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 combination 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 speaking, by adjoining a formal variable T corresponding to the infinite sum ∑ 1/n), and that these extended double shuffle (EDS) relations apparently suffice to describe the 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 renormalization 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, automorphisms) 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 thirdnamed author did in the year 1988–1994 but never published. Since that time much work has been done by other writers (Goncharov, Minh, Petitot, Boutet de Monvel, Écalle, 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(k_{1}, k_{2}, …, k_{n}) = ∑_{m_{1}>m_{2}>···>m_{n}>0} \frac{1}{m_{1}^{k_{1}}m_{2}^{k_{2}} ⋯ m_{n}^{k_{n}}} ,  (1.1)

where k = (k_{1}, k_{2}, …, k_{n}) 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(k_{1}, k_{2}, …, k_{n}) = ∫ ⋯ ∫_{1>t_{1}>t_{2}>···>t_{k}>0} ω_{1}(t_{1})ω_{2}(t_{2}) ⋯ ω_{k}(t_{k}),  (1.2)

where k = k_{1} + k_{2} + ⋯ + k_{n} is the weight and ω_{i}(t) = dt/(1 − t) if i ∈ {k_{1}, k_{1} + k_{2}, …, k_{1} + k_{2} + ⋯ + k_{n}} and ω_{i}(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 noncommutative polynomial algebra over the rationals in two indeterminates x and y, and ℌ^{1} and ℌ^{0} its subalgebras ℚ + ℌy and ℚ + xℌy, respectively. Let Z : ℌ^{0} → ℝ be the ℚ-linear map (“evaluation map”) which assigns to each word (monomial) u_{1}u_{2} ⋯ u_{k} in ℌ^{0} the multiple integral

∫ ⋯ ∫_{1>t_{1}>t_{2}>···>t_{k}>0} ω_{u_{1}}(t_{1})ω_{u_{2}}(t_{2}) ⋯ ω_{u_{k}}(t_{k})  (1.3)

where ω_{x}(t) = dt/t, ω_{y}(t) = dt/(1−t) . We set Z(1) = 1. Since the word u_{1}u_{2} ⋯ u_{k} is in ℌ^{0}, we always have ω_{u_{1}}(t) = dt/t and ω_{u_{k}}(t) = dt/(1 − t), so the integral converges. By the Drinfel’d integral representation (1.2), we have

Z(x^{k_{1}−1}yx^{k_{2}−1}y ⋯ x^{k_{n}−1}y) = ζ(k_{1}, k_{2}, …, k_{n}).

The weight k = k_{1} + k_{2} + ⋯ + k_{n} of ζ(k_{1}, k_{2}, …, k_{n}) is the total degree of the corresponding monomial x^{k_{1}−1}yx^{k_{2}−1}y ⋯ x^{k_{n}−1}y, and the depth n the degree in y.

Let z_{k} := x^{k−1}y, which corresponds under Z to the Riemann zeta value ζ(k). Then ℌ^{1} is freely generated by z_{k} (k = 1, 2, 3, …). We define the harmonic product ∗ on ℌ^{1} inductively by

1 ∗ w = w ∗ 1 = w,
z_{k}w_{1} ∗ z_{l}w_{2} = z_{k}(w_{1} ∗ z_{l}w_{2}) + z_{l}(z_{k}w_{1} ∗ w_{2}) + z_{k+l}(w_{1} ∗ w_{2}),

for all k, l ≥ 1 and any words w, w_{1}, w_{2} ∈ ℌ^{1}, and then extending by ℚ-bilinearity. Equipped with this product, ℌ^{1} becomes a commutative algebra ([8]) and ℌ^{0} a subalgebra. We will denote these algebras by ℌ_{∗}^{1} and ℌ_{∗}^{0}. The first multiplication law of MZV’s can then be stated by saying that the evaluation map Z : ℌ^{0} → ℝ is an algebra homomorphism with respect to the multiplication ∗, i.e.

Z(w_{1} ∗ w_{2}) = Z(w_{1})Z(w_{2})  (1.4)

for all w_{1}, w_{2} ∈ ℌ^{0}. For instance, the harmonic product z_{k} ∗ z_{l} = z_{k}z_{l} + z_{l}z_{k} + z_{k+l} corresponds to the identity ζ(k)ζ(l) = ζ(k, l) + ζ(l, k) + ζ(k + l). Notice 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 m_{i} in 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, corresponding to the product of two integrals in (1.2), is defined on all of ℌ inductively by setting

1� w = w� 1 = w,
uw_{1}� vw_{2} = u(w_{1}� vw_{2}) + v(uw_{1}� w_{2}),

for any words w, w_{1}, w_{2} ∈ 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 ℌ^{1} and ℌ^{0} become subalgebras of ℌ_{�} , denoted by ℌ1_{�} and ℌ0_{�} respectively. By the standard shuffle product identity of iterated integrals, the evaluation map Z is again an algebra homomorphism for the multiplication � :

Z(w_{1}� w_{2}) = Z(w_{1})Z(w_{2}).  (1.5)

Again, this rule is a formal consequence of the formula (1.3) and would hold for the values defined by these integrals if ω_{x} and ω_{y} were 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:

Z(w_{1}� w_{2}) = Z(w_{1} ∗ w_{2})  (w_{1}, w_{2} ∈ ℌ^{0}).  (1.6)

The first example is

4ζ(3, 1) + 2ζ(2, 2) = 2ζ(2, 2) + ζ(4) ( = ζ(2)^{2} )

from which we deduce 4ζ(3, 1) = ζ(4). These “finite” double shuffle relations, 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.

§2. Regularizations of multiple zeta values

Proposition 1. We have two algebra homomorphisms

Z^{∗} : ℌ_{∗}^{1} ⟶ ℝ[T] and Z^{�} : ℌ1_{�} ⟶ ℝ[T]

which are uniquely characterized by the properties that they both extend the evaluation map Z : ℌ^{0} → ℝ and send y to T.

Proof. This is clear from the isomorphisms ℌ_{∗}^{1} ≃ ℌ_{∗}^{0}[y] and ℌ1_{�} ≃ ℌ0_{�} [y] ([8] and [15]) and the fact that the map Z is an algebra homomorphism for both harmonic and shuffle products. □

For an index k = (k_{1}, …, k_{n}) (not necessarily admissible, i.e., any ordered set of positive integers), the images under the maps Z^{∗} and Z^{�} of the corresponding word x^{k_{1}−1}y ⋯ x^{k_{n}−1}y are denoted by Z_{k}^{∗}(T) and Zk^{�} (T), respectively. If k is admissible, we have Z_{k}^{∗}(T) = Zk^{�} (T) = ζ(k). In general, we see by induction on s that, for k = (\underbrace{1, 1, …, 1}_{s}, k′) with k′ admissible and s ≥ 0 we have

Z_{k}^{∗}(T) = ζ(k′)\frac{T^{s}}{s!} + (terms of lower degree in T)

and similarly

Zk^{�} (T) = ζ(k′)\frac{T^{s}}{s!} + (terms of lower degree in T) ,

and also that the coefficients of T^{i} in Z_{k}^{∗}(T) and Zk^{�} (T) are ℚ-linear combinations of multiple zeta values of weight k − i (k = weight of k). Here are a few examples:

\begin{matrix}k & (1) & (1, 1) & (1, 2) \\ Z_{k}^{∗}(T) & T & \frac{1}{2} T^{2} − \frac{1}{2}ζ(2) & ζ(2) T − ζ(2, 1) − ζ(3) \\ Zk^{�} (T) & T & \frac{1}{2} T^{2} & ζ(2) T − 2ζ(2, 1) \end{matrix}

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:

A(u) = exp(∑_{n=2}^{∞} \frac{(−1)^{n}}{n} ζ(n) u^{n}) = Z(exp(yu − \frac{1}{x} log(1 + xu) y)) . (2.1)

Here Z acts coefficientwise on the power series. As is easily seen from the standard Taylor expansion of log Γ(x) at x = 1, this is the Taylor expansion of e^{γu}Γ(1 + u) (γ = Euler’s constant) near u = 0:

A(u) = e^{γu} Γ(1 + u) (|u| < 1).

Define an ℝ-linear map ρ : ℝ[T] → ℝ[T] by

ρ(e^{T u}) = A(u) e^{T u}.  (2.2)

Equivalently, ρ is determined by

ρ(\frac{T^{l}}{l!} ) = ∑_{k=0}^{l} γ_{k} \frac{T^{l−k}}{(l − k)!} (l = 0, 1, 2, …)

and the ℝ-linearity, where the coefficients γ_{0} = 1, γ_{1} = 0, γ_{2} = ζ(2)/2, … are given by the generating function

A(u) = ∑_{k=0}^{∞} γ_{k}u^{k} .

Note that, by (2.1), the coefficient γ_{k} is an element of weight k in the ℚ-algebra generated by Riemann zeta values.

Theorem 1. For any index set k, we have

Zk^{�} (T) = ρ(Z_{k}^{∗}(T)) .  (2.3)

Proof. For M > 0 and an index set k = (k_{1}, k_{2}, …, k_{n}), put

ζ_{M}(k_{1}, k_{2}, …, k_{n}) := ∑_{M>m_{1}>m_{2}>···>m_{n}>0} \frac{1}{m_{1}^{k_{1}}m_{2}^{k_{2}} ⋯ m_{n}^{k_{n}}} .

If k is admissible, i.e., k_{1} > 1, then ζ_{M}(k) converges to ζ(k) as M → ∞. Note that we can write the product ζ_{M}(k)ζ_{M}(k′) as a linear combination of ζ_{M}(k′′)’s by the same rule as in the case of harmonic product of the convergent multiple zeta values. For instance, we have

ζ_{M}(k)ζ_{M}(k′) = ζ_{M}(k, k′) + ζ_{M}(k′, k) + ζ_{M}(k + k′) .

With this fact and the classical formula

ζ_{M}(1) = 1 + \frac{1}{2} + ⋯ + \frac{1}{M − 1} = log M + γ + O(\frac{1}{M}),

we see by induction that for any index set k we have

ζ_{M}(k) = Z_{k}^{∗}(log M + γ) + O(M^{−1} log^{J} M) for some J as M → ∞,

where Z_{k}^{∗}(T) is the associated polynomial defined in Proposition 1.

Next, for k = (k_{1}, k_{2}, …, k_{n}) and 0 < t < 1, put

Li_{k}(t) = ∫ ⋯ ∫_{t>t_{1}>t_{2}>···>t_{k}>0} ω_{1}(t_{1})ω_{2}(t_{2}) ⋯ ω_{k}(t_{k}) ,

where k = k_{1} + k_{2} + ⋯ + k_{n} and ω_{i}(t) = dt/(1 − t) if i ∈ {k_{1}, k_{1} + k_{2}, …, k_{1} + k_{2} + ⋯ + k_{n}} and ω_{i}(t) = dt/t otherwise. Iterated integration shows that

Li_{k}(t) = ∑_{m_{1}>m_{2}>···>m_{n}>0} \frac{t^{m_{1}}}{m_{1}^{k_{1}}m_{2}^{k_{2}} ⋯ m_{n}^{k_{n}}} .  (2.4)

The product Li_{k}(t)Li_{k′}(t) is a linear combination of Li_{k′′}(t)’s via the shuffle product identity of iterated integrals, and the formula specializes at t = 1 to that (with the shuffle product � ) of ζ(k)ζ(k′) if k and k′ are admissible. Together with Li_{1}(t) = log \frac{1}{1−t}, we conclude by induction that, for each index set k, we have

Li_{k}(t) = Zk^{�} (log \frac{1}{1 − t}) + O((1 − t) log^{J}(\frac{1}{1 − t})) for some J as t ↗ 1 .

We shall compare the behaviors of ζ_{M}(k) and Li_{k}(t). For that we start with the equation (2.4) to obtain

Li_{k}(t) = ∑_{m_{1}>m_{2}>···>m_{n}>0} \frac{t^{m_{1}}}{m_{1}^{k_{1}}m_{2}^{k_{2}} ⋯ m_{n}^{k_{n}}}

= ∑_{m=1}^{∞} ( ∑_{m>m_{2}>···>m_{n}>0} \frac{1}{m^{k_{1}}m_{2}^{k_{2}} ⋯ m_{n}^{k_{n}}} ) t^{m}

= ∑_{m=1}^{∞} (ζ_{m+1}(k) − ζ_{m}(k)) t^{m}

= (1 − t) ∑_{m=1}^{∞} ζ_{m}(k)t^{m−1} .

To deduce the theorem from this, we use the following lemma:

Lemma.

(i) Let P (T) ∈ ℝ[T] and Q(T) = ρ(P (T)). Then

∑_{m=1}^{∞} P (log m + γ) t^{m−1} = \frac{1}{1 − t} Q(log \frac{1}{1 − t}) + O(log^{J}(\frac{1}{1 − t}))

for some J (= deg P − 1) as t ↗ 1. (ii) For l ≥ 0, we have

∑_{m=1}^{∞} \frac{log^{l} m}{m} t^{m−1} = O(log^{l+1}(\frac{1}{1 − t})) as t ↗ 1.

