R ´eunion d’ ´et ´e de la SMC - Sask ato on, 5 juin 2001

R ´eunion d’ ´et ´e de la SMC - Sask ato on, 5 juin 2001

P olylogarithms

and

Multiple Zeta V alues

by

Mic hel W ALDSCHMIDT

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

Sp ecial V alues of the Riemann Zeta F unction ζ ( s ) := 1 + 1 2 s + 1 3 s + ·· · + 1 n s + ·· · s ≥ 2 .

L. Euler:

ζ (2 m ) = − 1 2 · (2 π i ) 2 m B 2 m (2 m )! for m ≥ 1

with t e t − 1 = ∞ X

k =0 B k k ! · t k , | t | < 2 π .

In particular ζ (2 m ) π − 2 m ∈ Q for m ≥ 1 .

F. Lindemann: π is tr ansc endental.

R. Ap ´ery: ζ (3) 6∈ Q .

T. Riv oal: In finitely many ζ ( a ) with a odd ≥ 5 ar e irr ational.

K. Ball + T. Riv oal: The Q -ve ctor subsp ac e of R sp anne d by

ζ (3) ,ζ (5) , ·· · ,ζ ( a )

has dimension ≥ log a 1 + log 2 ¡ 1 + o (1) ¢ .

T. Riv oal, W. Zudilin: One at le ast of the 9 numb ers

ζ (5) ,ζ (7) , ·· · ,ζ (21)

is irr ational.

Exp ected : Euler’s relations are the only algebraic relation among the n um b ers ζ (2) ,ζ (3) ,ζ (4) ,. .. ,ζ ( s ) ,. ..

Conjecture. The n um b ers

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

algebraically indep enden t.

In other terms: F or n ≥ 0 and P ∈ Q [ X 0 ,X 1 ,. .. ,X n ] \ { 0 } ,

P ¡ π ,ζ (3) ,ζ (5) ,. .. ,ζ (2 n + 1) ¢ 6 = 0 .

Definition. F or s = ( s 1 ,. .. ,s k ) in Z k with s 1 ≥ 2 and s j ≥ 1 ( 2 k ), ζ ( s 1 ,. .. ,s k ) := X

n

> ··· >n

≥ 1 n − s

1 ·· · n − s

k .

Examples of algebr aic relations

1 ζ (2) 2 = 2 ζ (2 , 2) + ζ (4)

Pr oof. X

n ≥ 1 1 n 2 X

m ≥ 1 1 m 2 = X

n>m ≥ 1 1 n 2 m 2 + X

m>n ≥ 1 1 n 2 m 2 + X

n ≥ 1 1 n 4 .

2 ζ (2) 2 = 2 ζ (2 , 2) + 4 ζ (3 , 1)

Corollary . ζ (3 , 1) = 1 4 ζ (4) .

3 ζ (3) = ζ (2 , 1)

EZ-Face b y J . Borw ein, P . Lisonek and P . Irvine

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

Conjectures of Zagier, Drinfeld, Kon tsevic h, Gonc haro v, Ho ffman, Broadh urst, Cartier .. .

F or p ≥ 2 let Z p denote the Q -v ector subspace of R spanned b y the 2 p − 2

elemen ts ζ ( s ) for s = ( s 1 ,. .. ,s k ) of weight s 1 + ·· · + s k = p and s 1 ≥ 2 . Let d p b e the dimension of Z p .

Z 0 = Q with ζ ( s ) := 1 for k = 0 Z 1 = { 0 } Z 2 = h ζ (2) i Q Z 3 = h ζ (3) i Q since ζ (2 , 1) = ζ (3) Z 4 = h ζ (4) i Q = h ζ (2) 2 i Q = h π 4 i Q since

ζ (3 , 1) = 1 4 ζ (4) , ζ (2 , 2) = 3 4 ζ (4) , ζ (2 , 1 , 1) = ζ (4) = 2 5 ζ (2) 2 .

Hence d 0 = 1 , d 1 = 0 , d 2 = d 3 = d 4 = 1 .

Z 5 = h ζ (2) ζ (3) ,ζ (5) i Q , hence d 5 ≤ 2.

