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MHF Preprint Series

Kyushu University

21st Century COE Program

Development of Dynamic Mathematics with

High Functionality

Derivation and double shuffle

relations for multiple zeta values

K. Ihara, M. Kaneko

D. Zagier

MHF 2004-29

( Received October 27, 2004 )

Faculty of Mathematics

Kyushu University

Fukuoka, JAPAN

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DERIVATION AND DOUBLE SHUFFLE RELATIONS

FOR MULTIPLE ZETA VALUES

Kentaro Ihara, Masanobu Kaneko and Don Zagier

In recent years, there has been a considerable amount of interest in certain real

numbers called multiple zeta values (MZV’s). These numbers, first considered by

Euler in a special case, have arisen in various contexts in geometry, knot theory,

mathematical physics and arithmetical algebraic geometry. It is known that there

are many linear relations over Q among the MZV’s, but their exact structure remains

quite mysterious.

The MZV’s can be given both as sums (1.1) or as integrals (1.2). From each of

these representations one finds that the product of two MZV’s is a Z-linear combi-

nation of MZV’s, described by a so-called shuffle product, but the two expressions

obtained are different. Their equality gives a large collection of relations among

MZV’s which we call the double shuffle relations. These are not sufficient to imply

all relations among MZV’s, but it turns out that one can extend the double shuffle

relations by allowing divergent sums and integrals in the definitions (roughly speak-

ing, by adjoining a formal variable T corresponding to the infinite sum?1/n), and

ring of MZV’s completely. This observation, which was made by the third author

a number of years ago and has been found independently by a number of other

researchers in the field, is central to this paper. Our first goal (Sections 1, 2 and 3)

is to explain the EDS relations in detail. This requires introducing a certain renor-

malization map whose definition, initially forced on us by the asymptotic properties

of divergent multiple zeta sums and integrals, is later seen to have a purely algebraic

meaning. This is carried out in Sections 4–5, in which we also prove the equivalence

of a number of different versions of the basic conjecture on the sufficiency of the

EDS relations. In the next two sections we prove a number of further algebraic

properties of the ring of MZV’s which can be deduced from the EDS relations. In

particular, we introduce a number of derivations (and, by exponentiation, automor-

phisms) of the ring of formal MZV’s and use them to give new, and in several cases

conjecturally complete, sets of relations among MZV’s. These identities contain

previous results of Hoffman and Ohno as special cases. Finally, the last section

of the paper contain a reformulation of the EDS relations as a problem of linear

algebra and some general results concerning this problem.

Some of the results in this paper (in particular, in Sections 2 and 8 concerning

the double shuffle relations and renormalization) originated in work which the third-

named author did in the year 1988–1994 but never published. Since that time much

1

that these extended double shuffle (EDS) relations apparently suffice to describe the

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work has been done by other writers (Goncharov, Minh, Petitot, Boutet de Monvel,

´Ecalle, Racinet, ...) and there is a considerable amount of overlap with their results.

We nevertheless present a self-contained description of the work.

§1. Double shuffle relations (convergent case)

The multiple zeta value (MZV) is defined by the convergent series

ζ(k) = ζ(k1,k2,... ,kn) =

?

m1>m2>···>mn>0

1

mk1

1mk2

2···mkn

n

,

(1.1)

where k = (k1,k2,... ,kn) is an admissible index set (= ordered set of positive

integers whose first element is strictly greater than 1). This value has an integral

representation, known as the Drinfel’d integral, as follows:

ζ(k1,k2,... ,kn) =

?

···

?

1>t1>t2>···>tk>0

ω1(t1)ω2(t2)···ωk(tk),

(1.2)

where k = k1+ k2+ ··· + knis the weight and ωi(t) = dt/(1 − t) if i ∈ {k1,k1+

k2,... ,k1+ k2+ ··· + kn} and ωi(t) = dt/t otherwise. There are many linear

relations over Q among MZV’s of the same weight. The main goal of the theory is

to give as complete a description of them as possible.

The product of two MZV’s is expressible as a sum of MZV’s. We may see this

by using either the defining series (1.1) or the integral representation (1.2) of ζ(k),

but the multiplication rules obtained by the two methods are not the same; the

equality of the products which they give will be our main tool for obtaining linear

dependences among MZV’s. To describe these multiplication rules, it is convenient

to use the algebraic setup given in Hoffman [8]. Let H = Q?x,y? be the non-

commutative polynomial algebra over the rationals in two indeterminates x and y,

and H1and H0its subalgebras Q + Hy and Q + xHy, respectively. Let Z : H0→ R

be the Q-linear map (“evaluation map”) which assigns to each word (monomial)

u1u2···ukin H0the multiple integral

?

1>t1>t2>···>tk>0

···

?

ωu1(t1)ωu2(t2)···ωuk(tk) (1.3)

where ωx(t) = dt/t, ωy(t) = dt/(1−t). We set Z(1) = 1. Since the word u1u2···uk

is in H0, we always have ωu1(t) = dt/t and ωuk(t) = dt/(1 − t), so the integral

converges. By the Drinfel’d integral representation (1.2), we have

Z(xk1−1yxk2−1y ···xkn−1y) = ζ(k1,k2,...,kn).