Since ζ_{m}(k) = Z_{k}^{∗}(log m + γ) + O(m^{−1} log^{J} m), the lemma gives

(1 − t) ∑_{m=1}^{∞} ζ_{m}(k) t^{m−1} = Q(log \frac{1}{1 − t}) + O((1 − t) log^{J+1}(\frac{1}{1 − t}))

with Q(T) = ρ(Z_{k}^{∗}(T)), so we conclude Zk^{�} (T) = ρ(Z_{k}^{∗}(T)) . □

Proof of the lemma. We first prove (ii). For l = 0, the left-hand side is \frac{1}{t} log \frac{1}{1−t} which is clearly O(log \frac{1}{1−t}) as t → 1. We now proceed by induction on l. We have

log^{l+1} m ≤ C_{l} ∑_{n=1}^{m} \frac{log^{l} n}{n} (m ≥ 1, l ≥ 0)

for some constant C_{l} independent of m. (This is easily seen by comparing the sum with the corresponding integral ∫_{1}^{m} \frac{log^{l} x}{x} dx.) Hence for t < 1 we obtain

∑_{m=1}^{∞} \frac{log^{l+1} m}{m} t^{m−1} ≤ C_{l} ∑_{m=1}^{∞} \frac{t^{m−1}}{m} ∑_{n=1}^{m} \frac{log^{l} n}{n}

= C_{l} ∑_{n=1}^{∞} \frac{log^{l} n}{n} t^{n−1} ∑_{r=1}^{∞} \frac{t^{r−1}}{r + n − 1}

< C_{l} ( ∑_{n=1}^{∞} \frac{log^{l} n}{n} t^{n−1})(\frac{1}{t} log \frac{1}{1 − t}) .

The estimate in (ii) now follows for all l by induction.

For (i), it is enough by linearity to prove the identity for P (T) = (T − γ)^{l}. Put Q(T) = ρ((T − γ)^{l}). Then

Q(T) = \frac{d^{l}}{du^{l}} [A(u) e^{(T −γ)u}]_{u=0} = \frac{d^{l}}{du^{l}} [Γ(1 + u) e^{T u}]_{u=0} ,

and hence

\frac{1}{1 − t}Q(log \frac{1}{1 − t}) = \frac{d^{l}}{du^{l}} [\frac{Γ(1 + u)}{(1 − t)^{1+u}} ]_{u=0}

= \frac{d^{l}}{du^{l}} [ ∑_{m=1}^{∞} \frac{Γ(m + u)}{Γ(m)}t^{m−1}]_{u=0} (binomial theorem)

= ∑_{m=1}^{∞} \frac{Γ^{(l)}(m)}{Γ(m)}t^{m−1}.

From the standard integral representation

log Γ(x) = (x − \frac{1}{2}) log x − x + C − ∫_{0}^{∞} \frac{t − [t] − 1/2}{x + t} dt (x > 0)

we see by induction on l that

\frac{Γ^{(l)}(m)}{Γ(m)} = log^{l} m + O(\frac{log^{l−1} m}{m}) (m → ∞)

for all l ≥ 0, and from this and (ii) of the lemma we deduce that

∑_{m=1}^{∞} \frac{Γ^{(l)}(m)}{Γ(m)} t^{m−1} = ∑_{m=1}^{∞} log^{l} m t^{m−1} + O(log^{l} \frac{1}{1 − t})

= ∑_{m=1}^{∞} P (log m + γ) t^{m−1} + O(log^{l} \frac{1}{1 − t}).

This completes the proof. □

As an example of the theorem, by comparing the two entries for k = (1, 2) in the table at the beginning of the section, we find

ζ(2) T − 2ζ(2, 1) = ζ(2) T − ζ(2, 1) − ζ(3) ,

the left-hand side being Zk^{�} (T) and the right ρ(Z_{k}^{∗}(T)). (Note that ρ(T) = T.) This equality gives Euler’s formula

ζ(2, 1) = ζ(3)

and shows that the space of weight 3 MZV’s is one-dimensional, whereas using only the finite shuffle relations as in §1 we were not able to reduce the dimension below 2. Similarly, by comparing the two sides of (2.3) for k = (1, 3) , (1, 2, 1), we find

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

and hence the identities

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

which, together with the formula 4ζ(3, 1) = ζ(4) obtained in §1 as a finite double shuffle relation, reduce the dimension of weight 4 MZV’s to 1.

In each of these examples, the degree with respect to T was at most 1 and hence the effect of the automorphism ρ, which acts as the identity on the subspace ℝ + ℝT of ℝ[T], was not visible. As an example involving higher powers of T, take k = (1, 1, 2) in (2.3). Then

Z1,1,2^{�}(T) = \frac{1}{2}ζ(2) T^{2} − 2ζ(2, 1) T + 3ζ(2, 1, 1),
Z_{1,1,2}^{∗}(T) = \frac{1}{2}ζ(2) T^{2} − (ζ(3) + ζ(2, 1)) T + \frac{1}{2}ζ(4) + ζ(3, 1) + ζ(2, 1, 1) ,

so from Z1,1,2^{�}(T) = ρ(Z_{1,1,2}^{∗}(T)) and ρ(T^{2}) = T^{2}+ζ(2) we find (again) ζ(2, 1) = ζ(3) and also

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

This latter relation is a consequence of the relations obtained above, which already reduced the dimension of the MZV space in this weight to 1, but at the same time we see that here Z1,1,2^{�}(T) ≠ Z_{1,1,2}^{∗}(T) and hence that the presence of the automorphism ρ in (2.3) is really necessary to make this identity correct.

These examples show that in weight up to 4, Theorem 1, together with the homomorphism properties of the evaluation maps Z^{�} and Z^{∗}, suffices to give all relations of MZV’s, and that this still remains true even if we restrict our attention to index sets k with at most one leading 1. It is conjectured that both of these statements remain true in all weights. These conjectures will be formulated more precisely in the next section in the language of the algebra ℌ.

§3. Extended double shuffle relations

Denote by regT_{�} (resp. reg_{∗}^{T}) the map (actually an isomorphism) ℌ1_{�} → ℌ0_{�} [T] (resp. ℌ_{∗}^{1} → ℌ_{∗}^{0}[T]) defined by the properties that it is the identity on ℌ^{0}, maps y to T, and is an algebra homomorphism. The maps ℌ1_{�} → ℌ0_{�} and ℌ_{∗}^{1} → ℌ_{∗}^{0} obtained by specializing regT_{�} and reg_{∗}^{T} to T = 0 will be denoted by reg_{�} and reg_{∗}, respectively. If R is a commutative ℚ-algebra with 1 and Z_{R} is any map from ℌ^{0} to R which is a homomorphism with respect to both multiplications � and ∗, i.e.,

Z_{R}(w_{1}� w_{2}) = Z_{R}(w_{1} ∗ w_{2}) = Z_{R}(w_{1})Z_{R}(w_{2}),  (3.1)

we say that Z_{R} has the “finite double shuffle (FDS)” property. We can then extend Z_{R} to maps ZR^{�} : ℌ^{1} → R[T] and Z_{R}^{∗} : ℌ^{1} → R[T] in the same way (viz., they agree with Z_{R} on ℌ^{0}, send y to T, and are homomorphisms with respect to � or ∗), or equivalently, we can define ZR^{�} and Z_{R}^{∗} as the composites of the maps regT_{�} and reg_{∗}^{T} with the map Z_{R} ⊗ 1 : ℌ^{0}[T] = ℌ^{0} ⊗ ℚ[T] → R ⊗ ℚ[T] = R[T]. Finally, we define an R-module automorphism ρ_{R} of R[T], generalizing the map ρ in Theorem 1, by the formula

ρ_{R}(e^{T u}) = A_{R}(u)e^{T u}  (3.2)

(together with the requirement of R-linearity), where A_{R}(u) is the power series defined, in analogy with (2.1), by

A_{R}(u) = exp(∑_{n=2}^{∞} \frac{(−1)^{n}}{n} Z_{R}(x^{n−1}y) u^{n}) ∈ R[[u]] .  (3.3)

Theorem 2. Let (R, Z_{R}) be as above with the FDS property. Then the following properties are equivalent:

(i) (ZR^{�} − ρ_{R} ∘ Z_{R}^{∗})(w) = 0 for all w ∈ ℌ^{1}.

(ii) (ZR^{�} − ρ_{R} ∘ Z_{R}^{∗})(w)|_{T =0} = 0 for all w ∈ ℌ^{1}.

(iii) ZR^{�} (w_{1}� w_{0} − w_{1} ∗ w_{0}) = 0 for all w_{1} ∈ ℌ^{1} and all w_{0} ∈ ℌ^{0}.

(iii′) Z_{R}^{∗}(w_{1}� w_{0} − w_{1} ∗ w_{0}) = 0 for all w_{1} ∈ ℌ^{1} and all w_{0} ∈ ℌ^{0}.

(iv) Z_{R}(reg_{�} (w_{1}� w_{0} − w_{1} ∗ w_{0})) = 0 for all w_{1} ∈ ℌ^{1} and all w_{0} ∈ ℌ^{0}.

(iv′) Z_{R}(reg_{∗}(w_{1}� w_{0} − w_{1} ∗ w_{0})) = 0 for all w_{1} ∈ ℌ^{1} and all w_{0} ∈ ℌ^{0}.

(v) Z_{R}(reg_{�} (y^{m} ∗ w_{0})) = 0 for all m ≥ 1 and all w_{0} ∈ ℌ^{0}.

(v′) Z_{R}(reg_{∗}(y^{m}� w_{0} − y^{m} ∗ w_{0})) = 0 for all m ≥ 1 and all w_{0} ∈ ℌ^{0}.

The implications (i) ⇒ (ii), (iii) ⇒ (iv), and (iii′) ⇒ (iv′) are obvious (note Z_{R} ∘reg_{�} (resp. Z_{R} ∘reg_{∗}) is the specialization ZR^{�} |_{T =0} (resp. Z_{R}^{∗}|_{T =0})). For (i) ⇒ (iii), multiply Z_{R}(w_{0})(∈ R) on both sides of ZR^{�} (w_{1}) = ρ_{R}(Z_{R}^{∗}(w_{1})) and use the R-linearity of ρ_{R} to get ZR^{�} (w_{1}� w_{0}) = ρ_{R}(Z_{R}^{∗}(w_{1} ∗ w_{0})). Using (i) on the right, we obtain (iii). The implication (i) ⇒ (iii′) is proved similarly (multiply Z_{R}(w_{0}) on both sides of ρ_{R}^{−1}(ZR^{�} (w_{1})) = Z_{R}^{∗}(w_{1})). By the same arguments we can show (ii) ⇒ (iv) and (ii) ⇒ (iv′). The properties (v) and (v′) are obtained respectively from (iv) and (iv′) by setting w_{1} = y^{m} and noting (for (iv)) reg_{�} (y^{m}� w) = 0 which follows from reg_{�} (y^{m}) = 0 (since y^{m} = y^{� m}/m!). We have thus shown the following implications:

(i) ⇒(ii)  (i) ⇒ (ii)
⇓ ⇓  and  ⇓  ⇓
(iii) ⇒(iv) ⇒ (v)  (iii′) ⇒ (iv′) ⇒ (v′).

Note that these implications are formal and do not depend on the precise definition of the algebraic renormalization map ρ_{R} (i.e., they would remain true if A_{R}(u) in (3.2) were replaced by any other power series). The real content of the theorem is the implications (v) ⇒ (i) and (v′) ⇒ (i), which we will prove in §5 after some algebraic preliminaries.

Definition. The ℚ-algebra R and the map Z_{R} : ℌ^{0} → R with the FDS property have the extended double shuffle (EDS) property if the eight equivalent properties of Theorem 2 are satisfied.

The content of Theorem 1 is now precisely that (ℝ, Z) satisfies the EDS property, in the form (i) of Theorem 2, and hence also in the forms (ii)–(v) and (iii′)–(v′). In particular, we have

Z(reg_{�} (w_{1}� w_{0} − w_{1} ∗ w_{0})) = 0  (∀w_{1} ∈ ℌ^{1} , ∀w_{0} ∈ ℌ^{0})  (3.4)

and

Z(reg_{�} (y^{m} ∗ w_{0})) = 0  (∀m ≥ 1 , ∀w_{0} ∈ ℌ^{0}).  (3.5)

The main conjecture about multiple zeta values is that the relations (3.4) suffice to give all linear relations over ℚ among MZV’s. To state this formally, we introduce the universal EDS ring, as follows:

If (R, Z_{R}) has the EDS property and φ : R → R′ is a ℚ-algebra homomorphism, then (R′, φ ∘ Z_{R}) also has the EDS property. Clearly, there exists a universal algebra R_{EDS}, viz., the quotient of ℌ^{0} divided by the necessary relations, and a map φ_{R} : R_{EDS} → R for any (R, Z_{R}) with the EDS property which makes the diagram

ℌ^{0} →R_{EDS}
Z_{R} ↘ ↓ φ_{R}
R

commute. Now for (R, Z_{R}) = (ℝ, Z) we formulate the following

Conjecture. The map φ_{ℝ} is injective, i.e., the algebra of multiple zeta values is isomorphic to R_{EDS}.

We briefly discuss here several possible different versions of the conjecture and their experimental status. We can consider statements of various strength, namely:

(1) The FDS and EDS relations suffice to give all relations among MZV’s.

(2) The FDS relations alone suffice.

(3) Only FDS and EDS with ζ(1) suffice.

(4) Only double shuffle relations with ζ(n) (n = 1, 2, 3, …) against finite zetas suffice.

