Introduction to multiple zeta values
1 Multiple zeta values and multiple polylogarithms 1.1 Multiple zeta values: de nition and rst results The (Riemann) zeta function is a meromorphic function on C de ned for 1 by the formula: ζ(s) =
+∞ X 1 . ns
n=1
The multiple zeta values (MZV) are a generalisation of the zeta values by which we understand the value of the zeta function at integers greater or equal to 2.
De nition 1. Let k = (k1 , . . . , kp ) be a p-tuple of positive integer with k1 > 2. The multiple zeta values ζ(k1 , ..., kp ) is the real number de ned by: ζ((k)) = ζ(k1 , ..., kp ) =
1
X k1 n1 >...>np >0 n1
k
· · · np p
The depth of k is the length p of the tuple and its weight is n = k1 + · · · + kp . We know little about what these real numbers are. Although we can compute ζ(2) =
π2 6
as an undergraduate exercise, even the more general result ζ(2n) = C2n π 2n
C2n ∈ Q
Bk is di cult. Here the constant C2n is linked with the Bernoulli numbers Bk de ned by exx−1 = k! . Altohough we know that ζ(3) is irrational , for the ζ(2n + 1) with n greater than 2 it is only a conjecture. The conjecture says that each MZV is a trancendental number. There are lots of algebraic relations between the MZVs as alreday the following example shows:
P
ζ(k)ζ(l) =
+∞ +∞ X X
n=1 m=1
X X 1 1 1 = + + k l k l n m n m ml n k n>m>0
m>n>0
+∞ X
1 nk+l
= ζ(k, m) + ζ(l, k)ζ(k + l).
(1)
n=0 n=m
As we will see later there are also rational linear relations between the MZVs. The MZV appear in dierent geomtric contexts (Moduli spaces of curves, fondamental group of P1 \ {0, 1, ∞}, Hodge structure). They also can be seen as the value of some important functions at the point 1.
1
1.2 Multiple polylogarithms De nition 2. For a p-tuple (k1 , . . . , kp ) of positive integers, the multiple polylogarithm Lik1 ,...,kp is de ned
by
n
X
z1n1 · · · zp p
n1 >...>np >0
nk11 · · · npp
Lik1 ,...,kp (z1 , . . . , zp ) =
|zi | < 1 ∀i).
k
Remark 1. In contrast of the case of the MZV where we need the condition k1 > 2 in order to insure the convergence of the series, we don't need that restriction for |zi | < 1. The logarithm function appears in weight 1 as Li1 (z) =
+∞ X zn
n=1
n
The polylogarithm of weight k is Lik (z) =
= − log(1 − z).
+∞ X zn nk
n=1
but we are only interested in the case k > 1. The multiple zeta value ζ(k1 , ..., kp ) (k1 > 2) appears as a speci c value of the corresponding Li function ζ(k1 , ..., kp ) = Lik1 ,...,kp (1, . . . 1).
Looking at the particular case where z1 = . . . = zp = z we can de ne an important spacial case of the multiple polylogarithm on C by : z 7−→ Lik1 ,...,kp (z) =
+∞ X
z n1 +···+np
n1 >...>np >0
nk11 · · · npp
(|z| < 1).
k
The multiple polylogarimths satisfyes dierential equations. We give, in the following proposition what there are in the case of the function from C to C.
Proposition 1. Let (k1 , . . . , kp ) be a p-tuple of positive integers. Then we have: dLik1 ,...,kp (z) dz
and
=
Lik1 −1,k2 ,...kp (z) 1−z
if k1 > 2
Li (z) k2 ,...kp z
if k1 = 1
dLi1 (z) − log(1 − z) 1 = = . dz dz 1−z
The Lik1 ,...,kp are such that Lik1 ,...,kp (0) = 0.
