数理解析研究所講究録 1397 巻 2004 年 1-9
Remark on divergent multizeta
series
L. Boutet de Monvel
There are several manners of regularizing divergent zeta series. It is useful to
compare these, since they are used to describe relations among convergent zeta
series. Here we compare the two main ones, related to the two shuffle products.
The remark in this paper (\S 1) was described in a lecture at the CIRM, april 2000,
and used e.g. by G. Racinet [13]. \S 3 gives a more formal version; in Q2 I have briefly
recalled the basic notions about polylogarithms and zeta-numbers used in \S 3.
Logarithmically divergent series
1
1.1
series
denote the set of formal series $f= \sum f_{n}z^{n}$ with complex coefficients
Let
has an asymptotic expansion of the form
in which the coefficient
$\mathcal{K}\subset \mathbb{C}[[z]]$
$f_{n}$
$\infty$
.
(1)
$p_{k}({\rm
Log}n)n^{-k}$
n)n^{-k}$
$f_{n}\sim$ $1p_{k}({\rm
Log}
$f_{n} \sim\sum_{1}$
1
a polynomial. Polylogarithm series $f=Li_{a}$
is apolynomial.
for $narrow\infty$ , where for each ,
belong to and zeta values are special cases of convergent series $f(1)$ , cf. below.
has
$S_{n}(f)$ $=S_{n}=$
\sum_{0}^{n}f_{k}$ . Condition (1) is equivalent to the fact that
We set $S_{n}(f)=S_{n}=
aasimilar
similar asymptotic expansion:
$k$
$p_{k}$
$\mathcal{K}$
$S_{n}$
$\sum n0f_{k}$
(2)
$S_{n} \sim\sum_{0}^{\infty}P_{k}({\rm Log} n)n^{-k}$
$S_{n} \sim\sum_{0}P_{k}({\rm Log} n)n^{-k}$
.
This implies that the series is convergent for $|z|<1$ and its sum, still denoted
admits for $zarrow 1-0$ an asymptotic expansion:
(3)
$f(z) \sim\sum_{k=0}^{\infty}(1-z)^{k}Q_{k}({\rm Log}\frac{1}{1-z})$
$f(z) \sim\sum_{k=0}(1-z)^{k}Q_{k}({\rm Log})\overline{1-z}[perp]$
are also polynomials (the converse is not true).
We will denote the leading polynomials
where the
(4)
$Q_{k}$
$S_{f}=P_{0}$
The following result is immediate:
,
$M_{f}=Q_{0}$
$f(z)$
2
Lemma 1 The following statements are equivalent:
1) The numerical series
is convergent.
2) The numerical series
is absoltrtely convergent.
$S_{f}=P_{0}$
3) The polynomial
is constant.
4) The polynomial $M_{f}=Q_{0}$ is constant.
and
are then equal constants: $S_{f}=M_{f}= \sum f_{n}$ .
$\sum f_{n}$
$\sum f_{n}$
$M_{f}$
$s_{f}$
1.2
Products.
Let
be a field. On the set of formal series
commutative products defined by:
$k\in \mathbb{C}$
(5)
$f \cross g=\sum f_{p}g_{q}z^{p+q}’$.
$k[[T]]$
we have two associative and
$f*g= \sum f_{p}g_{q}z^{\sup(p,q)}$
the first is the usual product (corresponding to the shuffle product below), the
is the -th partial sum,
second is the unique product such that if $S_{n}(f)$
we have $S_{n}(f*g)=S_{n}(f)S_{n}(g)$ for all (corresponding to the shuflle product
below).
$\mathrm{m}$
$=$
$\sum n1f_{k}$
$n$
$n$
$*$
It is immediate that is a sub-algebra for both products, i.e. the coefficients of
and $f*g$ have a logarithmic asymptotic expansion as above if and do.
,
The map $farrow M_{f}(X)$ , resp. $farrow S_{f}(X)$ a homomorphism of the algebra
, to the polynomial algebra
(surjective if
resp.
).
$\mathcal{K}$
$f\cross g$
$f$
$g$
$(\mathcal{K}, \cross)$
$(\mathcal{K}, *)$
$k=\mathbb{C}$
$\mathbb{C}[T]$
Corollary 1 Let
be the subset of convergent series with vanishing sum: I is
an ideal of for both multiplication laws , it. It contains all elements of the $fom$
$I\subset \mathcal{K}$
$\mathcal{K}$
$f*g-f\cross g$
$\cross$
for ,
$f$
$g\mathrm{E}$
$\mathcal{K}_{f}f$
convergent.