Pr oof. ζ (2 , 1 , 1 , 1) = ζ (5) ,

ζ (3 , 1 , 1) = ζ (4 , 1) = 2 ζ (5) − ζ (2) ζ (3) ,

ζ (2 , 1 , 2) = ζ (2 , 3) = 9 2 ζ (5) − 2 ζ (2) ζ (3) , ζ (2 , 2 , 1) = ζ (3 , 2) = 3 ζ (2) ζ (3) − 11 2 ζ (5) .

Conjecture (D. Zagier). W e ha ve

d p = d p − 2 + d p − 3 for p ≥ 4 .

Equiv alen t form ulation: X

p ≥ 0 d p X p = 1 1 − X 2 − X 3 ·

Conjecture (M. Hoffman). F or p ≥ 1 , a basis for the Q -v ector space Z p is giv en b y the n um b ers ζ ( 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 b y all ζ ( s ) is graded for the w eigh t:

Z p Z p

⊂ Z p + p

. The length k de fines a filtration on Z . Denote b y F k Z p the Q -v ector subspace of R spanned b y all ζ ( s ) of w eigh t p and length k . F urther, let d p k denote the dimension of F k Z p / F k − 1 Z p .

Conjecture (D. Broadh urst). W e ha v e 

 X

p ≥ 0 X

k ≥ 0 d pk X p Y k   − 1

= 1 − X 2 − X 3 + X Y 2 (1 − Y 2 ) (1 + X 2 )(1 − Y 6 ) ·

Conjecture (A.B. Gonc haro v). As a Q -algebra, Z is the direct of Z p for p ≥ 0 .

Example. An y ζ ( s ) with s of w eigh t p ≤ 12 is a homogeneous p olynomial in the follo wing 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

3 ζ (8 , 2 , 1)

4 ζ (8 , 2

Classical p olylogarithms

Li s ( z ) := X

n ≥ 1 z n n s for | z | < 1 .

R ecursive de finition:

Li 1 ( z ) = X

n ≥ 1 z n n = − log (1 − z ) ,

z d dz Li s ( z ) = Li s − 1 ( z ) ( s ≥ 2)

with Li s (0) = 0 .

F or s ≥ 2, Li s (1) = ζ ( s ).

Multiple p olylogarithms in one v ariable

Li s ( z ) := X

n

>n

> ··· >n

≥ 1 z n

n s

1 ·· · n s

k ,

for s = ( s 1 ,. .. ,s k ) with s j ≥ 1 (1 ≤ j ≤ k ).

ζ ( s ) = Li s (1) for s 1 ≥ 2 .

R ecursive de finition:

z d dz Li ( s

,... ,s

) ( z ) = Li ( s

− 1 ,s

,... ,s

) ( z ) if s 1 ≥ 2

(1 − z ) d dz Li (1 ,s

,... ,s

) ( z ) = Li ( s

,... ,s

) ( z ) if s 1 = 1 .

Initial conditions Li s (0) = 0 . Pr oof: X

n

>n

z n

− 1 = z n

1 − z ·

Noncomm utativ e P olynomials

Let X := { x 0 ,x 1 } denote an alphab et with tw o letters. Let X ∗

denote the set of w ords on X .

The linear com binations of w ords with rational co efficien ts X

u c u u,

where { c u ; u ∈ X ∗ } is a set of rational n um b ers with finite supp ort, is the non-comm utativ e ring

H := Q h x 0 ,x 1 i .

The pro duct is concatenation, the unit is the empt y w ord e . An example of suc h a p olynomial is

3 e − 4 x 0 + 5 x 1 − x 0 x 1 + x 1 x 0 + 2 x 0 x 1 x 2 0 .

F or s ≥ 1 de fine x s := x s − 1 0 x 1 . F or s = ( s 1 ,. .. ,s k ) ∈ Z k with de fine y s := x s

·· · x s

. Hence

y s = x s

− 1 0 x 1 x s

− 1 0 x 1 ·· · x s

− 1 0 x 1 .

F or instance y 2 , 3 = x 0 x 1 x 2 0 x 1 .

The n um b er of letters in y s is the w eigh t p of s , while the n um b ers is the length k .