1.3 Integral representation for multiple zeta values Using the well known fact that if a function with a derivative is such that f (0) = 0 then f (t) = we have with k1 > 3: Z ζ(k1 , ..., kp ) = Lik1 ,...,kp (1) =
0
1
Lik1 −1,k2 ,...,kp (t) t
Z dt = 0
2
1
1 t
Z 0
t
Lik1 −2,k2 ,...,kp (u) u
! du dt
Rt 0
f 0 (t)dt
Iterating the process, there appears the Kontsevich interate integral representation for the MZV that we present here. To the tuple k = (k1 , . . . , kp ), with k1 > 2 and n = k1 + · · · + kp , we associate the n-tuple k = ( 0, . . . , 0 , 1, . . . , 0, . . . , 0 , 1) = (εn , . . . , ε1 ) | {z } | {z } k1 −1
times
times
kp −1
and the dierential form ωk = ωk = (−1)p
dt1 dtn ∧ ··· ∧ . t1 − ε 1 tn − ε n
Then, setting ∆n = {0 < t1 < . . . < tn < 1}, direct integration yields: Z
Z
ζ(k) =
ωk = ∆n
0
1
1 tn
tn
Z
Z
1
···
tn−1
0
tn −2
0
tn−k1 +1
Z
1 tn−k1 +1
0
Z
1 1 − tn−k1
tn −k1
···
dtn−1 dtn .
0
Thus, instead of writing ζ(k1 , ..., kp ) we will sometime write ζ(k1 , ..., kp ) = ζ(0,...,0,1,...,0...0,1) = ζk¯
or if we want letters (indeterminates) to appear instead of numbers (0 and 1) . . X} , Y ) ζ(k1 , ..., kp ) = ζ( X, . . . , X , Y, . . . , X | .{z | {z } k1 −1
times
kp −1
times
2 Convergent double shu e relations
2.1 Formal shu e product and shu e of multiple zeta values The shu e product of an n-tuple (e1 , . . . , en ) = e1 · e and an m-tuple (f1 , . . . , fm ) = f1 · f ,the ei and the fj being symbols, is de ned recursively by:
X
X
Example 1.
XY
e1 · e
Xf
1
· f = e1 · (e
X(e
0 1
· e0 )) + e01 · ((e1 · e)
Xe )
(2)
0
and e () = () e = e. Here the + is a formal sum, b · B means that we concatenate b at the beginning of the tuple B and we impose that · is linear in B .
X AB = XY AB + XAY B + XABY + AXY B + AXBY + ABXY
We will write sh((e1 , . . . , en ), (e01 , . . . , e0m )) the subset of the group Sn+m of the permutations of n + m elements such that: σ ∈ sh((e1 , . . . , en ), (e01 , . . . , e0m ))
⇔
σ ∈ Sn+m σ(1) < σ(2) < . . . < σ(n)
and
σ(n + 1) < σ(n + 2) < . . . < σ(n + m)
X
Let k and l be two tuples of integers as above. If σ is a term of the formal sum k l all coe cients being equal to 1, we will write σ ∈ k l. We will put an index σ on any object which naturally depends on a shu e. The following proposition yields the quadratic relations (3) known as the shu e relations.
X
Proposition 2. Let k = (k1 , . . . , kp ) and l = (l1 , . . . , lq ) with k1 , l1 > 2. Then : Z ζ(k)ζ(l) = ∆n
Z ωk
∆m
Z
X
ωl =
σ∈k
3
X
ωσ = l)
∆n+m
X σ∈k
X
ζσ . l
(3)
Proof. Let n = k1 + ... + kp and m = l1 + ... + lq .Let ∆ be the product {0 < t1 < . . . < tn < 1} × {0 < tn+1 < . . . < tn+m < 1} Then we have Z
Z
Z
∆n
ωk
∆m
ωl =
∆n
dtn dt1 ··· 1 − t1 tn
Z ∆m
dtn+m dtn+1 ··· 1 − tn+1 tn+m
Z = ∆
dtn dtn+1 dtn+m dt1 ··· ··· . 1 − t1 tn 1 − tn+1 tn+m
The set ∆ can be split into a union of simplexes ∆=
a
with ∆σ = {0 6 tσ(1) 6 tσ(2) 6 ... 6 tσ(n+m) 6 1},
∆σ
σ∈sh([[1,n]],[[n+1,m]])
where [[1, n]] denote the ordered sequence of integers from 1 to n. As the sets {tr = ts } are of codimension 1,the integral over ∆ is the sum of the integrals over the correponding simplex. But the integral over one of these simplices is, up to the numbering of the variables, X Z ωσ . exactly one term of the sum σ∈sh(k,l)
∆n+m