(Here (3) means that we need only FDS and the formula (v) of Theorem 2 with m = 1, and (4) means that we need only (iv) of Theorem 2 with w_{1} = z_{n} = x^{n−1}y.) Statement (1) is just the conjecture stated above. We checked up to k = 13 that the FDS and EDS relations suffice to give all expected relations, i.e., to reduce the dimension to the numerically and theoretically predicted value. (This is all one can hope to do, since actually proving the linear independence over ℚ of MZV’s is out of reach.) The more optimistic statement (2) is wrong, as the weight 3 and weight 4 examples in §2 showed. The intermediate statement (3) says that it is enough to use Theorem 1 for index sets k for which Zk^{�} (T) is at most linear in T (in which case the map “ρ” is not needed in the formula (2.3)). The examples in §2 verified this up to weight 4, and it has been checked up to weight 16 by Minh, Petitot et al. in Lille. (This of course also verifies the weaker conjecture (1) up to this weight.) Finally, statement (4), a different and particularly simple-looking strengthening of (1), holds up to weight 12 but fails at weight 13, where the relations in (4) suffice only to reduce the dimension of the space of MZV’s to 17 instead of the value 16, which is what is obtained in this weight if one uses all the double shuffle relations.

§4. Algebraic formulas

Recall that the algebra ℌ^{1} (with concatenation product) is free on the generators z_{k} = x^{k−1}y (k ≥ 1). Denote by 𝔷 the ℚ-linear span of the z_{k}.^{1} Proposition 2.

1We will not distinguish between direct sums and direct products, using the same letters ℌ^{1} and 𝔷 also for the completions and freely allowing infinite sums (of elements with grading going to infinity); this will not lead to any problems since ℌ^{1} is finite-dimensional in each weight.

(i) For z ∈ 𝔷 the map δ_{z} : ℌ^{1} → ℌ^{1} defined by

δ_{z}(w) := z ∗ w − zw  (z ∈ 𝔷, w ∈ ℌ^{1})  (4.1)

is a derivation, and these derivations all commute. (ii) The above derivation δ_{z} extends to a derivation on all of ℌ, with values on

the generators given by

δ_{z}(x) = 0 ,  δ_{z}(y) = (x + y)z  (z ∈ 𝔷) .  (4.2)

In particular, δ_{z} preserves ℌ^{0}.

Proof. (i) It is almost immediate from the definition of ∗ that we have

z ∗ (ww′) = (z ∗ w)w′ + w(z ∗ w′) − wzw′  (4.3)

for all w, w′ ∈ ℌ^{1}, and this is equivalent to the derivation property. Now if z, z′ ∈ 𝔷 and w ∈ ℌ^{1} then from (4.1) and (4.3) and the associativity of ∗ we have

δ_{z}(δ_{z′}(w)) = z ∗ (z′ ∗ w − z′w) − zδ_{z′}(w) = δ_{z∗z′}(w) − z′δ_{z}(w) − z δ_{z′}(w) ,

which by virtue of the commutativity of ∗ is symmetric in z and z′.

(ii) Define a derivation δ′_{z} on ℌ by the formulas (4.2). Then for k ≥ 1 we have

δ′_{z}(z_{k}) = δ′_{z}(x^{k−1}y) = x^{k−1}(x + y)z = z_{k} ∗ z − zz_{k} ,

so δ′_{z} agrees with δ_{z} on the generators of ℌ^{1} and hence on all of ℌ^{1}. □

Proposition 3. The vector space 𝔷 becomes a commutative and associative algebra with respect to the multiplication ∘ defined by

z ∗ z′ = zz′ + z′z + z ∘ z′  (z, z′ ∈ 𝔷) .  (4.4)

Proof. We find immediately that z_{k} ∘ z_{l} = z_{k+l}, from which these properties follow. Notice that the map

γ : X ℚ[[X]] → 𝔷 ,  γ(X^{k}) = z_{k} (k = 1, 2, …)  (4.5)

is an algebra isomorphism for ∘. We will use it occasionally later. □

We now have three different associative and commutative multiplications: the shuffle product � (defined on all of ℌ), the harmonic product ∗ (defined on ℌ^{1} ⊂ ℌ), and the “circle” product ∘ (defined on 𝔷 ⊂ ℌ^{1}). We denote by exp_{�} , exp_{∗} and exp_{∘} the corresponding exponential maps. The next two propositions describe some of their properties.

Proposition 4. For z ∈ 𝔷 we have

exp_{∗}(z) = (2 − exp_{∘}(z))^{−1} .

(The inverse on the right is with respect to the concatenation product.)

Proof. Define a power series f(u) ∈ 𝔷[[u]] by

f(u) = exp_{∘}(zu) − 1 = z u + z ∘ z \frac{u^{2}}{2} + ⋯

(here the map exp_{∘} has been extended to 𝔷[[u]] in the obvious way, in accordance with the footnote above). Then f′(u) = z ∘ (1 + f(u)), so

z ∗ \frac{1}{1 − f(u)} = z ∗ ∑_{n≥0} f(u)^{n} = ∑_{α,β≥0} f(u)^{α}z f(u)^{β} + ∑_{α,β≥0} f(u)^{α}(z ∘ f(u))f(u)^{β}

= ∑_{α,β≥0} f(u)^{α} f′(u) f(u)^{β} = \frac{d}{du}(∑_{n≥0} f(u)^{n}) = \frac{d}{du}(\frac{1}{1 − f(u)}) .

(For the second equality we have used the identity

z ∗ w_{1}w_{2} ⋯ w_{n} = ∑_{i=0}^{n} w_{1} ⋯ w_{i}zw_{i+1} ⋯ w_{n} + ∑_{i=1}^{n} w_{1} ⋯ w_{i−1}(z ∘ w_{i})w_{i+1} ⋯ w_{n}

for z, w_{i} ∈ 𝔷, which follows from the derivation property of δ_{z} and the formula δ_{z}(w) = wz + z ∘ w (z, w ∈ 𝔷) .) It follows that the function F (u) := (1 − f(u))^{−1} satisfies F′(u) = z ∗ F (u) and F (0) = 1, so F (u) = exp_{∗}(zu) . □

Corollary 1. For all z ∈ 𝔷 we have

exp_{∗}(log_{∘}(1 + z)) = \frac{1}{1 − z} .  (4.6)

As an example, putting z = z_{k} in (4.6) and applying the evaluation map Z, we obtain

Corollary 2. For k ≥ 2, we have the identity

exp(∑_{n=1}^{∞}(−1)^{n−1}ζ(nk)\frac{u^{n}}{n} ) = 1 + ∑_{n=1}^{∞} ζ(\underbrace{k, k, …, k}_{n})u^{n}.

Proposition 5. For z, z′ ∈ 𝔷, w ∈ ℌ^{1} we have the identities

exp(δ_{z})(z′) = (exp_{∘}(z) ∘ z′) exp_{∗}(z) ,  (4.7)
exp(δ_{z})(w) = (exp_{∗}(z))^{−1} (exp_{∗}(z) ∗ w) .  (4.8)

Proof. Using the derivation property of δ_{z} and (4.4), we find

δ_{z}(z′w) = δ_{z}(z′)w + z′δ_{z}(w) = (z ∘ z′ + z′z)w + z′(z ∗ w − zw)
= (z ∘ z′)w + z′(z ∗ w)

for z, z′ ∈ 𝔷, w ∈ ℌ^{1}. Now replacing z′ and w by z^{∘α} ∘ z′ and z^{∗β} ∗ w (which again belong to 𝔷 and ℌ^{1}, respectively) we obtain

δ_{z}((z^{∘α} ∘ z′)(z^{∗β} ∗ w)) = (z^{∘(α+1)} ∘ z′) (z^{∗β} ∗ w) + (z^{∘α} ∘ z′) (z^{∗(β+1)} ∗ w)

and hence by induction

δ_{z}^{n}(z′w) = ∑_{α+β=n} \binom{n}{α} (z^{∘α} ∘ z′) (z^{∗β} ∗ w)

or, dividing by n! and summing over n,

exp(δ_{z})(z′w) = (exp_{∘}(z) ∘ z′) (exp_{∗}(z) ∗ w) .  (4.9)

Setting w = 1 in (4.9) gives equation (4.7). Observe that if δ is a derivation which increases the weight, then by Leibniz’s rule, exp(δ) is a well-defined automorphism. With this, dividing (4.9) by (4.7) gives (4.8). □

Combining the last two propositions, we obtain:

Proposition 6.

(i) For z ∈ 𝔷 define Φ_{z} : ℌ^{1} → ℌ^{1} by

Φ_{z}(w) := (1 − z) (\frac{1}{1 − z} ∗ w)  (z ∈ 𝔷, w ∈ ℌ^{1}) .  (4.10)

Then Φ_{z} is an automorphism of ℌ^{1} and we have the identity

Φ_{z}(w) = exp(δ_{t})(w) ,  where t = log_{∘}(1 + z) ∈ 𝔷.  (4.11)

The collection of all Φ_{z} (z ∈ 𝔷) forms a commutative subgroup of Aut(ℌ^{1}), with Φ_{z}Φ_{z′} = Φ_{z+z′+z∘z′} for z, z′ ∈ 𝔷. Equivalently, the map 1+X ℚ[[X]] → Aut(ℌ^{1}) mapping 1 + f(X) to Φ_{γ(f)}, with γ defined by (4.5), is a homomorphism of groups.

(ii) The automorphism Φ_{z} of ℌ^{1} extends to an automorphism of all of ℌ, with Φ_{z}(x) = x and Φ_{z}(x + y) = (x + y)(1 − z)^{−1} . In particular, Φ_{z} also induces an automorphism of ℌ^{0}.

Proof. (i) From equation (4.6) we have exp_{∗}(t) = (1 − z)^{−1}, and substituting this into (4.8) (with z replaced by t) immediately gives (4.11). The fact that Φ_{z} is an automorphism now follows from Proposition 2 (i) and the fact that the exponential of a derivation is an automorphism, as noted before, and the last statements of the proposition follow immediately from formula (4.11) since for t = log_{∘}(1 + z) and t′ = log_{∘}(1 + z′) we have δ_{t} + δ_{t′} = δ_{t+t′} and t + t′ = log_{∘}(1 + z + z′ + z ∘ z′) by the commutativity of ∘.

(ii) If we iterate δ_{z}, then we find δ_{z}^{n}(x) = 0 and δ_{z}^{n}(y) = (x + y)z^{∗n} for all n ≥ 1. (This is clear by induction, since δ_{z}((x+y)w) = (x+y)(zw +δ_{z}(w)) = (x+y)(z ∗w) for any w ∈ ℌ^{1} .) It follows that exp(δ_{z})(x) = x, exp(δ_{z})(x + y) = (x + y) exp_{∗}(z). Combining this with formulas (4.11) and (4.6) we obtain the assertion. □

Remark. Written out, the fact that Φ_{z} is a homomorphism says that

(z^{n}) ∗ (ww′) = ∑_{α+β=n}(z^{α} ∗ w) (z^{β} ∗ w′) − ∑_{α+β=n−1}(z^{α} ∗ w) z (z^{β} ∗ w′)

for all n ≥ 0, a generalization of (4.3) which can also be proved directly by a tedious induction using (4.3), (4.4) and the commutativity and associativity of ∗ and ∘.

The analogous (but simpler) result for the shuffle product is as follows.

Proposition 7. Define the map d : ℌ → ℌ by d(w) = y� w − yw. Then d is a derivation and we have

exp(du)(w) = (1 − yu) (\frac{1}{1 − yu} � w)  (w ∈ ℌ) .  (4.12)

(Here, u is a formal parameter.) In particular, we have

exp(du)(x) = x\frac{1}{1 − yu} and exp(du)(y) = y \frac{1}{1 − yu}  (4.13)

Proof. That the d is a derivation on ℌ is easily checked. For (4.12), we show by induction the identity

\frac{1}{m!} d^{m}(w) = y^{m}� w − y (y^{m−1}� w)  (m ≥ 1, w ∈ ℌ) .  (4.14)

The case m = 1 is the definition of d. Assuming the identity for m, we have

\frac{1}{(m + 1)!} d^{m+1}(w) = \frac{1}{m + 1}d(y^{m}� w − y(y^{m−1}� w))

= \frac{1}{m + 1}[y� (y^{m}� w − y(y^{m−1}� w)) − y(y^{m}� w − y(y^{m−1}� w))]

= \frac{1}{m + 1}[(m + 1)y^{m+1}� w − y^{2}(y^{m−1}� w) − y(y� y^{m−1}� w)

− y(y^{m}� w) + y^{2}(y^{m−1}� w)]

= y^{m+1}� w − y(y^{m}� w).

Multiplying (4.14) by u^{m} and summing over m gives (4.12). Putting w = x (resp. w = y) in (4.14), we have d^{m}(x)/m! = xy^{m} (resp. d^{m}(y)/m! = y^{m+1}), which gives (4.13). □

Corollary. Let Δ_{u} be the automorphism of ℌ defined by

Δ_{u} = exp(−du) ∘ Φ_{yu}  (4.15)

(Here ∘ denotes composition.) Then for w ∈ ℌ^{1} we have

\frac{1}{1 − yu} ∗ w = \frac{1}{1 − yu} � Δ_{u}(w) .  (4.16)

In particular, for w_{0} ∈ ℌ^{0} we have

reg_{�} (\frac{1}{1 − yu} ∗ w_{0}) = Δ_{u}(w_{0}).  (4.17)

The images of the generators x and y of ℌ under Δ_{u} are given by

Δ_{u}(x) = x(1 + yu)^{−1}, Δ_{u}(y) = y + x(1 + yu)^{−1}yu.  (4.18)

Proof. The first identity directly follows from (4.12) by replacing w with Δ_{u}(w) and dividing both sides on the left by 1 − yu. To get the second, we put w = w_{0} in the first and take reg_{�} of both sides, noting that reg_{�} (y^{m}) = 0 for m ≥ 1. For the images of x and y, we use Proposition 6 (ii) and (4.13) to obtain

Δ_{u}(x) = exp(−du)(Φ_{yu}(x)) = exp(−du)(x) = x(1 + yu)^{−1}

and

Δ_{u}(x + y) = (exp(−du) ∘ Φ_{yu})(x + y) = exp(−du)((x + y)(1 − yu)^{−1})
= x + y,

and so

Δ_{u}(y) = Δ_{u}(x + y) − Δ_{u}(x) = y + x(1 + yu)^{−1}yu.
□

§5. Regularization formulas

In this section we give various algebraic formulas for the regularization maps regT_{�} and reg_{�} (and also reg_{∗}^{T} and reg_{∗}). Using these, we complete the proof of Theorem 2. We also apply these formulas to show that the well-known sum formula for MZV’s is a formal consequence of the extended double shuffle relations.

