(x_{0}x_{1}) = ζ(2) ≠ 0.

Conjecture of

Zagier, Drinfeld, Kontsevich and Goncharov.

For p ≥ 2 let d_{p} denote the dimension of the ℤmodule in MZV_{conv} spanned by the 2^{p−2} elements Ze(\underline{s}) for \underline{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(\underline{s}) with \underline{s} convergent of weight p; set Z_{0} = ℚ. Then the sum of Z_{p} (p ≥ 0) is direct, and the conjecture means

∑_{p≥0} q^{p} dim_{ℚ} Z_{p} = \frac{1}{1 − q^{2} − q^{3}} ·

Remark: d_{p} → ∞ by Rivoal’s result.

6. Further Results

Écalle: for weight ≤ 10, independent generators are

Ze(2), Ze(3), Ze(5), Ze(7), Ze(9),

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

Polylogarithms \underline{Classical}: for s ≥ 1 and |z| < 1,

Li_{s}(z) = ∑_{n≥1} \frac{z^{n}}{n^{s}} .

\underline{Higher dimension}: for \underline{s} = (s_{1}, …, s_{k}) with s_{i} ≥ 1,

Li\underline{s}(z) = ∑_{n_{1}>···>n_{k}≥1} \frac{z^{n_{1}}}{n_{1}^{s_{1}} ⋯ n_{k}^{s_{k}}}

Then

Li_{u⊔⊔v}(z) = Li_{u}(z)Li_{v}(z)

If s_{1} ≥ 2, then Li\underline{s}(1) = ζ(\underline{s}).

Work of Petitot (Lille): the functions Li_{w}, with w in X^{∗}, are linearly independent over ℂ.

7. Related Topics

\underline{Further connections with}: Combinatoric (theory of quasisymmetric functions, Radford’s Theorem and Lyndon words) Lie and Hopf algebras Resurgent series (Écalle’s theory) Mixed Tate motives on Specℤ (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-Teichmüller 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).