Example 2. Using the decoposition given by the shu e product one nd ζ(2)ζ(2) = ζ(XY )ζ(XY ) = ζ(XY XY ) + ζ(XXY Y ) + ζ(XXY Y ) + ζ(XXY Y ) + ζ(XXY Y ) + ζ(XY XY )
(4)
ζ(2)ζ(2) = 2ζ(2, 2) + 4ζ(3, 1)
2.2 Formal stu e and stu e of multiple zeta values The stu e product of a p-tuple k = (k1 , . . . , kp ) = k0 · kp and a q -tuple l = (l1 , . . . , lq ) = l0 · lp is de ned recursively by the formula: (5)
(k) ∗ (l) = (k ∗ l0 · lq + k0 ∗ l) · kp + k0 ∗ l0 · (kp + lq )
and k ∗ () = () ∗ k = k. Here the + is a formal sum, A · a means that we concatenated a at the end of the tuple A and we impose that · is linear in A. Let k and l be two such tuples of integers. If σ is a term of the formal sum k ∗ l with all coe cients being equal to 1, we will write σ ∈ st(k, l). The shu e product was based on the integral representation of the MZV and on splitting the integration domain. Here, with the de nition of the MZV as series, we will use essentially the idea. In order to multiply two multiple zeta values ζ(k) and ζ(l), we split the summation domain of the product ζ(k)ζ(l) {0 < n1 < . . . < np } × {0 < m1 < . . . < mq }
into all the domains that preserve the order of the ni and the mj and the boundary domains where some ni are equal to some mj . We obtain the following well-known proposition, giving the quadratic relations (6) between multiple zeta values known as the stu e relations :
Proposition 3. Let k = (k1 , . . . , kp ) and l = (l1 , . . . , lq ) as above with k1 , l1 > 2. Then we have :
ζ(k)ζ(l) =
X
n1 >...>np >0
X
1
m1 >...>mq >0
m1l1 · · · mqq
1 k
nk11 · · · npp
l
X
=
σ∈st(k,l)
We have already seen an example of such a decomposition in the equation (1).
4
ζ(σ).
(6)
2.3 Double shu e relation The product of MZV is decomposed, using the shu e product or the stu e product, into two dierent linear conbinations of other MZV. For example comparing (4) and (1) for n = m = 2 one nds: ζ(4) = 4ζ(3, 1).
As the product of two MZV is uniquely de ned as the product of real numbers, comparing the shu e product and the stu e product leads to rational linear relations between the MZV. Those relations are the double shu e relations.
3 Regularized double shu e relation 3.1 Regularisations Let A be the set of all possible words in the alphabet {X, Y }, and A be the polynomial algebra Q < X, Y > in two noncommutative variables, i.e. the monomial are the elements in A and the product · is the bilinear concatenation. We have previously de ne ζ(W ) for all words W begin with X and ending with Y , W ∈ X · A · Y . In this section we will see two theorems making it possible to consitently de ne values ζ(W ) even when W is not in X · A · Y in two dierent ways ζ X (W ) and ζ ∗ (W ), respectively corresponding to the shu e regularisation and to the stu e regularisation.
Shu e statement
Theorem 4 (Shu e regularisation). There exist a collection of real numbers ζ X (W ) for any word W such that the collection of these real numbers satis es :
∈A
1. If W ∈ AC = X · A · Y then ζ X (W ) = ζ(W ),
2. For all W1 and W2 in A,
ζ X (W1 )ζ X (W2 ) =
ζ X (W )
X W ∈W1
3. ζ X (Y ) = 0
X
W2
Remark 2. There are in fact many choices of collections of real numbers that satis y 1) and 2) but only one that also satis es the third condition.
Stu e statement The stu e regularisation is de ned only for words in
A · Y , that is ending with Y .
