Polylogarithms and Multiple Zeta Values
Michel Petitot
University of Lille I
Algorithms Seminar
April 19, 1999
[summary by Bruno Salvy]
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The polylogarithm is defined by the series
The convergence of this series at 1 is granted when s_{1}>1, and the
limit is denoted z(s_{1},...,s_{k}) and is called a multiple
zeta value since it extends the
classical Riemann zeta function. The number ås_{i} is
called the weight of the polylogarithm or multiple zeta.
Many polynomial identities relating multiple zeta values at integers
are known. For instance, reorganizing double sums yields the following
identity between multiple zetas of weight 4:

z(2,2)= 


= 

æ ç ç è 


ö ÷ ÷ ø 

 



= 

(z(2)^{2}z(4)).
(1) 
This could be simplified further using the wellknown values of
z at even integers.
This is a very active and diverse area. The reader is encouraged to
consult [1, 2] for surveys of
many beautiful results, generalizations and conjectures.
One of the most famous conjectures is the following.
Conjecture 1 [Zagier] The set of multiple zeta
values z(s_{1},...,s_{k}) with s_{i} positive integers, s_{1}³2
and s_{1}+...+s_{k}£ n generates a vector space over Q whose
dimension d_{n} obeys
d_{n+3}=d_{n+1}+d_{n}, d_{1}=0, d_{2}=d_{3}=1.
Note that since even the irrationality of z(5) is still
unproven, this conjecture is completely out of reach. Even a proof
that this sequence gives an upper bound is still to be found.
1 Shuffle and Stuffle
The manipulation leading to identity (1) is a
special case of a more general mechanism involving products of
multiple sums. By considering how indices in multiple sums can be
reorganized, it is natural to define the stuffle product
of two words
over N. (Stuffle
is a contraction of ``shuffle'' and ``stuff''.) Using lowercase symbols
to denote letters and capital symbols to denote words,
this is the formal sum defined recursively by
e* W=W*e=W,
aS* bT=a(S* bT)+b(aS* T)+(a+b)(S* T).
This definition is motivated by the following important stuffle relation:
A simple example is z(2)z(3)=z(2,3)+z(3,2)+z(5).
Another one is the identity (1) which is obtained
from (2)*(2)=2(2,2)+2(4).
In the same way as the stuffle product arises in the reorganization of
multiple sums, multiple integrals lead to considering the shuffle
product of words over the alphabet X={x_{0},x_{1}}. This is defined by
the same formula as the stuffle product except that
the last term in the sum is omitted.
A bijection between words over N^{*} and words of X^{*}x_{1} is provided by
the encoding
(s_{1},...,s_{k})« x 

x_{1}··· x 

x_{1}. 
This makes it possible to extend the shuffle product to these words.
For instance,

(2) W (2)® x_{0}x_{1} W
x_{0}x_{1}=2x_{0}x_{1}x_{0}x_{1}+4x_{0}x_{0}x_{1}x_{1}®2(2,2)+4(3,1), 


(2) W (3)® x_{0}x_{1} W
x_{0}x_{0}x_{1}=6x_{0}^{3}x_{1}^{2}+3x_{0}^{2}x_{1}x_{0}x_{1}+x_{0}x_{1}x_{0}^{2}x_{1}®6(4,1)+3(3,2)+(2,3). 

The following integral representation is then proved by induction

L 

(z)=log 

= 
ó õ 


,
L_{w}(z)= 
ì ï ï ï ï í ï ï ï ï î 

if w=x_{0}w', 

if w=x_{1}w'. 


