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**Derivation and double shuffle relations for multiple zeta values.**
*(English)*
Zbl 1186.11053

From the text: Derivation and extended double shuffle (EDS) relations for multiple zeta values (MZVs) are proved. Related algebraic structures of MZVs, as well as a ‘linearized’ version of EDS relations are also studied.

In recent years, there has been a considerable amount of interest in certain real numbers called multiple zeta values (MZVs). 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 \(\mathbb Q\) among the MZVs, but their exact structure remains quite mysterious.

The MZVs can be given both as sums \[ \zeta(\mathbf k)=\zeta(k_1,k_2,\dots,k_n)=\sum_{m_1>m_2>\dots>m_n>0}\frac 1{m_1^{k_1}m_2^{k_2}\cdots m_n^{k_n}}, \] or as integrals \[ \zeta(k_1,k_2,\dots,k_n)={\int\cdots\int}_{1>t_1>t_2>\cdots>t_k>0} \omega_1(t_1) \omega_2(t_2)\cdots\omega_k(t_k). \] From each of these representations one finds that the product of two MZVs is a \(\mathbb Z\)-linear combination of MZVs, described by a so-called shuffle product, but the two expressions obtained are different. Their equality gives a large collection of relations among MZVs which we call the double shuffle relations. These are not sufficient to imply all relations among MZVs, 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 \(\sum 1/n\)), and that these extended double shuffle (EDS) relations apparently suffice to describe the ring of MZVs 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 MZVs 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 MZVs and use them to give new, and in several cases conjecturally complete, sets of relations among MZVs. 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 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.

In recent years, there has been a considerable amount of interest in certain real numbers called multiple zeta values (MZVs). 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 \(\mathbb Q\) among the MZVs, but their exact structure remains quite mysterious.

The MZVs can be given both as sums \[ \zeta(\mathbf k)=\zeta(k_1,k_2,\dots,k_n)=\sum_{m_1>m_2>\dots>m_n>0}\frac 1{m_1^{k_1}m_2^{k_2}\cdots m_n^{k_n}}, \] or as integrals \[ \zeta(k_1,k_2,\dots,k_n)={\int\cdots\int}_{1>t_1>t_2>\cdots>t_k>0} \omega_1(t_1) \omega_2(t_2)\cdots\omega_k(t_k). \] From each of these representations one finds that the product of two MZVs is a \(\mathbb Z\)-linear combination of MZVs, described by a so-called shuffle product, but the two expressions obtained are different. Their equality gives a large collection of relations among MZVs which we call the double shuffle relations. These are not sufficient to imply all relations among MZVs, 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 \(\sum 1/n\)), and that these extended double shuffle (EDS) relations apparently suffice to describe the ring of MZVs 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 MZVs 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 MZVs and use them to give new, and in several cases conjecturally complete, sets of relations among MZVs. 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 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.

### MSC:

11M32 | Multiple Dirichlet series and zeta functions and multizeta values |