Proposition 8. For w_{0} = xw′_{0} ∈ ℌ^{0} we have the regularization formula

regT_{�} (\frac{1}{1 − yu}w_{0}) = exp(−du)(w_{0}) e^{T u} = x (\frac{1}{1 + yu} � w′_{0}) e^{T u} .  (5.1)

In particular, for all m ≥ 0 we have

reg_{�} (y^{m}w_{0}) = \frac{(−1)^{m}}{m!} d^{m}(w_{0}) = (−1)^{m} x (y^{m}� w′_{0}) .  (5.2)

Proof. Putting w = exp(−du)(w_{0}) in Proposition 7 and multiplying (1−yu)^{−1} from the left, we have (note the obvious identity (1 − yu)^{−1} = exp_{�} (yu))

\frac{1}{1 − yu}w_{0} = \frac{1}{1 − yu} � exp(−du)(w_{0})
= exp_{�} (yu) � exp(−du)(w_{0}).  (5.3)

Taking regT_{�} of this gives the first equation of (5.1). For the second, use the first equality in (4.13) with u replaced by −u and Proposition 7 applied to w = xw′_{0} to get

exp(−du)(w_{0}) = exp(−du)(x) exp(−du)(w′_{0})

= (x \frac{1}{1 + yu})((1 + yu)(\frac{1}{1 + yu} � w′_{0}))

= x (\frac{1}{1 + yu} � w′_{0}) .

This proves equation (5.1). Comparing the coefficients of u^{m} on both sides of this equation, we can rewrite it more explicitly as

regT_{�} (y^{m}w_{0}) = \frac{1}{m!} ∑_{l=0}^{m}(−1)^{l}\binom{m}{l} d^{l}(w_{0}) T^{m−l} = ∑_{l=0}^{m}(−1)^{l} x (y^{l}� w′_{0}) \frac{T^{m−l}}{(m − l)!} .

Equation (5.2) is the special case T = 0. □

Using this result, we can now complete the proof begun in §3.

Proof of Theorem 2. We only have to prove the implications (v) ⇒ (i) and (v’) ⇒ (i), since the other parts of the theorem were proved after its statement in §3. Noting Φ_{yu}(w_{0}) ∈ ℌ^{0} if w_{0} ∈ ℌ^{0}, replace w_{0} in Proposition 8 by Φ_{yu}(w_{0}) to get

regT_{�} (\frac{1}{1 − yu}Φ_{yu}(w_{0})) = (exp(−du) ∘ Φ_{yu})(w_{0})e^{T u} = Δ_{u}(w_{0})e^{T u}

and thus

ZR^{�} (\frac{1}{1 − yu}Φ_{yu}(w_{0})) = Z_{R}(Δ_{u}(w_{0}))e^{T u}  (5.4)

On the other hand, the definition of Φ_{yu} and equation (4.6) give

\frac{1}{1 − yu}Φ_{yu}(w_{0}) = \frac{1}{1 − yu} ∗ w_{0} = exp_{∗}(log_{∘}(1 + yu)) ∗ w_{0} .  (5.5)

17 Now we observe that

Z_{R}^{∗}( exp_{∗}(log_{∘}(1 + yu))) = Z_{R}^{∗}(exp_{∗}(∑_{n=1}^{∞} \frac{(−1)^{n−1}}{n} z_{n} u^{n}))

= exp(T u − ∑_{n=2}^{∞} \frac{(−1)^{n}}{n} Z_{R}(z_{n}) u^{n}) = A_{R}(u)^{−1} e^{T u}  (5.6)

with A_{R}(u) as in (3.3). (This is the only point where the specific definition of the power series A_{R} is used.) Hence applying ρ_{R} ∘ Z_{R}^{∗} to (5.5) yields

ρ_{R} ∘ Z_{R}^{∗}(\frac{1}{1 − yu}Φ_{yu}(w_{0})) = e^{T u} Z_{R}(w_{0}).  (5.7)

From (5.4) and (5.7) we get

(ZR^{�} − ρ_{R} ∘ Z_{R}^{∗})(\frac{1}{1 − yu}Φ_{yu}(w_{0}))

= Z_{R}((Δ_{u} − 1)(w_{0}))e^{T u}

= ( ∑_{m=1}^{∞} Z_{R}(reg_{�} (y^{m} ∗ w_{0}))u^{m})e^{T u} (by (4.16)).

Since Φ_{yu} acts as an automorphism of ℌ^{0} and since the components of \frac{1}{1 − yu} ℌ^{0}

span ℌ^{1}, this shows the equivalence of statements (i) and (v) of Theorem 2. The equivalence of (i) and (v′) is shown in a similar manner. Applying Z_{R} ∘ reg_{∗} to equation (4.6) (with z = yu) and using (5.6) with T = 0, we have

Z_{R}(reg_{∗}(\frac{1}{1 − yu})) = A_{R}(u)^{−1}.  (5.8)

Now replace w_{0} in (5.4) and (5.7) by Δ_{u}^{−1}(w_{0}) and take the difference to get

(ZR^{�} − ρ_{R} ∘ Z_{R}^{∗})(\frac{1}{1 − yu} exp(du)(w_{0})) = Z_{R}(w_{0} − Δ_{u}^{−1}(w_{0}))e^{T u}.

Multiplying A_{R}(u)^{−1} on both sides of this and using (5.8), we obtain

A_{R}(u)^{−1}(ZR^{�} − ρ_{R} ∘ Z_{R}^{∗})(\frac{1}{1 − yu} exp(du)(w_{0}))

= Z_{R}(reg_{∗}(\frac{1}{1 − yu}) ∗ (w_{0} − Δ_{u}^{−1}(w_{0})))e^{T u}

= Z_{R}(reg_{∗}(\frac{1}{1 − yu} ∗ w_{0} − \frac{1}{1 − yu} ∗ Δ_{u}^{−1}(w_{0})))e^{T u}

= Z_{R}(reg_{∗}(\frac{1}{1 − yu} ∗ w_{0} − \frac{1}{1 − yu} � w_{0}))e^{T u} (by (4.16)).

As before, this gives the equivalence of statements (i) and (v’) of Theorem 2. □

As a second application of Proposition 8, we show that the “sum formula” for MZV’s ([6],[19]), which states that the sum of all MZV’s of fixed weight and depth is equal to the Riemann zeta value of that weight, is a consequence of the extended double shuffle relations.

Proposition 9. Denote by S(k, m) the sum of all monomials in ℌ^{0} of weight k and depth m. For any k and m with k > m + 1 ≥ 2, we have

(−1)^{m}reg_{�} (y^{m} ∗ x^{k−m−1}y) = S(k, m + 1) − S(k, m).

Corollary. If (R, Z_{R} : ℌ^{0} → R) has the EDS property, then Z_{R}(S(k, m)) = Z_{R}(x^{k−1}y) for 0 < m < k. In particular, the sum of all multiple zeta values of weight k and depth m is equal to ζ(k) for each value m = 1, 2, …, k − 1.

Proof. Apply (v) of Theorem 2 to the statement of the proposition. □

Proof of Proposition 9. The harmonic product y^{m} ∗ x^{k−m−1}y, which corresponds to the product ζ(\underbrace{1, 1, …, 1}_{m})ζ(k − m) of MZV’s, is easily computed as

y^{m} ∗ x^{k−m−1}y = ∑_{i=0}^{m} y^{i}x^{k−m−1}y^{m+1−i} + ∑_{j=0}^{m−1} y^{j}x^{k−m}y^{m−j}.

By (5.2), we then obtain

reg_{�} (y^{m} ∗ x^{k−m−1}y)

= ∑_{i=0}^{m}(−1)^{i}x(y^{i}� x^{k−m−2}y^{m+1−i}) + ∑_{j=0}^{m−1}(−1)^{j}x(y^{j}� x^{k−m−1}y^{m−j})

= x^{k−m−1}y^{m+1} + ∑_{i=1}^{m}(−1)^{i}x {(y^{i}� x^{k−m−2}y^{m−i})y + (y^{i−1}� x^{k−m−2}y^{m+1−i})y}

+ x^{k−m}y^{m} + ∑_{j=1}^{m−1}(−1)^{j}x {(y^{j}� x^{k−m−1}y^{m−1−j})y + (y^{j−1}� x^{k−m−1}y^{m−j})y}

= ∑_{i=0}^{m}(−1)^{i}x(y^{i}� x^{k−m−2}y^{m−i})y + ∑_{i=0}^{m−1}(−1)^{i+1}x(y^{i}� x^{k−m−2}y^{m−i})y

+ ∑_{j=0}^{m−1}(−1)^{j}x(y^{j}� x^{k−m−1}y^{m−1−j})y + ∑_{j=0}^{m−2}(−1)^{j+1}x(y^{j}� x^{k−m−1}y^{m−1−j})y

= (−1)^{m}x(y^{m}� x^{k−m−2})y + (−1)^{m−1}x(y^{m−1}� x^{k−m−1})y

= (−1)^{m}(S(k, m + 1) − S(k, m)).

An alternative way of deducing this identity is by making use of the automorphism Δ_{u} (note the relation (4.17)) and the formula (4.18), as follows:

(−1)^{m}reg_{�} (y^{m} ∗ x^{k−m−1}y) = the degree k component of Δ_{−1}(x^{k−m−1}y)

= the degree k component of (x\frac{1}{1 − y} )^{k−m−1}(y − x\frac{y}{1 − y} )

= the degree k component of (x\frac{1}{1 − y} )^{k−m−1}y − (x\frac{1}{1 − y} )^{k−m}y

= S(k, m + 1) − S(k, m).

□

We end this section with a collection of formulas for the regularization map regT_{�} and its harmonic analogue reg_{∗}^{T} .

Proposition 10. For w_{0} ∈ ℌ^{0} we have

regT_{�} (\frac{1}{1 − yu} w_{0}) = reg_{�} (\frac{1}{1 − yu} w_{0}) e^{T u}

= (exp_{�} (−yu) � (\frac{1}{1 − yu} w_{0})) e^{T u}

(5.9)

and

reg_{∗}^{T} (\frac{1}{1 − yu} w_{0}) = reg_{∗}(\frac{1}{1 − yu} w_{0}) e^{T u}

= (exp_{∗}(−yu) ∗ (\frac{1}{1 − yu} w_{0})) e^{T u} .

(5.10)

Corollary (Explicit regularization formula). Let w ∈ ℌ^{1} and write w = y^{m}w_{0} with m ≥ 0 and w_{0} ∈ ℌ^{0}. Then

reg_{�} (w) = ∑_{i=0}^{m}(−1)^{i} y^{i}� y^{m−i}w_{0} , reg_{∗}(w) = ∑_{i=0}^{m} \frac{(−1)^{i}}{i!} y^{∗i} ∗ y^{m−i}w_{0} ,

and conversely

w = ∑_{i=0}^{m} reg_{�} (y^{m−i}w_{0})� y^{i} , w = ∑_{i=0}^{m} \frac{1}{i!} reg_{∗}(y^{m−i}w_{0}) ∗ y^{∗i} .

Proof. Equation (5.9) is essentially contained in Proposition 8 and its proof, since equations (5.1) and (5.3) give

reg_{�} (\frac{1}{1 − yu} w_{0}) = exp(−du)(w_{0}) = exp_{�} (−yu) � (\frac{1}{1 − yu} w_{0}) .

For (5.10), we first combine (4.10) and (4.6) to get

\frac{1}{1 − yu}w_{0} = \frac{1}{1 − yu} ∗ Φ_{yu}^{−1}(w_{0}) = exp_{∗}(log_{∘}(1 + yu)) ∗ Φ_{yu}^{−1}(w_{0})
= exp_{∗}(yu) ∗ [exp_{∗}(log_{∘}(1 + yu) − yu) ∗ Φ_{yu}^{−1}(w_{0})] .  (5.11)

Since the expression in square brackets belongs to ℌ^{0}[[u]] by virtue of Proposition 6 (ii), we have

reg_{∗}(\frac{1}{1 − yu} w_{0}) = exp_{∗}(log_{∘}(1 + yu) − yu) ∗ Φ_{yu}^{−1}(w_{0})

= exp_{∗}(−yu) ∗ \frac{1}{1 − yu}w_{0} .

From this (5.10) follows. Finally, the corollary is obtained by specializing (5.9) and (5.10) to T = 0 and comparing the coefficients of u^{m} on both sides either before or after •-multiplying both sides by exp_{•}(yu), where • = � or ∗, and noting that y^{� i} = i! y^{i}. □

§6. Derivation and double shuffle relations

Let Der(ℌ) be the Lie algebra of derivations of ℌ (with respect to the concatenation product, Lie algebra structure being defined by [∂, ∂′] := ∂∂′ − ∂′∂, as usual). Clearly, an element of Der(ℌ) is uniquely determined by the images of x and y. Examples of elements of Der(ℌ) include the maps δ_{z} (z ∈ 𝔷) and d introduced in Proposition 2 and Proposition 7. Let τ : ℌ → ℌ be the involutory anti-automorphism which interchanges x and y. If ∂ ∈ Der(ℌ), then \overline{∂} := τ∂τ is also an element of Der(ℌ). The involution τ preserves ℌ^{0}, and the standard duality theorem for MZV’s can be stated as Z((1 − τ)(w_{0})) = 0 for any w_{0} ∈ ℌ^{0}.

