Multiple Polylogarithms: An Introduction

M. Waldschmidt Dedicated to Professor R.P. Bambah on his 75th birthday

Multiple polylogarithms in a single variable are defined by

X zn1 Li(s ,...,s )(z) = , 1 k s1 sk n ···nk n1>n2>···>nk≥1 1

when s1,...,sk are positive integers and z a complex number in the unit disk. For k = 1, this is the classical polylogarithm Lis (z). These multiple polylogarithms can be defined also in terms of iterated Chen integrals and satisfy shuffle relations. Multiple polylogarithms in several variables are defined for si ≥ 1 and |zi | < 1(1 ≤ i ≤ k) by

n n X z 1 ···z k 1 k Li(s ,...,s )(z ,...,zk) = , 1 k 1 s1 sk n ···nk n1>n2>···>nk≥1 1

and they satisfy not only shuffle relations, but also stuffle relations. When one specializes the shuffle relations in one variable at z = 1 and the stuffle relations in several variables at z1 =···=zk = 1, one gets linear or quadratic dependence relations between the Multiple Zeta Values X 1 ζ(s ,...,sk) = 1 s1 sk n ···nk n1>n2>···>nk≥1 1

which are defined for k,s1,...,sk positive integers with s1 ≥ 2. The Main Diophantine Conjecture states that one obtains in this way all algebraic relations between these MZV.

Mathematics Subject Classification: 11J91, 33E30.

0. Introduction

A long term project is to determine all algebraic relations among the values

π, ζ(3), ζ(5),...,ζ(2n + 1),... of the Riemann zeta function X ζ(s) = 1 . ns n≥1 58 M. Waldschmidt

So far, one only knows that the first number in this list, π, is transcendental, that the second one, ζ(3), is irrational, and that the other ones span a Q-vector space of infinite dimension [Ri1], [BR]. See also [Ri2], [Zu1] and [Zu2]. The expected answer is disappointingly simple: it is widely believed that there are no relations, which means that these numbers should be algebraically independent: (?) For any n ≥ 0 and any nonzero polynomial P ∈ Z[X0,...,Xn],

P(π,ζ(3), ζ(5),...,ζ(2n + 1)) 6= 0.

If true, this property would mean that there is no interesting algebraic structure. The situation changes drastically if we enlarge our set so as to include the so- called Multiple Zeta Values (MZV, also called Euler-Zagier numbers or Polyzeta–ˆ see [Eu], [Z] and [C]):

X 1 ζ(s ,...,sk) = , 1 s1 sk n ···nk n1>n2>···>nk≥1 1 which are defined for k,s1,...,sk positive integers with s1 ≥ 2. It may be hoped that the initial goal would be reached if one could determine all algebraic relations between the MZV. Now there are plenty of relations between them, providing a rich algebraic structure. One type of such relations arises when one multiplies two such series: it is easy to see that one gets a linear combination of MZV. There is another type of algebraic relations between MZV, coming from their expressions as integrals. Again the product of two such integrals is a linear combination of MZV. Following [B3], we will use the name stuffle for the relations arising from the series, and stuffle for those arising from the integrals. The Main Diophantine Conjecture (Conjecture 5.3 below) states that these relations are sufficient to describe all algebraic relations between MZV.One should be careful when stating such a conjecture: it is necessary to include some relations which are deduced from the stuffle and shuffle relations applied to divergent series (i.e. with s1 = 1). There are several ways of dealing with the divergent case. Here, we use the multiple polylogarithms

n X z 1 Li(s ,...,s )(z) = 1 k s1 sk n ···nk n1>n2>···>nk≥1 1 which are defined for |z| < 1 when s1,...,sk are all ≥ 1, and which are also defined for |z|=1ifs1 ≥ 2. These multiple polylogarithms can be expressed as iterated Chen integrals, and from this representation one deduces shuffle relations. There is no stuffle rela- tions for multiple polylogarithms in a single variable, but one recovers them by Multiple Polylogarithms: An Introduction 59 introducing the multivariables functions n n X z 1 ···z k 1 k Li(s ,...,s )(z ,...,zk) = ,(si ≥ 1, 1 ≤ i ≤ k) 1 k 1 s1 sk n ···nk n1>n2>···>nk≥1 1 which are defined not only for |z1| < 1 and |zi|≤1 (2 ≤ i ≤ k), but also for |zi|≤1(1 ≤ i ≤ k) if s1 ≥ 2.

