Algebraic relations for multiple zeta values

Transcription

1 Russian Math. Surveys 58: 29 Uspekhi Mat. Nauk 58: 3 32 c 2003 RAS(DoM) and LMS DOI 0.070/RM2003v058n0ABEH Algebraic relations for multiple zeta values V. V. Zudilin [W. Zudilin] Abstract. The survey is devoted to the multidimensional generalization of the Riemann zeta function as a function of a positive integral argument. Contents. Introduction 2. Multiple zeta values 2 3. Identities: the method of partial fractions 3 4. Algebra of multiple zeta values 7 5. Shuffle algebra of generalized polylogarithms 8 6. Duality theorem 0 7. Identities: the generating-function method 8. Quasi-shuffle products 3 9. Functional model of the stuffle algebra 6 0. Hoffman s homomorphism for the stuffle algebra 8. Derivations 9 2. Ihara Kaneko derivations and Ohno s relations 2 3. Open questions q-analogues of multiple zeta values 25 Bibliography 28. Introduction In the domain Res >, the Riemann zeta function can be defined by the convergent series ζ(s) n s. () n An interesting and still unsolved problem concerns the polynomial relations over Q between the numbers ζ(s), s 2, 3, 4,... Thanks to Euler, we know the formula ζ(s) (2πi)s B s 2s! for s 2, 4, 6,..., (2) AMS 2000 Mathematics Subject Classification. Primary M06; Secondary G55, 6W25.

2 2 V. V. Zudilin [W. Zudilin] which expresses the values of the zeta function at even points in terms of the number π 4 n0 ( ) n n + and the Bernoulli numbers B s Q defined by the generating function t e t t 2 + t s B s s!, B s 0 forodd s 3. (3) s2 The relation (2) yields the coincidence of the rings Q[ζ(2),ζ(4),ζ(6),ζ(8),...]and Q[π 2 ], and hence, by Lindemann s theorem [7] on the transcendence of π, wecan conclude that each of the rings is of transcendence degree over the field of rational numbers. Much less is known about the arithmetic nature of values of the zeta function at odd integers s 3, 5, 7,...Apéryproved [] that the number ζ(3) is irrational and Rivoal recently showed [22] that there are infinitely many irrational numbers in the list ζ(3),ζ(5),ζ(7),... Conjecturally, each of these numbers is transcendental, and the above question on the polynomial relations over Q for the values of the series () at the integers s 2 has the following simple answer. Conjecture. The numbers π, ζ(3), ζ(5), ζ(7), ζ(9),... are algebraically independent over Q. This conjecture can be regarded as a part of mathematical folklore (see, for instance, [7] and [28]). In this survey we discuss a generalization of the problem of algebraic independence for values of the Riemann zeta function at positive integers (the so-called zeta values). Namely, we speak of an object studied extensively during the last decade in connection with problems concerning not only number theory but also combinatorics, algebra, analysis, algebraic geometry, quantum physics, and many other branches of mathematics. At the same time, no papers in this direction have been published in Russian until now (we only mention the paper [25] in press). By means of the present publication, we hope to attract the attention of Russian mathematicians to problems connected with multiple zeta values. The author is deeply indebted to the referee for several valuable remarks that have essentially improved the presentation. 2. Multiple zeta values The series () admits the following multidimensional generalization. For positive integers s,s 2,...,s l, where s >, we consider the values of the l-tuple zeta function ζ(s) ζ(s,s 2,...,s l ): ; (4) n s ns2 2 nsl n >n 2> >n l in what follows, the corresponding multi-index s (s,s 2,...,s l )issaidtobe admissible. The quantities (4) are called multiple zeta values [30] (and abbreviated l