Indeed
is multiplicative for (resp.
for ) and $M_{f}=0\Leftrightarrow S_{f}=0.$ The
second assertion follows from the fact that, if is convergent, setting $a=f(1)=$
, we have $f*g-f\cross g=(f-a)*$
$(f-a)$
since $f-a$ is convergent
with vanishing sum, the same is true for both terms:
, $S_{(f-a)*g}=0.$
A typical regularized zeta value is obtained by applying the character
(resp. or
) to a divergent polygarithm series, for some number .
$M_{f}$
$S_{f}$
$\cross$
$*$
$f$
$\sum f_{n}$
$g$
–
$\cross g;$
$M_{(f-a)\mathrm{x}g}=0$
$f\mapsto M_{f}(\theta)$
$f\mapsto S_{f}(\theta)$
1.3
$\theta$
Relation between
$M_{f}$
and
$Q_{f}$
Corollary 1 shows that the polynomials
et
determine each other for $f\in$ C. We
will explicit the formula which links them. We use the generating series $F=F(T, z)$ ,
with is a formal parameter:
$M_{[}$
$S_{f}$
$T$
(6)
so that
$F= \sum\frac{T^{n}}{n!}(\log\frac{1}{1-z})^{n}=(1-z)^{-T}$
$M_{F}(X)=e^{TX}$
. Then for
$F$
we get
$\sum S_{n}zn=\frac{F}{1-z}=(1-z)^{-T-1}=\sum(1+T)$
.
..
$(1+ \frac{T}{n})z^{n}$
i.e.
$S_{n}=(1+T)$
$(1+ \frac{T}{n})\sim\frac{e^{T1\mathrm{o}\mathrm{g}n}}{T!}$
\ldots
Taking limits (the limit holds in any reasonable sense, in particular the coefficients
converge) we get
of the
$T^{k}$
$S_{F}(X)= \frac{1}{T!}e^{TX}=\frac{1}{\frac{d}{dX}!}e^{TX}$
Theorem 1 For any
f
$\in \mathcal{K}$
we have:
$S_{f}(X)= \frac{1}{\frac{d}{dX}!}Qf(X)$
If we explicit the Taylor-series expansion of
$S_{f}(X)=\exp$
with here the convention
2
[
1
this can be rewritten
$\sum_{1}^{\infty}\frac{(-1)}{j}\dot{\mathrm{y}}-1\zeta(j)(\frac{d}{dX})^{j}]$
$\zeta(1)=\gamma$
$Q_{f}(X)$
, the Euler constant.
Polylogarithms.
In this section we recall basic facts about polylogarithms. Let
, or ).
acteristic 0 (
$k=\mathbb{Q}$
$\mathbb{R}$
$k$
be a field of char-
$\mathbb{C}$
Polylogarithm functions.
2.1
,
Let
denotes the free algebra with two generators
if there is a risk of confusion). It has a canonical basis $W$ consisting of words
(monomials): a word of degree $N$ is $w=X_{i_{0}}\ldots$ $XiN$ , $i_{k}=0$ or 1; $w=1$ if $N=0$
(empty word).
$\circ$
$X_{0}$
$\mathcal{L}=k\langle X_{0}, X_{1}\rangle$
$X_{1}(\mathcal{L}_{k}$
denotes the completion of : its elements are formal series $f=E$ w. We
identify with the graded dual of (it might be better to use another notation;
the shuffle products below live on the
the free (concatenation) product lives on
dual.
Duality endows with a structure of left or right -module, via inner prod. Since 1 is a generator of the left or right ffee module
ucts, e.g.
such that
, it is a cogenerator in the dual, i.e. if $a\in L:,$ $a\neq 0$ there exists
(or
).