1. Multiple Polylogarithms in One Variable and Multiple Zeta Values

Let k,s1,...,sk be positive integers. Write s in place of (s1,...,sk). One defines a complex function of one variable by n X z 1 Lis(z) = . s1 sk n ···nk n1>n2>···>nk≥1 1

This function is analytic in the open unit disk, and, in the case s1 ≥ 2, it is also continuous on the closed unit disk. In the latter case we have

ζ(s) = Lis(1).

One can also define in an equivalent way these functions by induction on the number p = s1 +···+sk (the weight of s) as follows. Plainly we have d (1.1) z Li(s ,...,s )(z) = Li(s − ,s ,...,s )(z) if s ≥ 2 dz 1 k 1 1 2 k 1 and d (1.2) (1 − z) Li( ,s ,...,s )(z) = Li(s ,...,s )(z) if k ≥ 2. dz 1 2 k 2 k

Together with the initial conditions

(1.3) Lis(0) = 0, the differential equations (1.1) and (1.2) determine all the Lis. Therefore, as observed by M. Kontsevich (cf. [Z]; see also [K] Chap. XIX, § 11 for an early reference to H. Poincaré, 1884), an equivalent definition for Lis is given by integral formulae as follows. Starting(∗) with k = s = 1, we write Z z dt Li1(z) =−log(1 − z) = , 0 1 − t

(∗) This induction could as well be started from k = 0, provided that we set Li∅(z) = 1. 60 M. Waldschmidt where the complex integral is over any path from 0 to z inside the unit circle. From the differential equations (1.1) one deduces, by induction, for s ≥ 2, Z Z Z Z Z z z t ts− ts− dt dt1 1 dt2 2 dts−1 1 dts Lis(z) = Lis−1(t) = ··· . 0 t 0 t1 0 t2 0 ts−1 0 1 − ts

In the last formula, the complex integral over t1, which is written on the left (and which is the last one to be computed), is over any path inside the unit circle from 0toz, the second one over t2 is from 0 to t1,...,and the last one over ts on the right, which is the first one to be computed, is from 0 to ts−1. Chen iterated integrals (see [K] Chap. XIX, § 11) provide a compact form for such expressions as follows. For ϕ1,...,ϕp differential forms and x,y complex numbers, define inductively

Z y Z y Z t ϕ1 ···ϕp = ϕ1(t) ϕ2 ···ϕp. x x x

For s = (s1, ···sk), set

s − s − ω = ω 1 1ω ···ω k 1ω , s 0 1 0 1 where dt dt ω (t) = and ω (t) = . 0 t 1 1 − t Then the differential equations (1.1) and (1.2) with initial conditions (1.3) can be written Z z (1.4) Lis(z) = ωs. 0

Example. Given a string a1,...,ak of integers, the notation {a1,...,ak}n stands for the kn-tuple (a1,...,ak,...,a1,...,ak), where the string a1,...,ak is repeated n times. For any n ≥ 1 and |z| < 1wehave

1 n (1.1) Li{ } (z) = (log(1/(1 − z))) , 1 n n! which can be written in terms of generating series as X∞ n −x Li{1}n (z)x = (1 − z) . n = 0 (z) The constant term Li{1}0 is 1. Multiple Polylogarithms: An Introduction 61

2. Shuffle Product and the First Standard Relations

∗ Denote by X ={x0,x1} the alphabet with two letters and by X the set of words on X. A word is nothing else than a non-commutative monomial in the two letters x0 and x1. The linear combinations of such words with rational coefficients X cuu, u

∗ where {cu; u ∈ X } is a set of rational numbers with finite support, is the non- commutative ring H = Qhx0,x1i. The product is concatenation, the unit is the ∗ empty word e. We are interested with the set X x1 of words which end with x1. The linear combinations of such words is a left ideal of H which we denote by 1 Hx1. Also we denote by H the subalgebra Qe + Hx1 of H. s y y = xs−1x s = (s ,...,s ) For a positive integer, set s s 0 1. Next, for a tuple 1 k y = y ···y X∗x of positive integers, define s s1 sk . Hence the set 1 is also the set of words ys, where k,s1,...,sk run over the set of all positive integers. We define b ∗ b Liu(z) for u ∈ X x1 by Liu(z) = Lis(z) when u = ys. By linearity we extend the b 1 definition of Liu(z) to H : X X b b Liu(z) = cuLiu(z) for v = cuu u u