3 Algebraic relations for multiple zeta values 3 as MZVs), or multiple harmonic series [0], or Euler sums. Thesums(4)forl 2 were originated by Euler [5] who obtained a family of identities connecting double and ordinary zeta values (see the corollary to Theorem below). In particular, Euler proved the identity ζ(2, ) ζ(3), (5) which has been rediscovered on numerous occasions since then. The quantities (4) were introduced by Hoffman in [0] and independently by Zagier in [30] (with the opposite order of summation on the right-hand side of (4)); moreover, in [0] and [30] some Q-linear and Q-polynomial relations were established and several conjectures were stated (some of which were later proved) concerning the structure of algebraic relations for the family (4). Hoffman also suggested [0] the alternative definition ζ(s) ζ(s,s 2,...,s l ): (6) n s ns2 2 nsl l n n 2 n l of the Euler sums with non-strict inequalities in the summation. Clearly, all relations for the series (6) can readily be rewritten for the series (4) (see, for instance, [0] and [25]), although several identities possess a compact form in the terms of multiple zeta values (6) (see the relations (38) in 7below). For each number (4) we introduce two characteristics, namely, the weight (or the degree) s : s + s s l and the length (or the depth) l(s) :l. We note [3] that the series on the right-hand side of (4) converges absolutely in the domain given by Re s > and l k Re s k >l; moreover, the multiple zeta function ζ(s) defined in the domain by the series (4) can be continued analytically to a meromorphic function on the whole space C l with possible simple poles at the hyperplanes s and j k s k j + m, wherej, <j l, andm, m, are integers. The problems concerning the existence of a functional equation for l> and the localization of non-trivial zeros (an analogue of Riemann s conjecture) for the function ζ(s) remain open. 3. Identities: the method of partial fractions In this section we present examples of identities (for multiple zeta values) that are proved by an elementary analytic method, namely, the method of partial fractions. Theorem (Hoffman s relations [0], Theorem 5.). The identity l ζ(s,...,s k,s k +,s k+,...,s l ) k l k s k 2 s k 2 j0 ζ(s,...,s k,s k j, j +,s k+,...,s l ) (7) holds for any admissible multi-index s (s,s 2,...,s l ).

4 4 V. V. Zudilin [W. Zudilin] Proof. For any k, 2,...,l we have n k>n k+> >n l and hence Therefore n sk+ + k n sk+ k+ nsl l n sk k mnsk+ k+ nsl l n k m>n k+> >n l n k n k>n k+> >n l mn k++ n k>m>n k+> >n l mn sk k nsk+ k+ nsl l, n sk k mnsk+ k+ nsl l ζ(s,...,s k,s k +,s k+,...,s l )+ζ(s,...,s k,s k,,s k+,...,s l ) n s n > >n k>n k+> >n l nsk+ k n sk+ k+ nsl l + n s nsk k mnsk+ k+ nsl l n > >n k>m>n k+> >n l n k n > >n k>n k+> >n l mn k++ n >n 2> >n l n s ns2 2 nsl l n k mn k++ mn s nsk k nsk+ k+ nsl l m. l ( ζ(s,...,s k,s k +,s k+,...,s l )+ζ(s,...,s k,s k,,s k+,...,s l ) ) k n >n 2> >n l m sl m,m 2,...,m l m,m 2,...,m l n s ns2 2 nsl l n m m (m + m 2 ) sl (m + + m l ) s ( M sl M sl 2 M s l m l+ m l+ M l+ m + +m l m m ), (8) where M k stands for m + m m k provided that k,...,l+ (clearly, M k n l+ k provided that k,...,l). We now note the following partialfraction expansion (with respect to the parameter u): u(u + v) s s v s u, u,v R; (9) v j+ (u + v) s j j0