$\circ\hat{\mathcal{L}}$
$f_{w}$
$\mathcal{L}$
$\hat{\mathcal{L}}$
$\mathcal{L}$
$\hat{\mathcal{L}},\cdot$
$\mathcal{L}^{0}$
$\mathcal{L}$
$\circ$
$\langle a\lrcorner g, g\rangle=\langle a, gf\rangle$
$f\in\hat{\mathcal{L}}$
$\mathcal{L}$
$a_{\lrcorner}f=1$
$f_{\llcorner}a=1$
To each
we associate linearly a polylogarithm function $Li_{a}(z)$ so that
the -valued the formal series $L=L(z)= \sum_{w\in W}Li_{w}(z)w$ satisfies the differential
equation
$\circ$
$a\in \mathcal{L}$
$\mathcal{L}$
(7)
$L\{z$
)
$=( \frac{dz}{z}X_{0}+\frac{dz}{1-z}X_{1})$
L,
$L^{reg}(0)=1$
4
are holomorphic functions on ] , 1 [, and extend as ramified
The coefficients
complex
plane. They have logarithmic singularities at 0, 1,
functions to the whole
if is a local parameter ( $u=z$ , $1-z$ or ) and
. . of the form
is holomorphic in a neighborhood of the singular point. The regularized value at
$z=0$ is defined by $\phi^{reg}(0)=$
.
$0$
$Li_{w}$
$\infty$
$\phi=\sum \mathit{7}$
$\mathrm{i}$
$\mathrm{e}$
$\phi_{k}(u)(\log u)^{k}$
$u$
$\frac{1}{z}$
$\phi_{k}$
$\mathrm{x}_{0}(0)$
If
$w=X_{i_{1}}$
...
$X_{i_{N}}$
, we have
,
$P_{0}f(z)=( \int_{0}^{z}f(\mathrm{t})\frac{dt}{t})^{\mathrm{r}eg}$
(8)
$Li_{w}(x)=/\cdots$
..
$Li_{w}=P_{i_{1}}$
$P_{1}f(z)=t_{0^{z}}t(t) \frac{dt}{1-t}$
$\int_{x>t_{1}>}$
$\omega_{i}1$
$(t_{1})\ldots$
.
are the operators
where ,
In particular for $0<x<1$
$P_{i_{N}}(1)$
$\omega_{i_{N}}(t_{N})$
... $t,>0$
$P_{0}$
,
with
$P_{1}$
$\omega_{0}=\frac{dt}{t}$
,
$\omega_{1}=\frac{dt}{1-t}$
; we choose as base point
Let denote the fundamental group of
the path followed by (the opposite of
1+0, and in what follows we denote
the usual order). is a ffee group generated by ) ) , with , a small loop around
0, and
a small loop around 1 preceded and followed by the real path $[+0,1-0]$ .
$L(z)$ is a ramified function;
should be thought of as a homotopy class of paths
avoiding 0, 1, , with origin 1+0.
, we get, for
Since the polar part of $dLL^{-1}$ at $z=0,$ resp. 1, is , resp.
$\mathbb{C}\backslash \{0;1\}$
$\Gamma$
$\circ$
$c_{1}c_{0}$
$c_{1}$
$c_{0}$
$\Gamma$
$\mathrm{C}\mathrm{h}$
$\gamma_{0}$
$1$
$\mathrm{Y}1$
$z$
$\infty$
$- \frac{X_{1}}{z-1}$
$\frac{X}{z}\alpha$
$\gamma\in\Gamma$
$L(z\gamma=L(z)\phi(\gamma)$
where
$\phi:\Gammaarrow$
$\mathrm{j}^{\mathrm{X}}$
is the group homomorphism such that
$\phi(’ 0)=e^{2i\pi X_{0}}$
0
,
$(71)=L^{\mathrm{r}eg}(1-0)^{-1}e^{-2i\pi X_{1}}L^{reg}(1-0)$
.
Equivalently
(9)
for any
$Li_{a}(z\gamma)=Li_{w\lrcorner\phi\gamma}(z)$
$\gamma\in\Gamma$
.
is dense in , and the algebras genIt follows in particular that the image
acting by right interior product, are
erated by the fundamental group, or by Xo,
$a\in
L,$
, LiaLf is a finite linear combination of
the same. In particular for any
branches of $Lia$ .