∗ b where u ranges over a finite subset of {e}∪X x1 and cu ∈ Q, while Lie(z) = 1. The ∗ set of convergent words is the set, denoted by {e}∪x0X x1, of words which start with x0 and end with x1 together with the empty word e. The Q-vector subspace 0 1 0 they span in H is the subalgebra H = Qe + x0Hx1 of H , and for v in H we set ˆ b ζ(v) = Liv(1) so that ζˆ : H0 → R is a Q-linear map and

ζ(yˆ s) = ζ(s)

∗ for ys in x0X x1. Definition. The shuffle product of two words in X∗ is the element in H which is defined inductively as follows:

exu = uxe = u for any u in X∗, and

(xiu)x(xj v) = xi(ux(xj v)) + xj ((xiu)xv) for u, v in X∗ and i, j equal to 0 or 1. 62 M. Waldschmidt

This product is extended by distributivity with respect to the addition to H and defines a commutative and associative law. Moreover H0 and H1 are stable under x. We denote by Hx0 ⊂ Hx1 ⊂ Hx the algebras where the underlying sets are H0 ⊂ H1 ⊂ H respectively and the product is x. Radford’s Theorem gives the structure of these algebras: they are (commutative) polynomials algebras on the set of Lyndon words (see for instance [R]). b b Computing the product Liu(z)Liu0 (z) of the two associated Chen iterated inte- grals yields (see [MPH], Th. 2): 0 Proposition 2.1. For u and u in Hx1 ,

b b b Liu(z)Liu0 (z) = Liuxu0 (z).

For instance from x x(x x ) = x x x + x x2 1 0 1 1 0 1 2 0 1 we deduce

(2.2) Li1(z)Li2(z) = Li(1,2)(z) + 2Li(2,1)(z).

Setting z = 1, we deduce from Proposition 2.1: 0 0 (2.3) ζˆ (u)ζ(uˆ ) = ζ(uˆ xu )

0 for u and u in Hx0 . These are the first standard relations between multiple zeta values.

3. Shuffle Product for Multiple Polylogarithms in Several Variables

The functions of k complex variables(∗) n n X z 1 ···z k 1 k Lis(z ,...,zk) = 1 s1 sk n ...nk n1>n2>···>nk≥1 1 have been considered as early as 1904 by N. Nielsen, and rediscovered later by A.B. Goncharov [G1, G2]. Recently, J. Ecalle´ [E]´ used them for zi roots of unity

∗ ( )Our notation for (z ,...,z ) Li(s1,...,sk) 1 k is the same as in [H], [W] or [C], but for Goncharov’s [G2] it corresponds to (z ,...,z ). Li(s1,...,sk) k 1 Multiple Polylogarithms: An Introduction 63

(in case s1 ≥ 2): these are the decorated multiple polylogarithms. Of course one recovers the one variable functions Lis(z) by specializing z2 =···=zk = 1. For simplicity we write Lis(z), where z stands for (z1,...,zk). There is an integral formula which extends (1.4). Define ( zdt if z 6= 0, ω (t) = 1−zt z dt t if z = 0. From the differential equations ∂ z (z) = (z) s ≥ 1 Lis Li(s1−1,s2,...,sk) if 1 2 ∂z1 and ∂ ( − z ) (z) = (z z ,z ,...,z ) 1 1 Li(1,s2,...,sk) Li(s2,...,sk) 1 2 3 k ∂z1 generalizing (1.1) and (1.2), we deduce Z 1 s −1 s −1 sk−1 s(z) = ω 1 ωz ω 2 ωz z ···ω ωz ...z . Li 0 1 0 1 2 0 1 k 0

Because of the occurrence of the products z1 ...zj (1 ≤ j ≤ k), it is convenient (see for instance [G1] and [B3L]) to perform the change of variables y −1 −1 j−1 yj = z ···zj (1 ≤ j ≤ k) and zj = (1 ≤ j ≤ k) 1 yj with y0 = 1, and to introduce the differential forms

ω0 (t) =−ω (t) = dt , y y−1 t − y ω0 = ω ω0 =−ω so that 0 0 and 1 1. Also define   s ,...,s λ 1 k = ( /y ,y /y ,...,y /y ) y ,...,y Lis 1 1 1 2 k−1 k 1 k  −s X X Yk Xk j −νj   = ··· yj νi ν ≥ ν ≥ j= i=j 1 1 Z k 1 1 p s − 0 s − 0 = (−1) ω 1 1ω ···ω k 1ω . 0 y1 0 yk 1p With this notation some formulae are simpler. For instance the shuffle relation is 0 easier to write with λ: the shuffle is defined on words on the alphabet {ωy ; y ∈ C}, (including y = 0), inductively by 0 0 0 0 0 0 (ωyu)x(ωy0 v) = ωy(ux(ωy0 v)) + ωy0 ((ωyu)xv). 64 M. Waldschmidt