For each integer n ≥ 1, define the derivations ∂_{n} and D_{n} in Der(ℌ) by

∂_{n}(x) = x(x + y)^{n−1}y,  ∂_{n}(y) = −x(x + y)^{n−1}y ,
D_{n}(x) = 0,  D_{n}(y) = x^{n}y .

Each of these derivations preserves ℌ^{1} and ℌ^{0}. As is easily seen by checking the images of the generators x and y, the derivations in each of the three families {∂_{n}} , \{D_{n}\}, and \{\overline{D}_{n}\} commute with one another: [∂_{m}, ∂_{n}] = [D_{m}, D_{n}] = [\overline{D}_{m}, \overline{D}_{n}] = 0 for any m, n ≥ 1. For each m ≥ 0, define the linear maps σ_{m} and \overline{σ}_{m} : ℌ → ℌ as homogeneous components of degree m of the homomorphisms σ := exp(∑_{n=1}^{∞} \frac{D_{n}}{n}) and \overline{σ} := exp(∑_{n=1}^{∞} \frac{\overline{D}_{n}}{n}) (= τστ):

σ = exp(∑_{n=1}^{∞} \frac{D_{n}}{n}) = ∑_{m=0}^{∞} σ_{m}, \overline{σ} = exp(∑_{n=1}^{∞} \frac{\overline{D}_{n}}{n}) = ∑_{m=0}^{∞} \overline{σ}_{m}.

We can use these endomorphisms to give two further collections of relations which are equivalent to the extended double shuffle relations.

Theorem 3. Assume that R, Z_{R} : ℌ^{0} → R satisfy the FDS. Then the following three properties are equivalent:

(i) (R, Z_{R}) satisfies the EDS.

(ii) Z_{R}(∂_{n}(w_{0})) = 0 for all w_{0} ∈ ℌ^{0} and all n.

(iii) Z_{R}((σ_{m} − \overline{σ}_{m})(w_{0})) = 0 for all w_{0} ∈ ℌ^{0} and all m.

Combining this result (for R = ℝ) with Theorem 1, we obtain the following “derivation relations” for multiple zeta values.

Corollary. One has Z(∂_{n}(w_{0})) = 0 for all n ≥ 1 and all w_{0} ∈ ℌ^{0} .

This corollary is a generalization of Hoffman’s relation [7] which is equivalent to the case n = 1. An alternative proof is given in [9].

Part (iii) of the theorem enables us to understand Ohno’s relation [13] in the light of the EDS relations. To see this we describe the map σ_{m} more concretely as follows. Put D = ∑_{n=1}^{∞} \frac{D_{n}}{n}. Since D(x) = 0 and D(y) = (x + \frac{x^{2}}{2} + \frac{x^{3}}{3} + ⋯)y = (− log(1 − x))y, we have D^{n}(x) = 0, D^{n}(y) = (− log(1 − x))^{n}y and hence σ(x) = x and σ(y) = ∑_{n=0}^{∞} \frac{1}{n!}(− log(1 − x))^{n}y = (1 − x)^{−1}y. From this, we have

σ(x^{k_{1}−1}yx^{k_{2}−1}y ⋯ x^{k_{n}−1}y)

= x^{k_{1}−1}(1 − x)^{−1}yx^{k_{2}−1}(1 − x)^{−1}y ⋯ x^{k_{n}−1}(1 − x)^{−1}y

= ∑_{m=0}^{∞} ∑_{{e_{1}+e_{2}+···+en=m \atop e_{i}≥0}} x^{k_{1}+e_{1}−1}yx^{k_{2}+e_{2}−1}y ⋯ x^{k_{n}+e_{n}−1}y,

namely,

σ_{m}(x^{k_{1}−1}yx^{k_{2}−1}y ⋯ x^{k_{n}−1}y) = ∑_{{e_{1}+e_{2}+···+en=m \atop e_{i}≥0}} x^{k_{1}+e_{1}−1}yx^{k_{2}+e_{2}−1}y ⋯ x^{k_{n}+e_{n}−1}y.

Ohno’s relation [13] then states that Z((σ_{m}−σ_{m}τ)(w_{0})) = 0 for all m ≥ 1, w_{0} ∈ ℌ^{0}. Under duality, this follows from Theorem 3 (iii) and Theorem 1.

The crucial identities needed for the proof of Theorem 3 are given in the following result, which will be proved in a more general form in the next section.

Theorem 4. i) The automorphism Δ_{u} defined in (4.15) is given in terms of the derivations ∂_{n} by

Δ_{u} = exp(∑_{n=1}^{∞}(−1)^{n} \frac{∂_{n}}{n} u^{n}) .  (6.1)

ii) Set ∂ = ∑_{n=1}^{∞} \frac{∂_{n}}{n}. Then we have

exp(∂) = \overline{σ}σ^{−1} .  (6.2)

Proof of Theorem 3. The equivalence of Theorem 3 (ii) and Theorem 2 (v) follows from (6.1) because (Δ_{u} − 1)(w_{0}) = ∑_{m=1}^{∞} reg_{�} (y^{m} ∗ w_{0})u^{m} by (4.17). By (6.2), we have ∂ = log(1 − (σ − \overline{σ})σ^{−1}) and σ − \overline{σ} = (1 − exp(∂))σ. This gives the equivalence (ii) ⇔ (iii) of Theorem 3. □

§7. Various derivations and automorphisms of ℌ

In this section we study the various derivations and automorphisms being considered in more detail and more systematically, and give a proof of Theorem 4.

Set δ_{n} := δ_{z_{n}}. For the convenience of the reader, we give a table of the various derivations which have been introduced (in Proposition 2, Proposition 7 and Section 6). In this table and for the rest of this section, unlike the previous sections, z denotes x + y.

d  \overline{d}  D_{n}  \overline{D_{n}}  δ_{n}  \overline{δ_{n}}  ∂_{n} = −\overline{∂_{n}}
x  xy  x^{2}  0  xy^{n}  0  xy^{n−1}z  xz^{n−1}y
y  y^{2}  xy  x^{n}y  0  zx^{n−1}y  0  −xz^{n−1}y
z  zy  xz  x^{n}y  xy^{n}  zx^{n−1}y  xy^{n−1}z  0

Note that each of these derivations belongs not merely to Der(ℌ) = Lie algebra of all derivations of ℌ, but to the subspace Der^{+}(ℌ) consisting of derivations which increase the weight, or equivalently which induce the zero derivation on the associated graded space Gr(ℌ) = ⨁(ℌ^{(k)}/ℌ^{(k+1)}) of ℌ, where ℌ^{(k)} denotes the subspace generated by all monomials in x and y of weight ≥ k. The space Der^{+}(ℌ) is isomorphic via the exponential map to the subgroup Aut^{1}(ℌ) of Aut(ℌ) consisting of all automorphisms which induce the identity automorphism on Gr(ℌ) (i.e., automorphisms ϕ of ℌ such that ϕ(x) − x and ϕ(y) − y contain only monomials of weight ≥ 2). We want to understand the automorphisms corresponding to all the above derivations. In particular, the fact that the derivations in each of the four families \{D_{n}\}, \{\overline{D_{n}}\}, {∂_{n}}, and {δ_{n}} commute with one another is more naturally interpreted in terms of the commutation of the corresponding automorphisms, since the set of automorphisms is closed under composition and the set of derivations isn’t. As a first step, we generalize the notations slightly. We define D, \overline{D}, ∂, δ : X ℚ[[X]] → Der^{+}(ℌ) as the ℚ-linear maps sending X^{n} (n ≥ 1) to D_{n}, \overline{D_{n}}, ∂_{n} and δ_{n}, respectively, or alternatively as the maps sending f(X) ∈ X ℚ[[X]] to the derivations D_{f}, \overline{D}_{f}, ∂_{f}, δ_{f} defined on generators by

\begin{matrix}D_{f}(x) = 0, D_{f}(y) = f(x)y;  \overline{D}_{f}(y) = 0, \overline{D}_{f}(x) = xf(y) ; \\ ∂_{f}(z) = 0, ∂_{f}(x) = x \frac{f(z)}{z} y;  δ_{f}(x) = 0, δ_{f}(z) = z \frac{f(x)}{x} y . \end{matrix}  (7.1)

(Note that δ_{f} is just δ_{γ(f)}, with γ as in (4.5), and that \overline{D}_{f} = \overline{D_{f}}.) The corresponding automorphisms are described by the following proposition.

Proposition 11. For h(X) ∈ 1 + X ℚ[[X]] let σ_{h}, \overline{σ}_{h}, Δ_{h}, Ψ_{h} ∈ Aut^{1}(ℌ) be the automorphisms defined by the following action on generators:

σ_{h}(x) = x ,  σ_{h}(y) = h(x) y ,  (7.2)
\overline{σ}_{h}(y) = y ,  \overline{σ}_{h}(x) = x h(y) ,  (7.3)
Δ_{h}(z) = z ,  Δ_{h}(x) = x (1 + \frac{h(z) − 1}{z} y)^{−1} ,  (7.4)

Ψ_{h}(x) = x ,  Ψ_{h}(z) = z (1 − \frac{h(x) − 1}{x} y)^{−1} .  (7.5)

Then each of the maps h ↦ σ_{h}, \overline{σ}_{h}, Δ_{h}, Ψ_{h} is a homomorphism from 1+X ℚ[[X]] to Aut(ℌ) , and they are related to the derivations D_{f}, \overline{D}_{f}, ∂_{f} and δ_{f} by

σ_{h} = exp(D_{f}), \overline{σ}_{h} = exp(\overline{D}_{f}), Δ_{h} = exp(−∂_{f}), Ψ_{h} = exp(δ_{f})  (7.6)

for all h ∈ 1 + X ℚ[[X]] , where f = log(h) ∈ X ℚ[[X]] .

Proof. Let g(X) and h(X) be two elements of 1 + X ℚ[[X]]. It is easy to verify the equations σ_{gh} = σ_{g}σ_{h}, \overline{σ}_{gh} = \overline{σ}_{g}\overline{σ}_{h}, and σ_{h} = exp(D_{f}), \overline{σ}_{h} = exp(\overline{D}_{f}) with

23 f = log(h). If we use the map γ in (4.5) then γ(X^{k}) = x^{k−1}y implies γ(h(X)−1) = \frac{h(x)−1}{x}y and hence we have Ψ_{h} = Φ_{γ(h−1)}. The proposition for Ψ_{h} then follows from (i) of Proposition 6, and this implies also the statement for Δ_{h} because of the identities Δ_{h} = εΨ_{h}ε and −∂_{f} = εδ_{f}ε, where ε is the involution of ℌ interchanging x and z and sending y to −y. □

We can now relate the above derivations and automorphisms to the ones previously studied and give the proof of Theorem 4. First, notice that the automorphism Δ_{1+uX} is nothing other than the automorphism Δ_{u} in Theorem 4 (i) defined by Δ_{u} = exp(−du)∘Φ_{yu} in Section 4, as is seen by comparing (4.18) and (7.4). The formula of Theorem 4 (i) then follows from the third equation of (7.6) with h = 1+uX and f = log(1 + uX). On the other hand, the automorphisms denoted σ and \overline{σ} in Section 6 are σ_{1−X}^{−1} and \overline{σ}_{1−X}^{−1} in our notations. Part (ii) of Theorem 4 then follows from the special case u = −1 of part (i) and the formula Δ_{1+uX} = \overline{σ}_{1+uX}^{−1} · σ_{1+uX}, whose proof is immediate from the definitions (7.2)–(7.4) by checking the action on generators.

In the rest of the section, we show that the Lie algebra generated by the four derivations d, \overline{d}, \overline{d} − D_{1}, and d − \overline{D_{1}} contain all derivations considered so far, and discuss some properties of this Lie algebra.

Proposition 12. For all n ≥ 1 we have the commutation formulas

[ \overline{d}, δ_{n}] = n δ_{n+1} ,  [ d, \overline{δ_{n}}] = n \overline{δ_{n+1}} ,  (7.7)
[\overline{d}, D_{n}] = n D_{n+1} ,  [d, \overline{D_{n}}] = n \overline{D_{n+1}} .  (7.8)

Proof. Since in each case both sides of the equation to be proved are derivations on all of ℌ, it suffices to show that they agree on the generators x and y. We have

[ \overline{d}, δ_{n}](x) = \overline{d}(0) − δ_{n}(x^{2}) = 0 = n δ_{n+1}(x)

and

[ \overline{d}, δ_{n}](y) = \overline{d}(zx^{n−1}y) − δ_{n}(xy)
= xzx^{n−1}y + (n − 1)zx^{n}y + zx^{n}y − xzx^{n−1}y
= nzx^{n}y = n δ_{n+1}(y) .

The proofs of the other three identities are similar and will be omitted. □

Theorem 5. Let 𝔏_{4} be the Lie subalgebra of Der(ℌ) generated by the four elements d, \overline{d}, D = \overline{d} − D_{1}, and \overline{D} = d − \overline{D_{1}}, and 𝔏_{3} the Lie subalgebra of 𝔏_{4} generated by d, \overline{d}, and D − \overline{D}. Then

(i) 𝔏_{3} = \overline{𝔏_{3}}, 𝔏_{4} = \overline{𝔏_{4}}.

