Réunion d’été de la SMC - Saskatoon, 5 juin 2001

Polylogarithms and Multiple Zeta Values by Michel WALDSCHMIDT

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

Special Values of the Riemann Zeta Function

ζ(s) := 1 + \frac{1}{2^{s}} + \frac{1}{3^{s}} + ⋯ + \frac{1}{n^{s}} + ⋯ s ≥ 2.

L. Euler:

ζ(2m) = −\frac{1}{2} · \frac{(2πi)^{2m}B_{2m}}{(2m)!} for m ≥ 1

with

\frac{t}{e^{t} − 1} = ∑_{k=0}^{∞} \frac{B_{k}}{k!} · t^{k}, |t| < 2π.

In particular ζ(2m)π^{−2m} ∈ ℚ for m ≥ 1. F. Lindemann: π is transcendental. R. Apéry: ζ(3) ∉ ℚ.

T. Rivoal: Infinitely many ζ(a) with a odd ≥ 5 are irrational. K. Ball + T. Rivoal: The ℚ-vector subspace of ℝ spanned by

ζ(3), ζ(5), ⋯, ζ(a)

has dimension

≥ \frac{log a}{1 + log 2}(1 + o(1)).

T. Rivoal, W. Zudilin: One at least of the 9 numbers

ζ(5), ζ(7), ⋯, ζ(21)

is irrational.

\underline{Expected}: Euler’s relations are the only algebraic relation among the numbers

ζ(2), ζ(3), ζ(4), …, ζ(s), …

Conjecture. The numbers

π, ζ(3), ζ(5), …, ζ(2n + 1), …

algebraically independent. In other terms: For n ≥ 0 and P ∈ ℚ[X_{0}, X_{1}, …, X_{n}] \ {0},

P (π, ζ(3), ζ(5), …, ζ(2n + 1)) ≠ 0.

Definition. For \underline{s} = (s_{1}, …, s_{k}) in ℤ^{k} with s_{1} ≥ 2 and s_{j} ≥ 1 (2 ≤ j ≤ k),

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

Examples of algebraic relations

1  ζ(2)^{2} = 2ζ(2, 2) + ζ(4)

Proof.

∑_{n≥1} \frac{1}{n^{2}} ∑_{m≥1} \frac{1}{m^{2}} = ∑_{n>m≥1} \frac{1}{n^{2}m^{2}} + ∑_{m>n≥1} \frac{1}{n^{2}m^{2}} + ∑_{n≥1} \frac{1}{n^{4}} .

2  ζ(2)^{2} = 2ζ(2, 2) + 4ζ(3, 1)

Corollary. ζ(3, 1) = \frac{1}{4}ζ(4).

3  ζ(3) = ζ(2, 1)

EZ-Face by J. Borwein, P. Lisonek and P. Irvine

     http://www.cecm.sfu.ca/projects/EZFace/index.html

Conjectures of Zagier, Drinfeld, Kontsevich, Goncharov, Hoffman, Broadhurst, Cartier…

For p ≥ 2 let Z_{p} denote the ℚ-vector subspace of ℝ spanned by the 2^{p−2} elements ζ(\underline{s}) for \underline{s} = (s_{1}, …, s_{k}) of weight s_{1} + ⋯ + s_{k} = p and s_{1} ≥ 2. Let d_{p} be the dimension of Z_{p}.

Z_{0} = ℚ  with ζ(\underline{s}) := 1 for k = 0
Z_{1} = {0}
Z_{2} = ⟨ζ(2)⟩_{ℚ}
Z_{3} = ⟨ζ(3)⟩_{ℚ}  since ζ(2, 1) = ζ(3)
Z_{4} = ⟨ζ(4)⟩_{ℚ} = ⟨ζ(2)^{2}⟩_{ℚ} = ⟨π^{4}⟩_{ℚ}

since

ζ(3, 1) = \frac{1}{4}ζ(4), ζ(2, 2) = \frac{3}{4}ζ(4), ζ(2, 1, 1) = ζ(4) = \frac{2}{5}ζ(2)^{2}.

Hence

d_{0} = 1, d_{1} = 0, d_{2} = d_{3} = d_{4} = 1.

Z_{5} = ⟨ζ(2)ζ(3), ζ(5)⟩_{ℚ}, hence d_{5} ≤ 2.

Proof.

\begin{aligned}ζ(2, 1, 1, 1) & = ζ(5), \\ ζ(3, 1, 1) = ζ(4, 1) & = 2ζ(5) − ζ(2)ζ(3), \\ ζ(2, 1, 2) = ζ(2, 3) & = \frac{9}{2}ζ(5) − 2ζ(2)ζ(3), \\ ζ(2, 2, 1) = ζ(3, 2) & = 3ζ(2)ζ(3) − \frac{11}{2}ζ(5). \end{aligned}

Conjecture (D. Zagier). We have

d_{p} = d_{p−2} + d_{p−3} for p ≥ 4.