4. Stuffle Product and the Second Standard Relations

The functions Lis(z) satisfy not only shuffle relations, but also stuffle relations arising from the product of two series: X 0 00 (4.1) Lis(z)Lis0 (z ) = Lis00 (z ), s00 00 00 00 where the notation is as follows: s runs over the tuples (s1,...,sk00 ) obtained s = (s ,...,s ) and s0 = (s0 ,...,s0 ) from 1 k 1 k0 by inserting, in all possible ways, 0 0 some 0 in the string (s1,...,sk) as well as in the string (s1,...,sk0 ) (including in front and at the end), so that the new strings have the same length k00, with max {k,k0}≤k00 ≤ k + k0, and by adding the two sequences term by term. 00 00 00 For each such s , the component zi of z is zj if the corresponding si is just sj 0 0 00 0 (corresponding toa0ins ),itisz` if the corresponding si is s` (corresponding to 0 00 0 a0ins), and finally it is zj z` if the corresponding si is sj + s`. For instance

s s1 s2 0 s3 s4 ... 0 s0 s0 s0 s0 ... s0 0 1 2 0 3 k0 s00 s s + s0 s0 s s + s0 ... s0 1 2 1 2 3 4 3 k0 z00 z z z0 z0 z z z0 ... z0 . 1 2 1 2 3 4 3 k0 00 Of course the 0’s are inserted so that no si is zero. Examples. For k = k0 = 1 the stuffle relation (4.1) yields 0 0 0 0 (4.2) Lis(z)Lis0 (z ) = Li(s,s0)(z, z ) + Lis0,s(z ,z)+ Lis+s0 (zz ), while for k = 1 and k0 = 2wehave

0 0 0 0 0 0 Lis(z)Li(s0 ,s0 )(z ,z ) = Li(s,s0 ,s0 )(z, z ,z ) + Li(s0 ,s,s0 )(z ,z,z ) 1 2 1 2 1 2 1 2 1 2 1 2 0 0 0 0 0 0 (4.3) +Li(s0 ,s0 ,s)(z ,z ,z)+ Li(s+s0 ,s0 )(zz ,z ) + Li(s0 ,s+s0 )(z ,zz ). 1 2 1 2 1 2 1 2 1 2 1 2 The stuffle product ? is defined on X∗ inductively by e?u= u?e= u for u ∈ X∗, xn ?w= w?xn = ωxn 0 0 0 for any n ≥ 1 and w ∈ X∗, and 0 0 0 0 (ysu)?(yt u ) = ys(u?(yt u )) + yt ((ysu)?u) + ys+t (u?u) for u and u0 in X∗,s ≥ 1,t ≥ 1. This product is extended by distributivity with respect to the addition to H and defines a commutative and associative law. Moreover H0 and H1 are stable Multiple Polylogarithms: An Introduction 65

0 1 under ?. Wedenote by H? ⊂ H? ⊂ H? the corresponding harmonic algebras. Their structure has been investigated by M. Hoffman [H]: again they are (commutative) polynomials algebras over Lyndon words. z =···=z = z0 =···=z0 = Specializing (4.1) at 1 k 1 k0 1, we deduce 0 0 (4.4) ζˆ (u)ζ(uˆ ) = ζ(u?uˆ )

0 0 for u and u in H?. These are the second standard relations between multiple zeta values. For z = z0 = z0 = instance (4.3) with 1 2 1gives ζ(s)ζ(s0 ,s0 ) = ζ(s,s0 ,s0 ) + ζ(s0 ,s,s0 ) + ζ(s0 ,s0 ,s) 1 2 1 2 1 2 1 2 +ζ(s + s0 ,s0 ) + ζ(s0 ,s+ s0 ) 1 2 1 2 s ≥ ,s0 ≥ s0 ≥ for 2 1 2 and 2 1.