5 Algebraic relations for multiple zeta values 5 for the proof it suffices to use the fact that on the right-hand side we sum a geometric progression. By setting u m l+, v M l,ands s in (9), we obtain and hence m l+ M s l+ m l+ (m l+ + M l ) s ( ) s 2 M s l m l+ M l+ j0 M s l m l+ M j+ l M s j l+ s + j0 M j+ l M s j, l+. m l+ M s l+ Continuing the equality (8), we see that l ( ζ(s,...,s k,s k +,s k+,...,s l )+ζ(s,...,s k,s k,,s k+,...,s l ) ) k s 2 j0 m,m 2,...,m l+ + s 2 j0 m,m 2,...,m l+ M sl M sl 2 M s2 ζ(s j, j +,s 2,...,s l )+ l M j+ l M s j l+ M sl M sl 2 M s2 l m l+m s l+ m,m 2,...,m l+ M sl M sl 2 M s2 l m lm s l+ (0) (we have transposed the indices m l and m l+ in the last multiple sum). Using the identity (9) with u m k+, v M k M k+ m k+,ands s l+ k, we conclude that s l+ k M sl+ k k m k+ j0 and therefore m,m 2,...,m l+ s l+ k j0 + s l+ k j0 + M sl M j+ k M sl+ k j k+ M sl+ k k m,m 2,...,m l+ m,m 2,...,m l+ M sl m k+ M sl k k+2 M sl sl+2 k M M sl+2 k k +, k, 2,...,l, m k+ M sl+ k k+ M s l+ k M j+ k M sl+ k j k+ M sl k k+2 m k+ M sl+ k k+ M s l+ ζ(s,...,s l k,s l+ k j, j +,s l+2 k,...,s l ) m,m 2,...,m l+ M sl M sl+2 k k k, 2,...,l. m k M sl+ k k+ M s l+ M s l+, ()

6 6 V. V. Zudilin [W. Zudilin] Successively applying the identities () for the multiple sum on the right-hand side of the equality (0) in the inverse order (that is, beginning with k l and ending with k ), we obtain l ( ζ(s,...,s k,s k +,s k+,...,s l )+ζ(s,...,s k,s k,,s k+,...,s l ) ) k s 2 j0 ζ(s j, j +,s 2,...,s l ) l + + l k k j0 + s l+ k j0 m,m 2,...,m l+ s k 2 ζ(s,...,s l k,s l+ k j, j +,s l+2 k,...,s l ) m M sl 2 M sl 3 M s l+ ζ(s,...,s k,s k j, j +,s k+,...,s l ) l ζ(s,...,s k,s k,,s k+,...,s l ). (2) k After the necessary reductions on the left- and right-hand sides of the equality (2), we finally arrive at the desired identity (7). For l the statement of Theorem can be represented in the following form. Corollary (Euler s theorem). The identity s 2 ζ(s) ζ(s j, j) (3) j holds for any integer s 3. We also note that for s 3 the identity (3) is simply the relation (5). In [3] the following result was also proved by using the method of partial fractions. Theorem 2 (cyclic sum theorem). The identity l ζ(s k +,s k+,...,s l,s,...,s k ) k l k s k 2 s k 2 j0 ζ(s k j, s k+,...,s l,s,...,s k,j+) holds for any admissible multi-index s (s,s 2,...,s l ). Theorem 2 immediately proves that the sum of all multiple zeta values of fixed length and fixed weight does not depend on the length; this statement, as well as Theorem, generalizes Euler s theorem stated above.