Equivalent formulation:

∑_{p≥0} d_{p}X^{p} = \frac{1}{1 − X^{2} − X^{3}} ·

Conjecture (M. Hoffman). For p ≥ 1, a basis for the ℚ-vector space Z_{p} is given by the numbers ζ(\underline{s}) for (s_{1}, …, s_{k}) ∈ ℤ^{k} (k ≥ 1) satisfying

s_{1} + ⋯ + s_{k} = p and s_{j} ∈ {2, 3}.

The subalgebra Z of ℝ generated by all ζ(\underline{s}) is graded for the weight:

Z_{p}Z_{p′} ⊂ Z_{p+p′}.

The length k defines a filtration on Z. Denote by F^{k}Z_{p} the ℚ-vector subspace of ℝ spanned by all ζ(\underline{s}) of weight p and length k. Further, let d_{pk} denote the dimension of F^{k}Z_{p}/F^{k−1}Z_{p}.

Conjecture (D. Broadhurst). We have

(∑_{p≥0} ∑_{k≥0} d_{pk}X^{p}Y^{k})^{−1} = 1 − X^{2} − X^{3} + \frac{XY^{2}(1 − Y^{2})}{(1 + X^{2})(1 − Y^{6})}·

Conjecture (A.B. Goncharov). As a ℚ-algebra, Z is the direct sum of Z_{p} for p ≥ 0.

Example. Any ζ(\underline{s}) with \underline{s} of weight p ≤ 12 is a homogeneous polynomial in the following 11 MZV:

k p 2  3  5  7  8  9  10  11  12
1 ζ(2) ζ(3) ζ(5) ζ(7)  ζ(9)  ζ(11)
2  ζ(6, 2)  ζ(8, 2)  ζ(10, 2)
3  ζ(8, 2, 1)
4  ζ(8, 2, 1, 1)

Classical polylogarithms

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

Recursive definition:

Li_{1}(z) = ∑_{n≥1} \frac{z^{n}}{n} = − log(1 − z),

z \frac{d}{dz} Li_{s}(z) = Li_{s−1}(z) (s ≥ 2)

with Li_{s}(0) = 0.

For s ≥ 2, Li_{s}(1) = ζ(s).

Multiple polylogarithms in one variable

Li\underline{s}(z) := ∑_{n_{1}>n_{2}>···>n_{k}≥1} \frac{z^{n_{1}}}{n_{1}^{s_{1}} ⋯ n_{k}^{s_{k}}},

for \underline{s} = (s_{1}, …, s_{k}) with s_{j} ≥ 1 (1 ≤ j ≤ k).

ζ(\underline{s}) = Li\underline{s}(1) for s_{1} ≥ 2.

Recursive definition:

z \frac{d}{dz} Li_{(s_{1},...,s_{k})}(z) = Li_{(s_{1}−1,s_{2},...,s_{k})}(z) if s_{1} ≥ 2

(1 − z)\frac{d}{dz} Li_{(1,s_{2},...,s_{k})}(z) = Li_{(s_{2},...,s_{k})}(z) if s_{1} = 1.

Initial conditions Li\underline{s}(0) = 0. Proof:

∑_{n_{1}>n_{2}} z^{n_{1}−1} = \frac{z^{n_{2}}}{1 − z} ·

Noncommutative Polynomials

Let X := {x_{0}, x_{1}} denote an alphabet with two letters. Let X^{∗} denote the set of words on X.

The linear combinations of words with rational coefficients

∑_{u} c_{u}u,

where {c_{u} ; u ∈ X^{∗}} is a set of rational numbers with finite support, is the non-commutative ring

ℌ := ℚ⟨x_{0}, x_{1}⟩.

The product is concatenation, the unit is the empty word e.

An example of such a polynomial is

3e − 4x_{0} + 5x_{1} − x_{0}x_{1} + x_{1}x_{0} + 2x_{0}x_{1}x_{0}^{2}.

For s ≥ 1 define x_{s} := x_{0}^{s−1}x_{1}. For \underline{s} = (s_{1}, …, s_{k}) ∈ ℤ^{k} with s_{j} ≥ 1

define y\underline{s} := x_{s_{1}} ⋯ x_{s_{k}}. Hence

y\underline{s} = x_{0}^{s_{1}−1}x_{1}x_{0}^{s_{2}−1}x_{1} ⋯ x_{0}^{s_{k}−1}x_{1}.

For instance

y_{2,3} = x_{0}x_{1}x_{0}^{2}x_{1}.

The number of letters in y\underline{s} is the weight p of \underline{s}, while the numbers of x_{1} is the length k.

The set of words y\underline{s} with \underline{s} = (s_{1}, …, s_{k}) is nothing else than the set X^{∗}x_{1} of words which end with x_{1}. Define

L̂i_{y\underline{s}}(z) := Li\underline{s}(z)

for such a word.