The set of w ords y s with s = ( s 1 ,. .. ,s k ) is nothing else than the X ∗ x 1 of w ords whic h end with x 1 . De fine b Li y

( z ) := Li s ( z )

for suc h a w ord.

F urther de fine ˆ ζ ( y s ) := ζ ( s ) if s 1 ≥ 2 . Hence ˆ ζ ( w ) is de fined for w ord w ∈ x 0 X ∗ x 1 whic h starts with x 0 and ends with x 1 . F urthermore ˆ ζ ( w ) = b Li w (1) for w ∈ x 0 X ∗ x 1 .

F or a p olynomial w = X

u c u u de fine

b Li w ( z ) := X

u c u b Li u ( z )

if eac h u with u 6∈ { e } ∪ X ∗ x 1 has c u = 0. Hence b Li w ( z ) is de fined for

w ∈ H 1 := Q e + H x 1

Similarly set ˆ ζ ( w ) := X

u c u ˆ ζ ( u )

if eac h u with u 6∈ { e } ∪ x 0 X ∗ x 1 has c u = 0. F or

w ∈ H 0 := Q e + x 0 H x 1

w e ha v e ˆ ζ ( w ) = b Li w (1) .

Hence ˆ ζ : H 0 → R is a Q -linear map.

Definition. Sh uffle pro duct of tw o w ords in X ∗ : elemen t in H defined inductiv ely b y:

e x u = u x e = u for u and u 0 in X ∗ , and ( x i u ) x ( x j v ) = x i ¡ u x ( x j v ) ¢ + x j ¡ ( x i u ) x v ¢

for u , v in X ∗ and i , j in { 0 , 1 } . Examples. F or i 1 , i 2 , j , j 1 , j 2 in { 0 , 1 } ,

( x i

x i

) x x j = x i

x i

x j + x i

x j x i

+ x j x i

x i

.

( x i

x i

) x ( x j

x j

) = x i

x i

x j

x j

+ x i

x j

x i

x j

+ x i

x j

x j

x i

x j

x i

x i

x j

+ x j

x i

x j

x i

+ x j

x j

x i

x i

Extending b y distributivit y with resp ect to the addition to H de comm utativ e and asso ciativ e algebras

H 0 x ⊂ H 1 x ⊂ H x

R adfor d’s The or em: H 0 x , H 1 x and H x are (comm utativ e) p olynomials algebras on the set of Lyndon w ords. Conse quenc e:

H 1 x = H 0 x [ x 1 ] , H x = H 1 x [ x 0 ] = H 0 x [ x 0 ,x 1 ] . Prop osition. F or u and u 0 in H 1 x ,

b Li u ( z ) b Li u

( z ) = b Li u x u

( z ) . Consequence. F or u and u 0 in H 0 x ,

ˆ ζ ( u

) ˆ ζ ( u 0 ) = ˆ ζ ( u x u 0 ) .

Prop osition.  

 z d dz b Li x

u ( z ) = b Li u ( z )

(1 − z ) d dz b Li x

u ( z ) = b Li u ( z )

Notation. ω 0 ( z ) := dz z , ω 1 ( z ) := dz 1 − z · Definition. Define b Li w ( z ) for an y w ∈ X ∗ as follo ws.

Li e ( z ) := 1 , b Li x

( z ) := 1 s

! (log z ) s for s ≥ 1

and, for j ∈ { 0 , 1 } ,

b Li x

u ( z ) := Z z

0 b Li u ( z ) ω j ( z )

if the w ord x j u con tains a letter x 1 .

Generating series.

b Li ( z ) := X

w ∈ X

b Li w ( z ) w .

Knizhnik-Zamolo dc hik o v Differen tial Equation

d dz b Li ( z ) = ³ x 0 z + x 1 1 − z ´ b Li ( z ) . Initial condition: z 7→ b Li ( z ) z − x

is holomorphic near 0 in C \ © ( −∞ , 0] ∪ [1 , ∞ ) ª : 0 1

Theorem (H.N. Minh and M. P etitot). The functions b Li w ( z ) , for w in X ∗ are linearly indep enden t ov er C .

Drinfeld asso ciator:

Φ K Z = X

w ∈ X

ˆ ζ ( w ) w ∈ C hh x 0 ,x 1 ii .