(2) 
The recursive definition of the
shuffle now reads UV=òU'V+òUV', whence the
shuffle relation:
L_{A}(z)L_{B}(z)= 

L_{S}(z). 
Setting z=1 in these identities yields new identities concerning
multiple z values. Our examples above thus lead to
z(2)^{2}=2z(2,2)+4z(3,1),
z(2)z(3)=6z(4,1)+3z(3,2)+z(2,3).
Conjecture 2 All known relations concerning multiple zeta values
follow from the stuffle product of multiple zetas and the shuffle
product of polylogarithms specialized at 1.
This has been checked up to weight 12 [3], and the set of
identities thus obtained coincides with the bound provided by Zagier's
conjecture.
2 Monodromy and Consequences
A first step towards proving the conjecture above is provided by the
following theorem.
Theorem 1 [[4]] The ideal of
algebraic relations
between polylogarithms
at z is generated by the shuffle relations.
A Lyndon word is a nonempty word which precedes its strict
right factors in the lexicographic order. A classical theorem due to
Radford states that the Lyndon words form a basis of the shuffle
algebra. This leads to the following result.
Corollary 1 The polylogarithms indexed by Lyndon words form a
transcendence basis of the polylogarithms. In particular, the classical
polylogarithms Li_{k}=L_{x}_{0}^{k}_{x}_{1} are algebraically independent.
This theorem is proved for relations involving
polylogarithms of weight bounded by a fixed number. Using the
shuffle relations, any polynomial in polylogarithms
can be reduced to a linear combination of polylogarithms. Since the
shuffle relations form a Gröbner basis for the total degree order
(degrevlex), any polynomial which is not in the ideal is thus reduced
to a nonzero linear combination. The theorem is thus reduced to
proving that the polylogarithms are linearly independent. This
is done by computing the monodromy of polylogarithms as we now describe.
It turns out to be convenient to prove a more
general theorem where polylogarithms with indices ending in x_{0} are
allowed. Consistency with
the shuffle relations is achieved with
L 

(z):= 
ó õ 


=log 
z,
L 

(z):= 
ó õ 


= 

log^{m}z. 
At z=0, the situation is simple: a word ending with x_{1}
corresponds to an analytic polylogarithm, whence a trivial
monodromy. An easy induction on the weight shows that all words ending
in x_{0} can be rewritten as a sum of shuffles of powers of x_{0} and
words ending in x_{1}. Here are the corresponding relations up to
weight 3:
L 

=L 

L 

L 

,
L 

=L 

L 

L 

L 

, 

L 

=L 

L 

2L 

,
L 

=L 

L 

L 

L 

+L 

. 

Let M_{0}f(z) be f(ze^{2ip}); applying M_{0}Id
on the righthand sides of these identities only affects
the L_{x}_{0}^{k}. Their monodromy follows from ( M_{0}Id)L_{x}_{0}=2ip. Another
shuffle thus shows that
( M_{0}Id 
)L 

=2ip
L_{U}+ 

µ_{V}L_{V}, ( M_{0}Id 
)L 

=0, 
where the words V in the sum all have weight smaller than the weight of U.
We now proceed to prove the analogous property at 1 with M_{1}f(1z):=f((1z)e^{2ip}):
( M_{1}Id 
)L 

=2ip
L_{U}+ 

µ_{V}L_{V}, ( M_{1}Id 
)L 

=0.
(3) 
At z=1, words ending with x_{0} correspond to polylogarithms that are
analytic there, hence, have a trivial monodromy. This is the second
identity. The situation is slightly more complicated than at the
origin because of divergence.
As above, an induction on the weight shows that all words beginning
with x_{1} can be rewritten as a sum of shuffles of powers of x_{1} and
words beginning with x_{0}.
The monodromy of L_{x}_{1}^{k} follows from
that of the logarithm. The remaining words are those beginning
with x_{0} and ending with x_{1}. Consider the path consisting of a
straight line from z to a circle of radius e around 1,
turning around 1 in the anticlockwise direction and coming back
to z. Then Cauchy's theorem implies that
( M_{1}Id 
)L 

(z)= 

ó õ 


( M_{1}Id 
)L 

(t)
+ 

ó õ 


L 

(t). 
Another induction shows that the rightmost integral tends to 0, while
convergence of L_{Ux}_{1} at 1 reduces the first limit to
This makes it possible to compute all the monodromies of words ending
in x_{1} and proves (3). Here are the
corresponding relations up to
weight 3, using p to denote 2ip:
( M_{1}Id 
)L 

= 


,
( M_{1}Id 
)L 

=pL 

,
( M_{1}Id 
)L 

=pL 

, 

( M_{1}Id 
)L 

=p(L 

z 

)+ 

L 

,
( M_{1}Id 
)L 

=2L 

pL 

2pz 

. 