$\phi(\mathbb{Z}(\Gamma)$
$\mathcal{L}$
$X_{1}$
$f\in\hat{\mathcal{L}}$
is injective, i.e. the $LiW$ ) $w\in W$ are linearly indepenThe map $a\in$
); this is a simple special
dent over (in fact over the field of rational functions
$a\neq
0$
,
such that
case of the Riemann-Hilbert theorem: if
there exists
$=1$
,
of
and
does not
so LiaLf
is a linear combination branches of
vanish]. The range of this map is exactly the set of functions which are ramified at
0, 1, , of finite type and unipotent monodromy, and any branch of which at 0, 1 or
oo has a logarithmic singularity as above.
$\circ$
$\mathcal{L}_{\mathrm{C}}$
$\mapsto Li_{a}$
$\mathbb{C}$
$\mathbb{C}(z)$
$a\in \mathcal{L}_{\mathbb{C}}$
$a_{\lrcorner}f=1,$
$f\in \mathcal{L}_{\mathbb{C}}$
$Li_{a}$
$Li_{a}$
$\infty$
2.2
Polyzeta numbers
If
origin iff
$\circ$
$a\in \mathcal{L}$
$a$
, the function
is orthogonal to
$Li_{a}$
(or rather its first branch) is holomorphic at the
. The Taylor series of
then belongs
$\mathcal{L}X_{0}(a\lrcorner X_{0}=0)$
$Li_{a}$
5
orthogonal
the sub-algebra $A$
to the algebra of Ql. We denote $A$
.
of
$A$ is a free algebra with generators the $y_{j}=X_{0}^{j-1}X_{1}$ .
It has a basis consisting
of
. . , indexed by indexed by sequences $s=$
of monomials
).
(the concatenation product, in this order,
integers
$=k+\mathcal{L}X_{1)}$
$\subset \mathcal{L}$
$\mathcal{K}$
$\mathcal{L}X_{0}$
$j_{S}=y_{s_{1}}$
$(s_{1}, \ldots, s_{r})$
$y_{s_{r}}$
$y\emptyset=1$
$\geq 1$
The polylogarithm series are the Taylor series at $z=0$ of the
have
(10)
$Li_{y_{s}}(z)=$
$\sum$
$n_{1}>$
...n
$Li_{a}$
, $a\in
A:$
we
$\frac{z^{n_{1}}}{n_{1}^{s_{1}}..n_{r}^{s_{\mathrm{r}}}}$
$s\geq 1$
).
the sub-algebra $k+$ XOCXX (orthogonal to
We denote
$s_{r}(r>0)$ we have
iff $s_{1}>1.$ Then the polygarithm series
If $s=s_{1}$ ,
$Li_{s}(1)$ is obviously convergent (
(
.. has a limit for $zarrow 1-0$ ). The
corresponding zeta number is
$A^{0}\subset A$
$\circ$
$X_{0}\mathcal{L}+\mathcal{L}X_{1}$
$y_{s}\in A^{0}$
$\ldots$
$Li_{s}$
(11)
$z)= \int_{0}^{z}\frac{dx}{x}$
$\zeta_{s}=Li_{s}(1)=$
$\sum$
$n_{1}>$
...n
,
$\frac{1}{n_{1}^{s_{1}}..n_{r}^{s}}$
$s\geq 1$
Products.
2.3
Let us recall that two shuffle product, and , are defined on resp. $A$ .
is the unique product compatible with the usual product of holomorphic
.
functions, i.e.
(&L satisfies the differential equation
$\mathrm{m}$
$C$
$*$
$\circ \mathrm{m}$
$Li_{a\mathrm{I}\mathrm{I}\mathrm{I}b}=Li_{a}Li_{b}$
$L$
$d(L \otimes L)=[\frac{dz}{z}(X_{0}\otimes 1+1\otimes X_{0})+\frac{dz}{1-z}(X_{1}\otimes 1+1\otimes X_{1})](L\otimes L)$
,
$(L\otimes L)^{reg}(0)=1$
:
for $i=0,1$ (the completion takes
so the dual coproduct is the algebra homomorphism
$\Delta_{\mathrm{m}}(X_{i})=X_{i}\otimes 1+1\otimes$
$X_{i}$
).
bigger than
For the product law,
the monodromy elements
$\Delta_{\mathrm{m}}$
$\mathcal{L}arrow \mathcal{L}\otimes C$
$\hat{\mathcal{L}}$
to
$\mathcal{L}(\otimes \mathcal{L}$
such that
which is
-
$\hat{\mathcal{L}}\otimes\hat{\mathcal{L}}$
$\mathrm{m}$
$L$
is of group type, i.e.