(ii) 𝔏_{3} contains ∂_{n}, δ_{n}, \overline{δ_{n}} for all n ≥ 1.

(iii) 𝔏_{4} also contains D_{n} and \overline{D_{n}} for all n ≥ 1.

Proof. Property (i) is clear from the definition. We have

δ_{1} = \overline{d} − D + \overline{D}  (7.9)

(both sides are derivations sending x to 0 and y to zy) and D_{1} = \overline{d} − D (by the definition of D), so δ_{1} ∈ 𝔏_{3}, D_{1} ∈ 𝔏_{4}, and also \overline{δ_{1}} ∈ 𝔏_{3}, \overline{D_{1}} ∈ 𝔏_{4} by (i). Proposition 12 and induction over n then show that the derivations δ_{n}, \overline{δ_{n}} belong to 𝔏_{3} and D_{n}, \overline{D_{n}} to 𝔏_{4} for all n ≥ 1. Finally, combining equations (6.1), (4.15) and (4.11) we have

exp(∑_{n=1}^{∞} \frac{∂_{n}}{n} u^{n}) = Δ_{−u} = exp(du) exp(δ_{log_{∘}(1−yu)}) = exp(du) exp(− ∑_{n=1}^{∞} \frac{δ_{n}}{n} u^{n}) .

(7.10)

From this, we have ∂_{n} ∈ 𝔏_{3} for all n by the Baker-Campbell-Hausdorff formula. □

A more explicit formula for ∂_{n} is obtained as follows. We differentiate (7.10) with respect to u to get

(∑_{n=1}^{∞} ∂_{n}u^{n−1}) exp(∑_{n=1}^{∞} \frac{∂_{n}}{n} u^{n})

= d exp(du) exp(− ∑_{n=1}^{∞} \frac{δ_{n}}{n} u^{n}) − exp(du)(∑_{n=1}^{∞} δ_{n}u^{n−1}) exp(− ∑_{n=1}^{∞} \frac{δ_{n}}{n} u^{n})

and hence

∑_{n=1}^{∞} ∂_{n}u^{n−1} = d − exp(du)(∑_{n=1}^{∞} δ_{n}u^{n−1}) exp(−du)

= d − exp(ad(d)u)(∑_{n=1}^{∞} δ_{n}u^{n−1}).

Here, by (7.7) we have

∑_{n=1}^{∞} δ_{n} = ∑_{n=0}^{∞} \frac{ad(\overline{d})^{n}}{n!} (δ_{1}) = exp(ad(\overline{d}))(δ_{1}).

Hence we obtain

∑_{n=1}^{∞} ∂_{n} = d − exp(ad(d)) exp(ad(\overline{d}))(δ_{1}).

Written out, this gives an explicit description of ∂_{n} in terms of the generators of 𝔏_{3}:

∂_{n} = \{ \begin{matrix}d − \overline{d} + D − \overline{D} & if n = 1, \\ ∑_{i+j=n−1} \frac{ad(d)^{i}}{i!} \frac{ad(\overline{d})^{j}}{j!} (D − \overline{D} − \overline{d}) & if n ≥ 2. \end{matrix}

The following proposition gives further relations in the Lie algebra 𝔏_{4}. Since we do not use this proposition later (except that the relations do contribute to reduce the dimensions of 𝔏_{4}^{(n)} as in the table below), we omit the proof.

Proposition 13.

(i) The two derivations D and \overline{D} commute.

(ii) We have

\frac{ad(\overline{d})^{n}}{n!} (d) = [ad(d), \frac{ad(\overline{d})^{n−1}}{(n − 1)!} ](D) (n > 1),
\frac{ad(d)^{n}}{n!} (\overline{d}) = [ad(\overline{d}), \frac{ad(d)^{n−1}}{(n − 1)!} ](\overline{D}) (n > 1).

Here is a short table of the dimensions of the graded piece 𝔏_{3}^{(n)} and 𝔏_{4}^{(n)} of 𝔏_{3} and 𝔏_{4} for n ≤ 8 and, for comparison, the same data for the free Lie algebras 𝔉_{3} and 𝔉_{4} on 3 and 4 generators, respectively.

\begin{matrix}n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 𝔏_{3}^{(n)} & 3 & 3 & 5 & 8 & 14 & 23 & 43 & 75 \\ 𝔉_{3}^{(n)} & 3 & 3 & 8 & 18 & 48 & 116 & 312 & 810 \\ 𝔏_{4}^{(n)} & 4 & 5 & 8 & 13 & 22 & 37 & 66 & 115 \\ 𝔉_{4}^{(n)} & 4 & 6 & 20 & 60 & 204 & 670 & 2340 & 8160 \end{matrix}

We end this section by presenting some results which seems interesting to us, even though there is no obvious application to the structure of the double shuffle relations. Each of the four families \{δ_{n}\}, \{\overline{δ_{n}}\}, \{D_{n}\}, and \{\overline{D_{n}}\} has the property that their elements mutually commute and, by Proposition 12, that they have the form ad(X)^{n}(Y) for some X, Y ∈ Der(ℌ). We give two results, one generalizing and one reinterpreting this phenomenon.

Proposition 14. Let a, b be a basis of the 2-dimensional space spanned by x and y, and δ a derivation which sends a to 0 and b to any element of the form κb^{2} + λba + μab + νa^{2}. Suppose θ is another derivation sending a to a^{2} and b to the sum of ab and any linear combination of [a, b] and δ(b). Then the derivations δ′_{n} defined by δ′_{n} = ad(θ)^{n−1}(δ)/(n − 1)! are given on generators by δ′_{n}(a) = 0, δ′_{n}(b) = κba^{n−1}b + λba^{n} + μa^{n}b + νa^{n+1} and all commute with one another.

Proof. The proof is routine and is omitted. □

If we set a = z (= x + y), b = y, δ(b) = b^{2} − ab (= −xy), and θ(b) = ab − [a, b]/2 in the proposition, then we have δ′ = ∂_{1} and δ′_{n} = ∂_{n}, thereby proving not only that the elements {∂_{n}} mutually commute, as remarked at the beginning of §7, but also that they have the form ad(θ)^{n−1}(∂_{1})/(n − 1)!. Note however that the derivation θ in this case (defined by θ(x) = (xz + zx)/2, θ(y) = (yz + zy)/2) does not belong to the Lie algebra 𝔏_{4}.

Definition. Two elements X and Y in a Lie algebra over a field of characteristic 0 semi-commute if the power series exp(tX) exp(−tY) in (the completion of) the universal enveloping algebra commute with each other for different t, i.e., if all of the coefficients of this power series commute with each other. Clearly, if X and Y commute, then they semi-commute.

Proposition 15. Let X and δ be two elements in a Lie algebra over a field K of characteristic 0. Then the following three statements are equivalent:

(i) All of the elements ad(X)^{n}(δ) (n = 0, 1, 2, …) commute with one another. (ii) The elements X and X − δ semi-commute. (iii) Any two elements of X + Kδ semi-commute.

Proof. We put δ_{n} := ad(X)^{n−1}(δ)/(n − 1)! (n = 1, 2, 3, …). Assume (i). Then we have

exp(∑_{n=1}^{∞} \frac{δ_{n}}{n} t^{n}) = exp(tX) exp(−t(X − δ))

since both sides satisfy the differential equation f′(t) = exp(tX)δ exp(−tX)f(t) and f(0) = 1. This implies (ii). Conversely, assume (ii) and set

f(t) = exp(tX) exp(−t(X − δ)).

We calculate the derivative:

f′(t) = exp(tX)δ exp(−(X − δ)) = Z(t)f(t),

where

Z(t) = exp(tX)δ exp(−tX) = exp(ad(X)t)(δ) = ∑_{n=1}^{∞} δ_{n}t^{n−1}.

We want to show that if f(t) and f(u) commute for all t and u then Z(t) and Z(u) commute for all t and u (i.e., all δ_{n} commute). Differentiating the equation f(t)f(u) = f(u)f(t) with respect to u, we find f(t)Z(u) = Z(u)f(t). Differentiating this with respect to t, we obtain Z(t)Z(u) = Z(u)Z(t). Thus we have shown the equivalence of (i) and (ii). But then (i) and (iii) are also equivalent, since (i) is invariant under δ → −cδ for any c ∈ K^{∗}. □

Remark. (i) If X and Y are two elements of a Lie algebra which semi-commute with each other, then the commutation identity is

e^{uX}e^{−(u+v)Y} e^{vX} = e^{−vY} e^{(u+v)X}e^{−uY}

or more symmetrically,

e^{aX}e^{bY} e^{cX}e^{aY} e^{bX}e^{cY} = 1 if a + b + c = 0,

which is somewhat reminiscent of the so-called Yang-Baxter equation.

(ii) Let L_{0} be the quotient of the free Lie algebra on two letters X and δ by the relation that all ad(X)^{n}(δ) commute, or equivalently, setting δ_{n} = ad(X)^{n−1}(δ)/(n − 1)!, the Lie algebra with basis (X, δ_{1}, δ_{2} …) and brackets given by [X, δ_{n}] = nδ_{n+1}, [δ_{m}, δ_{n}] = 0. Since this is a graded Lie algebra (with X and δ of weight 1, and δ_{n} of weight n), it extends naturally to a slightly larger Lie algebra L_{1} = L_{0} + K · H where [H, X] = X and [H, δ_{n}] = nδ_{n}. Proposition 12 then says that there are several natural copies of L_{1} in the endomorphisms of ℌ, and Proposition 15 gives a nice interpretation of embeddings of L_{1} into any Lie algebra in terms of semicommuting elements. It is therefore of interest to note that the same Lie algebra L_{1} has occurred in the work of Connes and Moscovici in connection with questions about cyclic homology, foliations and, more recently, the so-called “Rankin-Cohen brackets” in the theory of modular forms ([2], [3]). But while our copies of L_{1} act on ℌ as derivations, and hence can be thought of as Hopf algebra representations of the standard Hopf algebra associated to the Lie algebra L_{1} (i.e., the Hopf algebra whose underlying algebra is the universal enveloping algebra of L_{1} and whose coproduct is defined by requiring the generators X, δ and H—and hence all elements of L_{1}—to be primitive), Connes and Moscovici consider a “twisted” Hopf algebra structure which has the same underlying algebra and in which H and δ are still primitive, but with Δ(X) = X ⊗ 1 + δ ⊗ H + 1 ⊗ X. It would be very interesting to discover whether there is any deeper reason for the occurrence of the same Lie algebra L_{1} in these very different contexts, and whether the twisted Hopf structure considered by Connes and Moscovici plays any role in the context of MZV’s.

§8. Linearized double shuffle relations

In this section, we fix the depth n and look at the (extended) double shuffle relation modulo elements of lower depth and products. This amounts to “linearizing” the double shuffle relation and reduces the problem of finding an upper bound (and conjecturally exact value) for the number of generators of the ℚ-algebra of MZV’s of given weight k and depth n to the solution of an elementary, but hard, problem of linear algebra. Some consequences of this reduction for general n are given at the end of this section. A more detailed discussion of the special cases n = 2 and 3 as well as some general results will be given in a subsequent paper.

Let Z = ⨁_{k≥0} Z_{k} be the graded algebra generated by MZV’s over ℚ, where Z_{k} is the ℚ-vector space generated by MZV’s of weight k. The space Z_{k} has a natural filtration Z_{k} = ⋃_{n≥0} Z_{k}^{(n)}, where Z_{k}^{(n)} is the ℚ-vector space spanned

by MZV’s of weight k and depth ≤ n, and setting Z^{(n)} = ⨁_{k≥0} Z_{k}^{(n)} gives a corresponding filtration Z = ⋃_{n≥0} Z^{(n)} on the whole algebra Z. Let I = ⨁_{k≥1} Z_{k} be the augumentation ideal of Z and I^{2} its square. The quotient space T = I/I^{2} also inherits the grading and filtration. All products of MZV’s become trivial in T , so the dimension of T_{k}, the weight k component of T , coincides with the number D_{k} of algebra generators of Z in weight k. We can also consider the bigraded vector space M associated to the graded filtered space T :

M = ⨁_{k, n≥1} M_{k}^{(n)}, M_{k}^{(n)} = T_{k}^{(n)}/T_{k}^{(n−1)} ≃ Z_{k}^{(n)}/(Z_{k}^{(n−1)} + Z_{k}^{(n)} ∩ I^{2}).

Then clearly the dimension D_{k,n} of M_{k}^{(n)} equals the number of algebra generators of Z of weight k and depth n, and we have D_{k} = ∑_{n=1}^{k−1} D_{k,n}.

Remark. There is a conjectural formula giving these dimensions D_{k,n}, due to Broadhurst and Kreimer. Since it is too beautiful to omit, but we did not want to interrupt the text here, we have reproduced this formula in an appendix at the end of the paper.

Our object is to introduce certain vector spaces DSh_{n}(d) (n, d > 0) whose dimensions give upper bounds for (and are conjecturally equal to) the numbers D_{n+d,n}, and then to discuss the calculation of these spaces. Let 𝔖_{n} denote the symmetric group of order n and R = R_{n} = ℤ[𝔖_{n}] its group ring. We denote by V_{n} the space ℚ[x_{1}, …, x_{n}] of polynomials in n variables with rational coefficients and by V_{n}(d) its subspace of homogeneous polynomials of degree d. The group 𝔖_{n} acts on these spaces in a natural way by (f|σ)(x_{1}, …, x_{n}) = f(x_{σ^{−1}(1)}, …, x_{σ^{−1}(n)}). Note that this is a right action, i.e. f|(στ) = (f|σ)|τ for all σ and τ in 𝔖_{n}. This action extends to an action of R in the standard way by f|(∑ a_{i}σ_{i}) = ∑ a_{i} (f|σ_{i}) .