There is a one to one map between A · Y and the set of tuples of positive integers:
Proposition 5 (Stu e in AY ). Any word W in A · Y can be written W = X k1 −1 Y X k2 −1 Y · · · X kp −1 Y
ki ∈ N∗ p > 0.
Here X 0 is intepreted as the empty letter, so that X 0 Y = Y . The following map is one to one between AY and the set of tuples of positive integers: W = X k1 −1 Y X k2 −1 Y · · · X kp −1 Y 7−→ kW = (k1 , k2 , . . . kp )
Using this one to one correspondence, the stu e W1 ∗ W2 of two words in A · Y is well de ned.
Theorem 6 (Stu e regularisation). There exist a collection of real numbers ζ ∗ (W ) for any word in A · Y such that the collection of these reals satis es : 5
1. If W ∈ AC = X · A · Y then ζ ∗ (W ) = ζ(W ), 2. For all W1 and W2 in AY ,
X
ζ ∗ (W1 )ζ ∗ (W2 ) =
ζ ∗ (W )
W ∈W1 ∗W2
3. ζ ∗ (Y ) = 0 Remark 3. There are in fact many choices of collections of real numbers that satis es 1) and 2) but only one that also satis es the third condition.
3.2 Explicit results and examples General expressions Those two dierent regularisations are dierent for words that are power of Y and there is a comparison theorem which allows one to know what are the dierences. In this section we will give some explicite expressions and some relations that one can nd using this extended double shu e. We will writte wh(W ) for the number of Y occuring in W .
Proposition 7. Let W = en · · · e1 be a word in A (ei ∈ {X, Y }). As for the convergent word we can associate to W a dierential form: ωW = (−1)wh(W )
εi = 0 if ei = X where εi = 1 if eiP= Y . wh(W ) = εi
dt1 dtn ∧ ··· ∧ t1 − ε 1 tn − ε n
For all ε > 0, setting ∆n (ε) = {ε < t1 < . . . < tn < 1 − ε} the integral: Z IW (ε) =
ωW ∆n (ε)
admits a series expansion around 0 in ε and log(ε), that is IW (ε) ∈ R[[ε, log(ε)]]. The constant term of this serie is the shu e regularisation ζ X (W ). Using that propositions we can easily compute the regularisation of certain class of words.
Proposition 8. For any convergent word W let W be the word obtained from W writting it in the reverse order and θ(W ) be the word where one has exchange X and Y in W , then: ζ X (θ(W )) = (−1)n ζ(W ) ¯ ) = (−1)n ζ(W ) ζ X (W
Remark 4. Those equalities are in fact true for every word W in A.
Theorem 9 (Furusho-The multiple zeta value algebra and the stable derivation algebra, prop 3.2.3). For all word W in A, the number ζ X (W ) is a rational linear combination of MZV for convergent words. e be the serie Proposition 10. Let Φ e= Φ
X (−1)n−1 n>1
n
ζ X (X n−1 Y )Y n +
X
(−1)wh(W ) ζ X (W )W.
W ∈A·Y
e Then for any word W in A · Y , the real number ζ ∗ (W ) is the coe cient of W in Φ
6
The second sum is closely linked to the Drinfel'd associator ΦKZ =
X
(−1)wh(W ) ζ(W )
W ∈A
and the rst one can be seen as a correction term. Remark 5. We see that the two regularisations are dierent only for words that are power of Y . The fact that the two regularisation have to be dierent is shown in the following example :
Example 3. The stu e product ζ(1)2 = 2ζ ∗ (1, 1) + ζ(2) leads to ζ ∗ (1, 1) = π2 /12 but the series expansion of the corresponding integral gives ζ X = 0.
Extended double shu e relations As ζ X (1) = ζ ∗ (1) = 0, the two dierent products give linear relations between MZV.
Example 4. We have: ζ(2)ζ(1) = ζ(2)ζ X (1) = 2ζ(2, 1) + ζ X (1, 2) = ζ(2)ζ ∗ (1) = ζ(2, 1) + ζ ∗ (1, 2) + ζ(3).
As ζ X (1, 2) = ζ ∗ (1, 2) we have: ζ(3) = ζ(2, 1).
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