$\phi(\gamma)$
,
$\gamma\in\Gamma$
.
is only defined on $A$ . The dual coproduct
ment that the generating series
0*
(12)
$\Delta_{*}$
$\mathrm{i}.\mathrm{e}$
. for the generators we have
$\Delta_{*}(y_{j})=\sum_{p+q=j}y_{p}\otimes y_{q}$
(& $L(z)$ ;
so are
is characterized by the require-
$u(T)=1+ \sum_{1}^{\infty}y_{j}T^{j}$
is of group type,
(13)
$\Delta_{\mathrm{m}}L(z)=L(z)$
(with the convention
$y_{0}=1$
)
$\epsilon$
To the word
(formal series)
one associates the elementary “quasi-symmetric” function
$y_{s}\in A$
$P_{s}(\mathrm{t}_{1}, .
.
, , t_{n}, \ldots)$
(14)
:
$P_{s}= \sum_{n_{1}>\cdots>n_{r}}t_{n_{1}}^{s_{1}}$
..
$t_{n_{r}}^{s_{r}}$
of quasiform a basis of the algebra
Let us recall that the
symmetric functions, which is a sub-algebra of the algebra all formal series in , . . .
Since $u(T)$ of group type in $4[[T1$ , so is the formal product
$\prime p$
$P_{B}$
$\subset k[[t_{1}, \ldots t_{n}, \ldots]]$
$t_{1}$
$($
U
(15)
$=\ldots$ $u(t_{n}) \ldots u(t_{1})=\sum P_{s}y_{s}$
(infinite concatenation product, in the indicated order) . So the linear map : $4arrow$
.
such that $\Psi(y_{s})=P_{s}$ is an algebra isomorphism: $(A, *)$
...
is obFor $f=Li_{y_{s}}(z)$ , (10), (11) show that series
$S_{N}(f)$
$U(f)$
, and the -th sum
is obtained by
in
tained by substituting
$=S_{N}(f)S_{N}(g)$
$S_{N}(f*g)$
$k\leq
N$
.
$t_{k}=0$
otherwise;
,
we
have
so
if
substituting
$\Psi$
$arrow \mathcal{P}$
$7$
$\zeta_{f}=\sum_{n_{1}>..n_{r}}n_{1}^{-s_{1}}$
$n_{r}^{-s_{r}}$
$\mathrm{N}$
$t_{k}= \frac{1}{k}$
$t_{k}= \frac{1}{k}$
Relations between polylogarithm series.
3
3.1
Main identity
-
second version.
for some integer and
Any monomial $a\in A$ is uniquely factorized as $y=y_{1}^{k}$
$b\in A^{0}$
. An easy induction then shows that for either law $A$ is a
some monomial
,
. The successive powers ,
are
polynomial algebra:
not equal, and the relation between them is given by the next result:
$k$
$b$
$A_{\mathrm{m}}=$
$4\mathrm{L}[y_{1}]$
$A_{*}=A_{*}^{0}[y_{1}]$
Theorem 2 We have the following identity
expm
(16)
where
expm,
$y_{1}T=$
exp’
$y_{1}^{*k}$
of formal series with coefficients
$r\cdot es$
This follows immediately from the equality
in
$4_{\mathrm{m}}[[T]]$
$y_{1}^{\mathrm{m}k}=k!y_{1\ldots 1}=k!X_{1}^{k}$
;
$U( \exp^{\mathrm{m}}y_{1}T))=\prod(1+t_{j}T)=\exp\sum_{k\geq 1,j}\frac{(-1)}{k}k-1t$
Since
$\sum t_{j}^{k}=U(y_{k})$
, resp.
\sum_{n_{1}>\cdots>n_{k}}t_{n_{1}}\ldots t_{n_{k}}=\sum_{n_{1}\neq\cdots\neq n_{k}}t_{n_{1}}\ldots t_{n_{k}}$
and
this implies (16).
in
$\sum_{1}^{\infty}\frac{(-1)}{k}yk-1kTk$
resp. exp* denote the exponential ser
$U(y_{1}^{\mathrm{m}k})=k!