The weight k = k1+ k2+ ··· + kn of ζ(k1,k2,...,kn) is the total degree of the

corresponding monomial xk1−1yxk2−1y ···xkn−1y, and the depth n the degree in y.

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Let zk := xk−1y, which corresponds under Z to the Riemann zeta value ζ(k).

Then H1is freely generated by zk(k = 1,2,3,...). We define the harmonic product

∗ on H1inductively by

1 ∗ w = w ∗ 1 = w,

zkw1∗ zlw2 = zk(w1∗ zlw2) + zl(zkw1∗ w2) + zk+l(w1∗ w2),

for all k, l ≥ 1 and any words w, w1, w2∈ H1, and then extending by Q-bilinearity.

Equipped with this product, H1becomes a commutative algebra ([8]) and H0a

subalgebra. We will denote these algebras by H1

law of MZV’s can then be stated by saying that the evaluation map Z : H0→ R is

an algebra homomorphism with respect to the multiplication ∗, i.e.

Z(w1∗ w2) = Z(w1)Z(w2)

for all w1, w2 ∈ H0. For instance, the harmonic product zk∗ zl = zkzl+ zlzk+

zk+l corresponds to the identity ζ(k)ζ(l) = ζ(k,l) + ζ(l,k) + ζ(k + l).

that this multiplication rule corresponds simply to the formal multiplication and

rearrangement of the terms of the sums (1.1), and would remain true if the numbers

miin these sums were to run over any other discrete subsets of R+, so long as the

series converged absolutely.

The other commutative product ?, referred to as the shuffle product, correspond-

ing to the product of two integrals in (1.2), is defined on all of H inductively by

setting

∗and H0

∗. The first multiplication

(1.4)

Notice

1?w = w?1 = w,

uw1?vw2 = u(w1?vw2) + v(uw1?w2),

for any words w, w1, w2∈ H and u, v ∈ {x,y}, and again extending by Q-bilinearity.

This product gives H the structure of a commutative Q-algebra ([15]) which we

denote by H?. Obviously the subspaces H1and H0become subalgebras of H?,

denoted by H1

iterated integrals, the evaluation map Z is again an algebra homomorphism for the

multiplication ?:

Z(w1?w2) = Z(w1)Z(w2).

Again, this rule is a formal consequence of the formula (1.3) and would hold for the

values defined by these integrals if ωxand ωywere replaced by any other differential

forms for which the integrals converged; it is only in the equality between the two

multiplication rules that the specific definition of MZV’s is important.

By equating (1.4) and (1.5), we get the double shuffle relations of MZV:

?and H0

?respectively. By the standard shuffle product identity of

(1.5)

Z(w1?w2) = Z(w1∗ w2)(w1,w2∈ H0).

(1.6)

The first example is

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

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from which we deduce 4ζ(3,1) = ζ(4).

however, do not suffice to obtain “all” relations. For instance, we have 1, 2 and 4

MZV’s in weights 2, 3 and 4 respectively, but the relation above of weight 4 is

obviously the only double shuffle relation in weight ≤ 4, so that we are only able to

reduce the dimensions to 1, 2, 3 rather than the correct 1, 1, 1. We therefore need

a larger supply of relations. This is the object of the “renormalization” procedure

discussed in the next section.

These “finite” double shuffle relations,

§2. Regularizations of multiple zeta values

Proposition 1. We have two algebra homomorphisms

Z∗: H1

∗−→ R[T]and

Z?: H1

?−→ R[T]

which are uniquely characterized by the properties that they both extend the evalua-

tion map Z : H0→ R and send y to T.

Proof. This is clear from the isomorphisms H1

[15]) and the fact that the map Z is an algebra homomorphism for both harmonic

and shuffle products.

∗? H0

∗[y] and H1

?? H0

?[y] ([8] and

?

For an index k = (k1,... ,kn) (not necessarily admissible, i.e., any ordered set

of positive integers), the images under the maps Z∗and Z?of the corresponding

word xk1−1y ···xkn−1y are denoted by Z∗

admissible, we have Z∗

that, for k = (1,1,... ,1

?

Z∗

s!+ (terms of lower degree in T)

k(T) and Z?

k(T), respectively. If k is

k(T) = Z?

,k?) with k?admissible and s ≥ 0 we have

k(T) = ζ(k). In general, we see by induction on s

???

s

k(T) = ζ(k?)Ts

and similarly

Z?

k(T) = ζ(k?)Ts

s!+ (terms of lower degree in T),

and also that the coefficients of Tiin Z∗

of multiple zeta values of weight k − i (k = weight of k). Here are a few examples:

k(T) and Z?

k(T) are Q-linear combinations

k

(1)(1,1) (1,2)

Z∗

Z?

k(T)

k(T)

T

T

1

2T2−1

2ζ(2)

ζ(2)T − ζ(2,1) − ζ(3)

ζ(2)T − 2ζ(2,1)

1

2T2

To state the main renormalization formula, we introduce the following power

series A(u) with coefficients in the subring of R generated by Riemann zeta values:

?∞

n=2

4

A(u) = exp

?

(−1)n

n

ζ(n)un

?

= Z?exp?yu −1

xlog(1 + xu)y??.

(2.1)