The proof of Theorem 1 is concluded by considering
the maximal weight involved in a minimal nontrivial linear
combination: applying both operators ( M_{0}Id) and ( M_{1}Id)
leads to linear relations of smaller weight, that have to be trivial.
3 Changes of Variables
The group of six rational functions z, 1z, 1/z, 1/(1z),
11/z, z/(1z) permutes the singularities 0, 1, ¥. If h
is an element of this group, then
L_{xU}(h(z))= 
ó õ 

L_{U}(t)w_{x}(t) dt 
= 
ó õ 

L_{U}(h(s))w_{x}(h(s))h'(s) ds. 
It turns out that for all h in the group and all xÎ{x_{0},x_{1}},
w_{x}(h(s))h'(s) can be rewritten as a linear combination of ds/s
and ds/(1s). Thus by induction, all polylogarithms at h(z) can be
rewritten in terms of polylogarithms at z.
For the classical dilogarithm Li_{2}=L_{x}_{1}_{x}_{0}, we
get
Li_{2}(1z)+Li 
_{2}(z)=L 

(z)L 

(z)+z(2),
Li_{2}(z)Li 
_{2}(1z^{1})=L 

(z)L 

(z)+z(2)+L 

(z). 
Setting z to 1/2,±f,± 1/f,1+f,11/f, where f
is the golden ratio,
yields the only known
values of Li_{2} in closed form.
4 Noncommutative Generating Function
All the inductions mentioned here are conveniently handled by
introducing the noncommutative generating
function L(z)=åL_{W}(z)W where the sum is over all words
of X={x_{0},x_{1}}^{*}.
The integral representation of polylogarithms is equivalent to a
linear differential equation:
A consequence of the rewriting of words ending by x_{0} is that all
polylogarithms except L_{x}_{0}^{k} tend to 0 at the origin. This leads
to the initial condition L(e)=e^{ln}^{e
x}_{0}+O(e^{1d}), for e®0,
where d is an arbitrarily small real number. The shuffle
relation then implies that this generating function is a Lie
exponential. A noteworthy consequence is that it can be factored as a
product of Lie exponentials indexed by Lyndon words, which turns out
to yield an efficient algorithm for computing
identities [3].
The inductions used in the monodromy computations translate very
explicitly into
M 
_{0}L(z)=L(z)e 

, M 
_{1}L(z)=L(z)Z^{1}e 

Z, 
where Z is very close to being the generating function of the
multiple zeta values: it is the unique Lie exponential such that
(Zx_{0})=(Zx_{1})=0, (Zx_{0}Wx_{1})=z 

,
WÎ X. 
Similarly, the changes of variables can be interpreted at the
level of L(z) [5].
References
 [1]

Borwein (Jonathan M.), Bradley (David M.), Broadhurst (David J.), and Lisonek (Petr). 
Special values of multiple polylogarithms. Transactions of the
American Mathematical Society, 1999. 
To appear.
 [2]

Lewin (Leonard) (editor). 
Structural properties of polylogarithms. 
American Mathematical Society, Providence, RI, 1991,
xviii+412p.
 [3]

Minh (Hoang Ngoc) and Petitot (Michel). 
Lyndon words, polylogarithms and the Riemann z function. Discrete Mathematics, To appear.
 [4]

Minh (Hoang Ngoc), Petitot (Michel), and Van der Hoeven (Joris). 
Shuffle algebra and polylogarithms. In Formal Power Series and
Algebraic Combinatorics. 
1998. Proceedings PFSAC'98, Toronto.
 [5]

Minh (Hoang Ngoc), Petitot (Michel), and Van der Hoeven (Joris). 
L'algèbre des polylogarithmes par les séries
génératrices. In Formal Power Series and Algebraic
Combinatorics. 
1999. Proceedings PFSAC'99, Barcelona.
This document was translated from L^{A}T_{E}X by
H^{E}V^{E}A.