$y_{1}^{\mathrm{m}k}$
$T^{k}$
$A_{*}[[T]]$
.
so we have
$A$
:
7
One can reformulate this identity as follows: let $D$ be the “left interior” product
operator by
in $A$ . This is a derivation, for both laws and since $X_{1}=y_{1}$ is
. . . $y_{s_{n}}=0$ if $>1$ ,
primitive for both coproducts Am, . We have
..
if $s_{1}=1$ and the fixed sub-algebra of $D$ is the subring $A^{0}=k+X_{0}\mathcal{L}X_{1}$ above,
corresponding to the “obviously” convergent series. Theorem 1 can be expressed as
follows:
$X_{1}$
$*$
$\Delta$
Corollary 2 Let
$\Phi(D)$
(17)
be the (infinite order)
$\Phi(D)=\mathrm{e}\mathrm{x}\mathrm{p}$
Then for
$f_{k}\in A^{0}$
$Dy_{s_{1}}$
’
$\mathrm{m}$
$s_{1}$
differential
$( \sum_{k\geq 2}\frac{(-1)}{k}y_{k}D^{k})k-1$
operator on
$y_{s_{r}}$
$j_{s_{2}}$
$(A, *)$
:
.
we have
(18)
$\sum \mathrm{j}_{k}$
”
$y_{1}^{\mathrm{m}k}=$
f $(D) \sum f_{k}*y_{1}^{*k}$
Relations
3.2
from
Regularized divergent 4 values are obtained by extending the character
t $A$ for one of the laws or ; for this one only needs to define the zeta-value
, so the extended character is
, resp.
with the notation of
, the Euler constant.
\S 1; the most natural choices are
The two extensions are not equal. However both extensions coincide on on the
elements $a\in A$ such that the corresponding series $f=Li_{a}= \sum f_{n}z^{n}$ is convergent
for $z=1$ ( . $S_{f}=M_{f}=$ constant, as in 51); for such an element the zeta number
$\zeta_{a}=Li_{a}(1)$ is still unambiguously defined. These elements form a sub-algebra, for
both laws and , which contains $A^{0}=k+X_{0}\mathcal{L}X_{1}$ .
By \S 1 the null-ideal is the same for both extended characters: it is the set
is not defined
of all $a\in A$ such that $Li_{a}(1)$ is convergent, and the sum is 0.
over (the coefficients of the extended character are not rational, and is not of
codimention 1 if is smaller than ). However certainly contains all elements of
$12-j1\mathrm{m}Tl2$
!l3-y12 hence
the form $a*b-amb$ with $b\in A^{0}$ . E.g. we have
(Euler).
is stable
Let
be the -vector space generated by the $a*b-a\mathrm{m}b(b\in A^{0})$ .
by $D$ ( $D(a*b-a\mathrm{m}b)=Da$ $*b-$ Damb). In fact it follows easily from theorem 2 that
are equal. Then $I=A*V=A\mathrm{m}V$ is stable
the two or ideals $A*V$ and
of $f= \sum f_{k}*y_{1}^{*k}$
, i.e. by the coefficients
by so it is generated by
) for all $f=a*b$ -cunb as above.
(or $f=E$
). The
The ideal I thus introduced is rational (everything is well defined if
, i.e.
of
$I=J,$
ideal
null
the
we have
now standard conjecture is that for
the only algebraic relations between zeta numbers are those that can be deduced
formally from the comparison of the two product laws. This seems rather out of
reach right now and present work deals with proving that all known relations can
be reduced to those above, and making explicit the ring structure of the quotient
. these relations.
ring 4
$a\mapsto\zeta_{a}$
$A^{0}$
$*$
$\mathrm{o}$
$\mathrm{m}$
$f\mapsto S_{f}(\theta)$
$\zeta_{y_{1}}=\theta$
$M_{f}(\theta)$
$\mathit{4}\mathit{1}=0^{\cdot}\mathrm{o}\mathrm{r}\theta=\gamma$
$\mathrm{i}.\mathrm{e}$
$*$
$\mathrm{m}$
$J$
$J$
$J$
$\mathbb{Q}$
$k$
$J$
$\mathbb{C}$
$y_{h}$
$=$
$*$
$\zeta_{3}-\zeta_{12}=0$
$V$
$V$
$\mathrm{k}$
$*$
$\mathrm{m}$
$A\mathrm{m}V$
$I\cap A^{0}$
$D$
$f_{j}\in A^{0}$
$f_{k}\mathrm{m}y_{1}^{\mathrm{m}k}$
$k=\mathbb{Q}$
$k=\mathbb{Q}$
$\mathrm{m}\mathrm{o}\mathrm{d}$
$\zeta$
8
References
[1] Arakawa, T.; Kaneko, M. Multiple zeta values, poly-Bernouilli numbers and
related zeta functions. Nagoya Math. J. 153 (1999) 189-209.