For each integer l with 1 ≤ l ≤ n−1 we define the l^{th} shuffle element sh_{l} = sh_{l}^{(n)} in the group ring R by

sh_{l}^{(n)} = ∑_{\begin{matrix}σ∈𝔖_{n} \\ σ(1)<···<σ(l) \\ σ(l+1)<···<σ(n) \end{matrix}}σ .

We denote by ℑ^{(n)} = sh_{1}R + ⋯ + sh_{n−1}R the right ideal in R generated by all of these elements and, for any (right) representation V of 𝔖_{n}, define the “shuffle subspace” Sh(V) of V by

Sh(V) = Ker(ℑ^{(n)}, V) = ⋂_{l=1}^{n−1} Ker(sh_{l}^{(n)}, V) ,

the space of elements of V annihilated by the ideal ℑ^{(n)}. In particular, we have the shuffle space Sh_{n} := Sh(V_{n}) ⊂ V_{n} and its homogeneous part of degree d, Sh_{n}(d) := Sh(V_{n}(d)) ⊂ V_{n}(d).

We remark that the dimension of Sh(V) can be computed for any 𝔖_{n}-module V in a nice way using the theory of representation of finite groups. The following proposition gives explicit formulas for this dimension.

Proposition 16. Let V be any representation of 𝔖_{n} and let C ∈ 𝔖_{n} be an element of order n (i.e., a cyclic permutation of 1, …, n). Then we have the two formulas

dim Sh(V) = \frac{1}{n} ∑_{m|n} μ(\frac{n}{m}) tr(C^{m}, V) ,  (8.1)

and

dim Sh(V) = \frac{1}{φ(n)} ∑_{m|n} μ(\frac{n}{m}) dim(V^{C^{m}}) ,  (8.2)

where μ and φ denote the Möbius function and the Euler function respectively. In particular,

dim Sh_{n}(d) = \frac{1}{n} ∑_{e|(n,d)} μ(e) \binom{n/e + d/e − 1}{d/e} .  (8.3)

Actually, we proved this in the reverse direction, first proving the special case (8.3) by identifying the space Sh_{n}(d) with the graded part of degree d and weight n + d of the free Lie algebra on infinitely many generators of weights 1, 2, …; then noticing that (8.1) coincides with (8.3) for V = V_{n}(d), and that this suffices for the general case because the representations V_{n}(d) (d = 0, 1, 2, …) span the Grothendieck group of representations of 𝔖_{n} over ℚ; and finally deducing (8.2) from (8.1) by using the formula dim(V^{G}) = |G|^{−1} ∑_{g∈G} tr(g, V) which holds for any representation V of a group G. In any case we omit the details of the proofs since we will not use any of these formulas and there must exist a simpler proof anyway, undoubtedly already in the literature.

The space we are actually interested in, however, is a subspace of Sh_{n}(d) which is much harder to compute. To define it, we first extend the action of 𝔖_{n} on V_{n} to an action of Γ_{n} = GL_{n}(ℤ) on V_{n} by setting

(f|S)(x_{1}, …, x_{n}) := f((x_{1}, …, x_{n}) · S^{−1}) (S ∈ Γ_{n}),

which agrees with the previous definition for 𝔖_{n} if the elements of the symmetric group are identified with permutation matrices in Γ_{n} in the usual way. In particular, we will be interested in the element P of Γ_{n} given by

P = ( \begin{matrix}1 \\ −1 & 1 \\ & ... & ... \\ & & −1 & 1 \end{matrix} ) , P^{−1} = ( \begin{matrix}1 \\ 1 & 1 \\ ⋮ & & ... \\ 1 & 1 & ... & 1 \end{matrix} ) .

We now define the double shuffle subspace DSh_{n} of V_{n} as the intersection

DSh_{n} = Sh_{n} ∩ Sh_{n}|P^{−1}
= {f ∈ V_{n} | f|sh_{l} = f^{♯}|sh_{l} = 0 for l = 1, …, n − 1} ,

where for any polynomial f ∈ V_{n} we have set f^{♯} = f|P or explicitly

f^{♯}(x_{1}, …, x_{n}) = f(x_{1} + x_{2} + ⋯ + x_{n}, x_{2} + ⋯ + x_{n}, …, x_{n−1} + x_{n}, x_{n}) .

For an element S in ℤ[GL_{n}(ℤ)], we define S^{♯} := P SP^{−1}. The condition f^{♯}|sh_{l} = 0 is then equivalent to f|sh_{l}^{♯} = 0. We write DSh_{n}(d) for the degree d part of DSh_{n}.

As already mentioned, the double shuffle space DSh_{n}(d) is much harder to compute than the single shuffle space Sh_{n}(d), the reason being that it is defined in terms of the action of an infinite rather than a finite group and therefore cannot be computed using the theory of characters of finite groups. Some information about the dimension of DSh_{n}(d) can be obtained by looking at finite subgroups or quotients of the subgroup of Γ_{n} generated by 𝔖_{n} and P and applying the theory of characters. A few results of this nature, both for general n and for the special cases n = 2 and n = 3, will be discussed at the end of the section and in more detail in the planned paper [10]. Before doing this, we prove a theorem which says that the generating function of MZV’s belong to the double shuffle space and, as a consequence, bounds the number of linearly independent MZV’s in terms of the dimensions of the vector spaces DSh_{n}(d).

Let k be any (not necessarily admissible) index set. Recall that the “regularized multiple zeta values” Z_{k}^{∗}(0) and Zk^{�} (0) are respectively the constant terms of the polynomials Z_{k}^{∗}(T) and Zk^{�} (T) defined in §2. For a fixed n, consider the two generating functions

F_{n}^{∗}(x_{1}, …, x_{n}) = ∑_{k} Z_{k}^{∗}(0)x_{1}^{k_{1}−1} ⋯ x_{n}^{k_{n}−1}

and

Fn^{�} (x_{1}, …, x_{n}) = ∑_{k} Zk^{�} (0)x_{1}^{k_{1}−1} ⋯ x_{n}^{k_{n}−1}

in Z[x_{1}, …, x_{n}], where both the sums run over all index sets k = (k_{1}, …, k_{n}) (allowing k_{1} = 1) of depth n. We regard the coefficients as elements in Z^{(n)} and consider their images in M, and look at the images of F_{n}^{∗} and Fn^{�} in M̃[x_{1}, …, x_{n}] = M̃ ⊗_{ℚ} V_{n}, where M̃ = ℚ ⊕ M as a ℚ-vector space and is regarded as a ℚ-algebra in a trivial way (only ℚ-multiplication is non-zero).

Theorem 6. The two polynomials F_{n}^{∗} and Fn^{�} agree as elements of M̃ ⊗ V_{n} and belong to the subspace M̃ ⊗ DSh_{n}.

Corollary. For all k > n > 0, we have the upper bound

D_{k,n} ≤ dim_{ℚ} DSh_{n}(k − n).

Proof. The statement of the theorem follows from the following three assertions:

(i) (F_{n}^{∗}|sh_{l})(x_{1}, …, x_{n}) = 0 in M̃ ⊗ V_{n} for 1 ≤ l ≤ n − 1.

(ii) ((Fn^{�} )^{♯}|sh_{l})(x_{1}, …, x_{n}) = 0 in M̃ ⊗ V_{n} for 1 ≤ l ≤ n − 1.

(iii) F_{n}^{∗}(x_{1}, …, x_{n}) = Fn^{�} (x_{1}, …, x_{n}) in M̃ ⊗ V_{n}.

(i) With an obvious notational convention, we know from Proposition 1 in §2 that the values Z_{k}^{∗}(0) satisfy Z_{k}^{∗}(0)Z_{k′}^{∗}(0) = Z_{k∗k′}^{∗}(0) for any index sets k and k′. From this we easily have 0 = F_{l}^{∗}(x_{1}, …, x_{l})F_{n−l}^{∗}(x_{l+1}, …, x_{n}) = (F_{n}^{∗}|sh_{l})(x_{1}, …, x_{n}) in M̃[x_{1}, …, x_{n}]. This proves (i).

(ii) Here we use the iterated integral expression. For small ε > 0, put

ζ_{ε}(k_{1}, k_{2}, …, k_{n}) =  ∫ ⋯ ∫_{1−ε>t_{1}>t_{2}>···>t_{k}>0} ω_{1}(t_{1})ω_{2}(t_{2}) ⋯ ω_{k}(t_{k}),  (8.4)

where the integrand is the same as in (1.2). First we have

∫ ⋯ ∫_{a>x_{1}>···>x_{r}>b} \frac{dx_{1}}{x_{1}} ⋯ \frac{dx_{r}}{x_{r}} = \frac{1}{r!}(log \frac{a}{b})^{r}

for any a > b > 0, since the integral would be the same if x_{1}, …, x_{r} were ordered in any other way and the integral over unordered r-tuples a > x_{1}, …, x_{r} > b is simply the r-th power of ∫_{b}^{a} \frac{dx}{x}. Applying this to (8.4) with (a, b, r) = (1 − ε, u_{1}, k_{1} − 1), (u_{1}, u_{2}, k_{2} − 1), …, (u_{n−1}, u_{n}, k_{n} − 1), where u_{1} = t_{k_{1}}, u_{2} = t_{k_{1}+k_{2}}, …, u_{n} = t_{k_{1}+k_{2}+···+k_{n}} = t_{k}, we find

ζ_{ε}(k_{1}, k_{2}, …, k_{n}) = \frac{1}{∏_{i=1}^{n}(k_{i} − 1)!}×

∫ ⋯ ∫_{1−ε>u_{1}>···>u_{n}>0}(log \frac{1 − ε}{u_{1}})^{k_{1}−1} \frac{du_{1}}{1 − u_{1}} (log \frac{u_{1}}{u_{2}} )^{k_{2}−1} \frac{du_{2}}{1 − u_{2}} ⋯ (log \frac{u_{n−1}}{u_{n}} )^{k_{n}−1} \frac{du_{n}}{1 − u_{n}} .

From this, we obtain for the corresponding generating function Fn,ε^{�}:

Fn,ε^{�}(x_{1}, …, x_{n}) := ∑_{k_{1},... ,k_{n}≥1} ζ_{ε}(k_{1}, …, k_{n})x_{1}^{k_{1}−1} ⋯ x_{n}^{k_{n}−1}

= (1 − ε)^{x_{1}} ∫ ⋯ ∫_{1−ε>u_{1}>···>u_{k}>0} \frac{u_{1}^{−x_{1}+x_{2}}}{1 − u_{1}} du_{1} \frac{u_{2}^{−x_{2}+x_{3}}}{1 − u_{2}} du_{2} ⋯ \frac{u_{n}^{−x_{n}}}{1 − u_{n}} du_{n}

and hence

(Fn,ε^{�})^{♯}(x_{1}, …, x_{n}) = Fn,ε^{�}(x_{1} + x_{2} + ⋯ + x_{n}, x_{2} + ⋯ + x_{n}, …, x_{n})

= (1 − ε)^{x_{1}+···+x_{n}} ∫ ⋯ ∫_{1−ε>u_{1}>···>u_{k}>0} \frac{u_{1}^{−x_{1}}}{1 − u_{1}} du_{1} \frac{u_{2}^{−x_{2}}}{1 − u_{2}} du_{2} ⋯ \frac{u_{n}^{−x_{n}}}{1 − u_{n}} du_{n}.

Now, it is clear that (Fn,ε^{�})^{♯}(x_{1}, …, x_{n}) satisfies the shuffle relation

(Fl,ε^{�})^{♯}(x_{1}, …, x_{l})(Fn−l,ε^{�})^{♯}(x_{l+1}, …, x_{n}) = ((Fn,ε^{�})^{♯}|sh_{l})(x_{1}, …, x_{n})

for any ε and 1 ≤ l ≤ n − 1. From this and the definition of the polynomial Zk^{�} (T), we obtain the desired assertion.

(iii) This follows from our fundamental relation (2.3) and the fact that the coefficient of ρ(T^{i}) is contained in the ring generated by the Riemann zeta values (MZV’s of depth 1). □

To prove the corollary, let {f_{i}}_{i=1,... ,r} (r = dim DSh_{n}(d)) be a basis of DSh_{n}(d) over ℚ and write each f_{i}(x_{1}, …, x_{n}) as ∑_{k} c_{i}(k)x_{1}^{k_{1}−1} ⋯ x_{n}^{k_{n}−1}, where k = (k_{1}, …, k_{n}) runs over the index sets of depth n and weight k. Then the r × \binom{n+d−1}{d} matrix {c_{i}(k)}_{i,k} has rank r, so all of its columns can be expressed as ℚ-linear combinations of r of them, say, those labeled by the index sets k_{1}, …, k_{r}. This implies that the k-th coefficient of any polynomial F ∈ A ⊗ DSh_{n}(d), for any index set k of depth n and weight k and any ℚ-algebra A, is a rational linear combination (with coefficients not depending on F) of its k_{1}-st, …, k_{r}-th coefficients. Applying this to the polynomial F defined as the homogeneous component of degree d = k − n of F_{n}^{∗}(x_{1}, …, x_{n}) = Fn^{�} (x_{1}, …, x_{n}), which belongs to M̃⊗DSh_{n}(d) by the theorem, we see that the image in M_{k}^{(n)} of each MZV ζ(k) of weight k and depth n is a rational linear combination of the r MZV’s ζ(k_{1}), …, ζ(k_{r}), so dim_{ℚ} M_{k}^{(n)} ≤ r. □

In the remainder of this section we give some estimates of the spaces DSh_{n}(d) and corollaries for MZV’s.