Further define ζ̂(y\underline{s}) := ζ(\underline{s}) if s_{1} ≥ 2. Hence ζ̂(w) is defined for any word w ∈ x_{0}X^{∗}x_{1} which starts with x_{0} and ends with x_{1}. Furthermore

ζ̂(w) = L̂i_{w}(1) for w ∈ x_{0}X^{∗}x_{1}.

For a polynomial w = ∑_{u} c_{u}u define

L̂i_{w}(z) := ∑_{u} c_{u}L̂i_{u}(z)

if each u with u ∉ {e} ∪ X^{∗}x_{1} has c_{u} = 0. Hence L̂i_{w}(z) is defined for

w ∈ ℌ^{1} := ℚe + ℌx_{1}

Similarly set

ζ̂(w) := ∑_{u} c_{u}ζ̂(u)

if each u with u ∉ {e} ∪ x_{0}X^{∗}x_{1} has c_{u} = 0. For

w ∈ ℌ^{0} := ℚe + x_{0}ℌx_{1}

we have

ζ̂(w) = L̂i_{w}(1).

Hence ζ̂: ℌ^{0} → ℝ is a ℚ-linear map.

Definition. Shuffle product of two words in X^{∗}: element in ℌ defined inductively by:

e� u = u� e = u for u and u′ in X^{∗},

and

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

for u, v in X^{∗} and i, j in {0, 1}.

Examples. For i_{1}, i_{2}, j, j_{1}, j_{2} in {0, 1},

(x_{i_{1}}x_{i_{2}})� x_{j} = x_{i_{1}}x_{i_{2}}x_{j} + x_{i_{1}}x_{j}x_{i_{2}} + x_{j}x_{i_{1}}x_{i_{2}}.

\begin{matrix}(x_{i_{1}}x_{i_{2}})� (x_{j_{1}}x_{j_{2}}) = & x_{i_{1}}x_{i_{2}}x_{j_{1}}x_{j_{2}} + x_{i_{1}}x_{j_{1}}x_{i_{2}}x_{j_{2}} + x_{i_{1}}x_{j_{1}}x_{j_{2}}x_{i_{2}}+ \\ x_{j_{1}}x_{i_{1}}x_{i_{2}}x_{j_{2}} + x_{j_{1}}x_{i_{1}}x_{j_{2}}x_{i_{2}} + x_{j_{1}}x_{j_{2}}x_{i_{1}}x_{i_{2}}. \end{matrix}

Extending by distributivity with respect to the addition to ℌ defines commutative and associative algebras

ℌ�^{0} ⊂ ℌ�^{1} ⊂ ℌ_{�}

Radford’s Theorem: ℌ�^{0} , ℌ�^{1} and ℌ_{�} are (commutative) polynomials algebras on the set of Lyndon words. Consequence:

ℌ�^{1} = ℌ�^{0} [x_{1}], ℌ_{�} = ℌ�^{1} [x_{0}] = ℌ�^{0} [x_{0}, x_{1}].

Proposition. For u and u′ in ℌ�^{1} ,

L̂i_{u}(z)L̂i_{u′}(z) = L̂i_{u� u′}(z).

Consequence. For u and u′ in ℌ�^{0} ,

ζ̂(u)ζ̂(u′) = ζ̂(u� u′).

Proposition.

z \frac{d}{dz} L̂i_{x_{0}u}(z)  = L̂i_{u}(z)
(1 − z)\frac{d}{dz} L̂i_{x_{1}u}(z) = L̂i_{u}(z)

Notation.

ω_{0}(z) := \frac{dz}{z}, ω_{1}(z) := \frac{dz}{1 − z} ·

Definition. Define L̂i_{w}(z) for any w ∈ X^{∗} as follows.

Li_{e}(z) := 1, L̂i_{x_{0}^{s}}(z) := \frac{1}{s!}(log z)^{s} for s ≥ 1

and, for j ∈ {0, 1},

L̂i_{x_{j}u}(z) := ∫_{0}^{z} L̂i_{u}(z)ω_{j}(z)

if the word x_{j}u contains a letter x_{1}. Generating series.

L̂i(z) := ∑_{w∈X^{∗}} L̂i_{w}(z)w.

Knizhnik-Zamolodchikov Differential Equation

\frac{d}{dz} L̂i(z) = (\frac{x_{0}}{z} + \frac{x_{1}}{1 − z} ) L̂i(z).

Initial condition: z ↦ L̂i(z)z^{−x_{0}} is holomorphic near 0 in

ℂ \ {(−∞, 0] ∪ [1, ∞)}: 0 1 Theorem (H.N. Minh and M. Petitot). The functions L̂i_{w}(z), for w in X^{∗} are linearly independent over ℂ. Drinfeld associator:

Φ_{KZ} = ∑_{w∈X^{∗}} ζ̂(w)w ∈ ℂ⟨⟨x_{0}, x_{1}⟩⟩.