[2] Bigotte, M.; Jacob, G.; Oussous, N. E.; Petitot, M. Generating power series
of coloured polylogarithm functions and Drinfeld associator. Computer mathematics (Chiang Mai, 2000), 39-48, Lecture Notes Ser. Comput., 8, World Sci.
Publishing, River Edge, NJ, 2000.
[3] Ecalle, J. La libre g\’en\’eration des multizeta et leur d\’ecomposition canonic0explicite en irr\’eductibles. S\’eminaire IHF, d\’ecembre 1999.
et l’arithm\’etique des multizetas et sommes d’Euler. col[4] Ecalle, J.
loque “Polylogarithmes, multizetas et la conjecture de Deligne-Ihar\"a, Luminy,
avril 2000, \‘a paraitre.
$\mathrm{A}\mathrm{R}\mathrm{I}/\mathrm{G}\mathrm{A}\mathrm{R}\mathrm{I}$
[5] Goncharov, A.B. Multiple (-values, Galois groups, and geometry of modular
varieties. European Congress of Mathematics, Vol. I (Barcelona, 2000), 361)
392, Progr. Math., 201, Birkhuser, Basel, 2001. (Reviewer: Jan
$\mathrm{N}\mathrm{e}\mathrm{k}\mathrm{o}\mathrm{v}\check{\mathrm{r}}$
[6] Goncharov, A.B. Multiple polylogarithms, cyclotomy and modular complexes.
. Letters 5 (1998) 497-5.l 6.
Meth.
${\rm Res}$
[7] Hoang Ngoc Minh; Jacob, G.; Petitot, M.; Oussous, N.E. De l’algbre des \langle de
Riemann multivaries l’algbre des \langle de Hurwitz multivaries. (French) [Rom the
multivariate Riemann zeta algebra to the multivariable Hurwitz zeta algebra]
Sm. Lothar. Combin. 44 (2000), Art. B44i, 21 pp. (electronic).
[8] Hoang Ngoc Minh, Petitot M. Lyndon words, polylogarithms, and the Riemann
function, Discrete Math. 217 (2000) 273-292. MR1766271 (2001i:33002)
$\zeta$
[9] Hoffman, M. The algebra of multiple harmonic series. J.
of Al. 1997.
[10] Hoffman, M.; Ohno, Y. Relations of multiple zeta values and their algebraic
expression. J. Algebra 262 (2003), no. 2, 332-347. (Reviewer: David Bradley)
[11] Hoffman, M. Quasi-shuffle products. J. Algebraic Combin. 11 (2000), no. 1,
49-68.
[12] Ohno, Y.; Zagier, D. Multiple zeta values of fixed weight, depth, and height.
Indag. Math. (N.S.) 12 (2001), no. 4, 483-487. (Reviewer: L. Skula) llM41
[13] Racinet, G. Series generatrices non-commutatives de polyzetas et associateurs
de Drinfeld. These, universite Jules Verne (Picardie), Amiens, d\’ecembre 2000.
[14] Racinet, G. Doubles mlanges des polylogarithmes multiples aux racines de
Funit. Publ. Math. Inst. Hautes tudes Sci. No. 95 (2002), 185-231.
$\epsilon$
[15] Reutenauer, C. Free Lie Algebras. London Math. Soc. Monographs,
7, Oxford Science PubL, 1993.
netn
series
[16] Waldschmidt, M. Valeurs zeta multiples. Une introduction. (Talence, 1999). J.
Thor. Nombres Bordeaux 12 (2000), no. 2, 581-595.
[17] Zagier, D. Values of zeta functions and their applications, in “First European
Congress of Math. 2497-512. Birkhduser, 1994.
L. Boutet de Monvel
Universite’ Pierre et Marie Curie
Analyse Algebrique, Institut de Math\’ematiques (UIVIR 7586)
case 82, 4 place Jussieu, 75252 Paris cedex 05, FRANCE
email: boutet@math.jussieu.fr