5. The Third Standard Relations and the Main Diophantine Conjectures

We start with an example. Combining the stuffle relation (4.2) for s = s0 = 1 with the shuffle relation (2.2) for z0 = z, we deduce

2 (5.1) Li(1,2)(z, 1) + 2Li(2,1)(z, 1) = Li(1,2)(z, z) + Li(2,1)(z, z) + Li3(z ). The two sides are analytic inside the unit circle, but not convergent at z = 1. We claim that X n n z 1 (1 − z 2 ) F(z) = ( , )(z, ) − ( , )(z, z) = Li 1 2 1 Li 1 2 2 n1n n1>n2≥1 2 tends to 0 as z tends to 1 inside the unit circle. Indeed for |z| < 1wehave

n2 n2−1 |1 − z |=|(1 − z)(1 + z +···+z )|

1 2 |F(z)|≤|1 − z|Li( , )(|z|) = |1 − z|(log(1/(1 −|z|))) . 1 1 2 Therefore, taking the limit of the relation (5.1) as z → 1 yields Euler’s formula

ζ(2, 1) = ζ(3). 66 M. Waldschmidt

This argument works in a quite general setting and yields the relations ˆ (5.2) ζ(x1 ?v− x1xv) = 0 for each v ∈ H0. These are the third standard relations between multiple zeta values. The Main Diophantine Conjectures below arose after the works of several mathe- maticians, including D. Zagier, A.B. Goncharov, M. Kontsevich, M. Hoffman, M. Petitot and Hoang Ngoc Minh, K. Ihara and M. Kaneko (see [C]). They imply that the three standard relations (2.3), (4.4) and (5.2) generate the ideal of algebraic relations between all numbers ζ(s). Here are precise statements. ∗ We introduce independent variables Zu, where u ranges over the set {e}∪Xx . P 1 1 For v = u cuu in H ,weset X Zv = cuZu. u

∗ In particular for u1 and u2 in x0Xx ,Zu xu and Zu ?u are linear forms in ∗ 1 ∗ 1 2 1 2 Zu,u ∈ x0X x1. Also, for v ∈ x0X x1,Zx xv−x ?v is a linear form in Zu, ∗ 1 1 u ∈ x0X x1. Denote by R the ring of polynomials with coefficients in Q in the variables Zu ∗ where u ranges over the set x0X x1, and by J the ideal of R consisting of all polynomials which vanish under the specialization map ˆ ∗ Zu 7→ ζ (u) (u ∈ x0X x1).

Conjecture 5.3. The polynomials Z Z − Z ,ZZ − Z Z , u1 u2 u1xu2 u1 u2 u1?u2 and x1?v−x1xv ∗ where u1, u2 and v range over the set of elements in x0X x1, generate the ideal J. This statement is slightly different from the conjecture in§3of[IK], where Ihara and Kaneko suggest that all linear relations between MZV’s are supplied by the regularized double shuffle relations

ζ(ˆ reg(u?v− uxv)) = 0, where u ranges over H1 and v over H0. Here, reg is the Q-linear map H → H0 which maps w to the constant term of the expression of w as a (commutative) 0 polynomial in x0 and x1 with coefficients in Hx. It is proved in [IK] that the linear polynomials Zreg(u?v−uxv) associated to the regularized double shuffle relations belong to the ideal J. On the other hand, at least for the small weights, one can check that the regularized double shuffle relations follow from the three standard relations. Hence our Conjecture 5.3 seems stronger than the conjecture of [IK], but we expect they are in fact equivalent. Multiple Polylogarithms: An Introduction 67

Denote by Zp the Q-vector subspace of R spanned by the real numbers ζ(s) with s of weight p, with Z0 = Q and Z1 ={0}. Using any of the first two standard relations (2.3) or (4.4), one deduces Zp · Zp0 ⊂ Zp+p0 . This means that the Q- vector subspace Z of R spanned by all Zp,p ≥ 0, is a subalgebra of R over Q which is graded by the weight. From Conjecture 5.3 one deduces the following conjecture of Goncharov [G1]:

Conjecture 5.4. As a Q-algebra, Z is the direct sum of Zp for p ≥ 0.

Conjecture 5.4 reduces the problem of determining all algebraic relations be- tween MZV to the problem of determining all linear such relations. The dimension dp of Zp satisfies d0 = 1, d1 = 0, d2 = d3 = 1. The expected value for dp is given by a conjecture of Zagier [Z]:

Conjecture 5.5. For p ≥ 3 we have

dp = dp−2 + dp−3.

An interesting question is whether Conjecture 5.3 implies Conjecture 5.5. For this question as well as other related problems, see [E]´ and [C].

References

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Insutitut de Mathématiques, Université P. et M. Curie (Paris VI) Théorie des Nombres Case 247, F-75013 Paris E-mail: [email protected] http://www.math.jussieu.fr/∼miv/