Theorem 7. If d is odd, then DSh_{n}(d) = {0} for every n > 0.

Corollary (Parity result). If k ≢ n mod 2, then D_{k,n} = 0. In other words, any MZV of weight k and depth n with k and n of opposite parity is a linear combination of MZV’s of smaller depth and products of MZV’s of lower weight.

This result, of which a different proof was given by Tsumura [17], generalizes classical results of Euler for n = 1 (that ζ(2j) is expressible as a product of smaller zeta-values for j ≥ 2) and n = 2 (that all double zeta values of odd weight can be expressed in terms of Riemann zeta values.)

To prove Theorem 7, we will construct a space ShC_{n}(d) containing DSh_{n}(d) and show that it is zero when d is odd. This larger space will be of interest also for d even, since it can be computed in terms of representation of finite groups and hence in principle gives a non-trivial upper bound for dim DSh_{n}(d) for all n and d. This subject will be treated in a later paper.

To define the space ShC_{n}(d), we introduce an action of 𝔖_{n+1} on the space ℚ[x_{1}, …, x_{n}] by identifying this with the space ℚ[y_{1}, …, y_{n+1}]/(y_{1} + ⋯ + y_{n+1}) via the obvious isomorphism

φ : ℚ[x_{1}, …, x_{n}] ≃ ℚ[y_{1}, …, y_{n+1}]/(y_{1} + ⋯ + y_{n+1})

given by

φ(x_{i}) = y_{i} (1 ≤ i ≤ n).

The action of 𝔖_{n+1} is then given by

(F |σ)(y_{1}, …, y_{n+1}) := F (y_{σ^{−1}(1)}, …, y_{σ^{−1}(n+1)}) (σ ∈ 𝔖_{n+1}).

Identifying 𝔖_{n} as the subgroup of 𝔖_{n+1} consisting of elements which fix n + 1, we can also regard the space ℚ[y_{1}, …, y_{n+1}]/(y_{1} + ⋯ + y_{n+1}) as an 𝔖_{n}-module; the map φ is then 𝔖_{n}-equivariant. As before, the actions of 𝔖_{n+1} extends naturally to an action of ℤ[𝔖_{n+1}] ⊃ ℤ[𝔖_{n}].

Let C_{n+1} = ( \begin{matrix}1 & ··· & n & n+1 \\ 2 & ··· & n+1 & 1 \end{matrix} ) ∈ 𝔖_{n+1} be the cyclic permutation and ε the involution (y_{1}, …, y_{n+1}) ↦ (−y_{1}, …, −y_{n+1}). Theorem 7 is a consequence of the following proposition.

Proposition 17. For n, d ≥ 1, define

ShC_{n}(d) := {f ∈ V_{n}(d)| f^{♯}|sh_{l} = 0 (1 ≤ l ≤ n − 1), φ(f^{♯})|C_{n+1} = εφ(f^{♯})}.

Then (i) DSh_{n}(d) ⊂ ShC_{n}(d), (ii) ShC_{n}(d) = {0} if d is odd.

Proof. We first prove (ii). For this, we use the easily checked identity

1 + sh_{1}^{(n)}C_{n+1} = C_{n+1}(1 + sh_{1}^{(n)}τ)  (8.5)

in ℤ[𝔖_{n+1}], where τ is the transposition 1 ↔ n + 1 and where the shuffle element sh_{1}^{(n)} ∈ ℤ[𝔖_{n}] is viewed as an element in ℤ[𝔖_{n+1}] in the way described above.

For f ∈ ShC_{n}(d), put F = φ(f^{♯}). Applying (8.5) to F and using the conditions F |sh_{1}^{(n)} = 0 and F |C_{n+1} = εF, we obtain

F = F |C_{n+1}(1 + sh_{1}^{(n)}τ) = εF |(1 + sh_{1}^{(n)}τ) = εF.

If d is odd, this gives F = 0 and hence f = 0.

To prove (i), we need a lemma.

Lemma. For 0 ≤ l ≤ n, let T_{l} = ( \begin{matrix}1 & ··· & l & l+1 & ··· & n \\ 1 & ··· & l & n & ··· & l+1 \end{matrix} ) ∈ 𝔖_{n}. Then we have the

relation

∑_{l=1}^{n−1}(−1)^{l−1}sh_{l} T_{l} = T_{0} + (−1)^{n},

or, since sh_{0} = sh_{n} =id, simply ∑_{l=0}^{n}(−1)^{l}sh_{l} T_{l} = 0.

Proof. For 1 ≤ l ≤ n − 1, we have

sh_{l} T_{l} = ∑_{\begin{matrix}σ∈𝔖_{n} \\ σ(1)<···<σ(l) \\ σ(l+1)>···>σ(n) \end{matrix}} σ = R_{l} + R_{l+1},

where

R_{i} = ∑_{\begin{matrix}σ∈𝔖_{n} \\ σ(1)<···<σ(i)>···>σ(n) \end{matrix}}σ ∈ ℤ[𝔖_{n}]  (i = 1, …, n).

Hence

∑_{l=1}^{n−1}(−1)^{l−1}sh_{l} T_{l} = R_{1} + (−1)^{n}R_{n} = T_{0} + (−1)^{n}. □

By the lemma, we have

f|T_{0} = f|T_{0}^{♯} = (−1)^{n−1}f for f ∈ DSh_{n}(d).  (8.6)

Let T′ = ( \begin{matrix}1 & ··· & n−1 & n & n+1 \\ n−1 & ··· & 1 & n+1 & n \end{matrix} ) ∈ 𝔖_{n+1}. It is straightforward to check the relations

T_{0}T′ = C_{n+1} (in 𝔖_{n+1}),  φ(f|P)|T′ = εφ(f|T_{0}P).  (8.7)

Now suppose f ∈ DSh_{n}(d). We only need to check that the second condition in the definition of ShC_{n}(d) is satisfied. With (8.6) and (8.7), we have

φ(f^{♯})|C_{n+1} = φ(f^{♯})|T_{0}T′ = φ(f|P)|T_{0}T′ = φ(f|P T_{0})|T′
= φ(f|T_{0}^{♯}P)T′ = εφ(f|T_{0}^{♯}T_{0}P) = εφ(f^{♯}).

This completes the proof of Proposition 17 and hence of Theorem 7. The corollary then follows immediately by virtue of the corollary to Theorem 6. □

One can sometimes get upper and lower bounds for DSh_{n}(d) using representation theory of finite groups. In particular, we can prove the following proposition and also some results for general n. We hope to discuss this more fully in [10].

Proposition 18. Assume d is even. Then we have

(i) dim_{ℚ} DSh_{2}(d) = [\frac{d}{6}] .

(ii) [\frac{d^{2} − 1}{48}] ≤ dim_{ℚ} DSh_{3}(d) ≤ [\frac{(d + 3)^{2}}{24} ] .

Appendix: The conjectural value of D_{k,n}.

In this appendix we describe the conjectural formula for D_{k,n} due to Broadhurst and Kreimer [1].

Let d_{k,n} = dim(Z_{k}^{(n)}/Z_{k}^{(n−1)}) be the “number of ℚ-linearly independent MZV’s of weight k and depth exactly n.” Denote by D(x, y) = ∑_{k,n} d_{k,n} x^{k}y^{n} the generating function of the numbers d_{k,n} and by D^{0}(x, y) the corresponding generating function for the “primitive part” Z^{0}, where Z = Z^{0} ⊗_{ℚ} ℚ[π^{2}]. Then

D(x, y) = (1 + \frac{x^{2}}{1 − x^{2}} y) D^{0}(x, y)

and the numbers D_{k,n} are given in terms of D^{0}(x, y) by the product expansion formula

∏_{k, n}(\frac{1}{1 − x^{k}y^{n}} )^{D_{k,n}} = \frac{1}{1 − x^{2}y} D^{0}(x, y)

or more explicitly by the formulas

D_{k,n} = ∑_{d|(k,n)} \frac{μ(d)}{d} · coefficient of x^{k/d}y^{n/d} in log D^{0}(x, y),

where μ(d) denotes the Möbius function. Knowing the numbers D_{k,n} is therefore equivalent to knowing the function D(x, y) or D^{0}(x, y). The Broadhurst-Kreimer conjecture says that the power series D^{0}(x, y) is given by

D^{0}(x, y) = \frac{1}{1 − O y + S y^{2} − S y^{4}} ,

where

O = ∑_{k>1, k odd} x^{k} = \frac{x^{3}}{1 − x^{2}}

corresponds to the odd zeta values ζ(3), ζ(5), … and

S = ∑_{k>0} dim S_{k}(SL_{2}(ℤ)) x^{k} = \frac{x^{12}}{(1 − x^{4})(1 − x^{6})}

is the generating series of the dimension of the graded vector space of cusp forms on the full modular group. The conjectural value for the full power series D(x, y) is then given by the rational function

D(x, y) = \frac{1 + E y}{1 − O y + S y^{2} − S y^{4}}

in x and y, where

E = ∑_{k>0, k even} x^{k} = \frac{x^{2}}{1 − x^{2}}

now counts the even zeta values ζ(2), ζ(4), … In particular, for the full number D_{k} = ∑_{n} D_{k,n} of weight k generators of the MZV algebra, this would imply

∏_{k≥2}(\frac{1}{1 − x^{k}} )^{D_{k}} = D(x, 1) = \frac{1}{1 − x^{2} − x^{3}} ,

in accordance with third-named author’s conjecture on the value of dim Z_{k}. This conjecture is still open, but it has been shown by Terasoma [16] and independently by Goncharov [5] that the coefficient of x^{k} in (1 − x^{2} − x^{3})^{−1} is an upper bound for dim Z_{k}.

The Broadhurst-Kreimer conjecture implies that D_{k,n} = 0 unless k ≥ 3n and k is congruent to n modulo 2, and gives the following table for n ≤ 6 and k −3n ≤ 14:

\begin{matrix}n⧹k − 3n & 0 & 2 & 4 & 6 & 8 & 10 & 12 & 14 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 2 & 0 & 1 & 1 & 1 & 2 & 2 & 2 & 3 \\ 3 & 0 & 1 & 2 & 2 & 4 & 5 & 6 & 8 \\ 4 & 1 & 1 & 3 & 5 & 7 & 11 & 16 & 20 \\ 5 & 1 & 2 & 5 & 9 & 15 & 23 & 36 & 50 \\ 6 & 1 & 3 & 7 & 14 & 27 & 45 & 73 & 113 \end{matrix}

It also implies the formulas

D_{k,1} = 1 ,  D_{k,2} = [\frac{d}{6}] ,  D_{k,3} = [\frac{d^{2} − 1}{48}]

for 1 ≤ n ≤ 3 and k − n = d ≡ 0 (mod 2). Each of these three formulas is known to give an upper bound for the true dimension ([18] for n = 2 and Goncharov [4] for n = 3; see also Proposition 18 of §8 of this paper).

BIBLIOGRAPHY

1. D. J. Broadhurst and D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, Physics Lett. B 393 (1997), 403–412.

2. A. Connes and H. Moscovici, Modular Hecke algebras and their Hopf symmetry, Moscow Mathematical Journal 4 (2004), 67–109.

3. A. Connes and H. Moscovici, Rankin–Cohen brackets and the Hopf algebra of transverse geometry, Moscow Mathematical Journal 4 (2004), 111–130.

4. A. B. Goncharov, Multiple Z-values, Galois groups and geometry of modular varieties, Progress in Math. 201 (2001), 361–392.

5. A. B. Goncharov, Multiple polylogarithms and mixed Tate motives, preprint math.AG/0103059.

6. A. Granville, A decomposition of Riemann’s Zeta-Function, Analytic Number Theory, London Mathematical Society Lecture Note Series 247 (1997), 95–101.

7. M. Hoffman, Multiple harmonic series, Pacific J. Math. 152 (1992), 275–290.

8. M. Hoffman, The algebra of multiple harmonic series, J. of Algebra 194 (1997), 477–495.

9. M. Hoffman and Y. Ohno, Relations of multiple zeta values and their algebraic expression, J. of Algebra 262 (2003), 332-347.

10. K. Ihara, M. Kaneko and D. Zagier, The double shuffle vector space, planned.

11. H. N. Minh, M. Petitot and J. V. D. Hoeven, Shuffle algebra and polylogarithms, Discrete Math. 225 (2000), 217–230.

12. L. Boutet de Monvel, Remarques sur les séries logarithmiques divergentes, lecture at the workshop “Polylogarithmes et conjecture de Deligne-Ihara”, C.I.R.M. (Luminy) (April 2000).

13. Y. Ohno, A generalization of the duality and sum formulas on the multiple zeta values, J. of Number Th. 74 (1999), 39–43.

14. G. Racinet, Doubles mélanges des polylogarithmes multiples aux racines de l’unité, Publ. Math. Inst. Hautes Études Sci. 95 (2002), 185–231.

15. C. Reutenauer, Free Lie Algebras, Oxford Science Publications, 1993.

16. T. Terasoma, Mixed Tate motives and multiple zeta values, Invent. Math. 149 (2002), 339–369.

17. H. Tsumura, Combinatorial relations for Euler-Zagier sums, Acta Arith. 111.1 (2004), 27–42.

18. D. Zagier, Periods of modular forms, trace of Hecke operators, and multiple zeta values, RIMS Kokyuroku 843 (1993), 162–170.

19. D. Zagier, Multiple zeta values, Unpublished manuscript, Bonn (1995).