7 Algebraic relations for multiple zeta values 7 Theorem 3 (sum theorem). The identity ζ(s,s 2,...,s l )ζ(s) s >,s 2,...,s l s +s 2+ +s ls holds for any integers s> and l. Theorems and 3 are special cases of Ohno s relations [2], which will be discussed in 2 below. 4. Algebra of multiple zeta values This section is based on the papers [] and [30]. To describe the known algebraic relations (that is, the numerical identities) over Q for the quantities (4), it is useful to represent ζ as a linear map of a certain polynomial algebra into the field of real numbers. Let us consider the coding of multi-indices s by words (that is, by monomials in non-commuting variables) on the alphabet X {x 0,x } by the rule s x s x s 0 x x s2 0 x x sl 0 x. We set ζ(x s ):ζ(s) (4) for all admissible words (that is, beginning with x 0 and ending with x ); then the weight (or the degree) x s : s coincides with the total degree of the monomial x s, whereas the length l(x s ):l(s) is the degree with respect to the variable x. Let Q X Q x 0,x be the Q-algebra of polynomials in two non-commuting variables which is graded by degree (where each of the variables x 0 and x is assumed to be of degree ); we identify the algebra Q X with the graded Q- vector space H spanned by the monomials in the variables x 0 and x. We also introduce the graded Q-vector spaces H Q Hx and H 0 Q x 0 Hx,where denotes the unit (the empty word of weight 0 and length 0) of the algebra Q X. Then the space H can be regarded as the subalgebra of Q X generated by the words y s x s 0 x,whereash 0 is the Q-vector space spanned by all admissible words. We can now regard the function ζ as the Q-linear map ζ : H 0 R defined by the relations ζ() and (4). Let us define the products (the shuffle product) onh and (the harmonic or stuffle product) onh by the rules for any word w, and w w w, w w w (5) x j u x k v x j (u x k v)+x k (x j u v), (6) y j u y k v y j (u y k v)+y k (y j u v)+y j+k (u v) (7) for any words u, v, any letters x j,x k, and any generators y j,y k of the subalgebra H, and then extend the rules (5) (7) to the whole algebra H and the whole subalgebra H by linearity. Sometimes it becomes useful to consider the stuffle product on the whole algebra H by formally adding to (7) the rule x j 0 w w xj 0 wxj 0 (8)

8 8 V. V. Zudilin [W. Zudilin] for any word w and any integer j. We note that induction arguments enable us to prove that each of the above products is commutative and associative (for the proof, see 8 below); the corresponding algebras H : (H, )andh : (H, ) (and also H : (H, )) are examples of the so-called Hopf algebras. The following two statements motivate the consideration of the above products and ; their proofs can be found in [], [3], and [28]. Theorem 4. The map ζ is a homomorphism of the shuffle algebra H 0 : (H 0, ) into R, that is, ζ(w w 2 )ζ(w )ζ(w 2 ) for all w,w 2 H 0. (9) Theorem 5. The map ζ is a homomorphism of the stuffle algebra H 0 : (H 0, ) into R, that is, ζ(w w 2 )ζ(w )ζ(w 2 ) for all w,w 2 H 0. (20) In what follows we present detailed proofs of these two theorems by using the differential-difference origin of the products and in suitable functional models of the algebras H and H 0. When proving Theorem 4 (see 5), we follow the scheme of the paper [27], whereas our proof of Theorem 5 (in 9) is new. Another family of identities is given by the following statement which is derived from Theorem in. Theorem 6. The map ζ satisfies the relations ζ(x w x w) 0 for all w H 0 (2) (in particular, the polynomials x w x w belong to H 0 ). All relations (both proved and experimentally obtained) known at present for the multiple zeta values follow from the identities (9) (2). Therefore, the following conjecture looks quite plausible. Conjecture 2 ([], [8], [27]). All algebraic relations over Q among the multiple zeta values are generated by the identities (9) (2); equivalently, ker ζ {u v u v : u H, v H 0 }. 5. Shuffle algebra of generalized polylogarithms To prove the shuffle relations (9) for multiple zeta values, we define the generalized polylogarithms z n Li s (z) :, z <, (22) n s n >n 2> >n l ns2 2 nsl l where l is a positive integer, for any l-tuple of positive integers s,s 2,...,s l. By definition, Li s () ζ(s), s Z l, s 2, s 2,..., s l. (23) By setting Li xs (z) :Li s (z), Li (z) :, (24) as above in the case of multiple zeta values, we extend the action of the map Li: w Li w (z) by linearity to the graded algebra H (rather than to H, because the multi-indices are coded by words in H ).

9 Algebraic relations for multiple zeta values 9 Lemma. Let w H be an arbitrary non-empty word and let x j be the first letter in its representation (that is, w x j u for some word u H ). Then d dz Li w(z) d dz Li x j u(z) ω j (z)li u (z), (25) where ω j (z) ω xj (z) : z z if x j x 0, if x j x. Proof. Assuming that w x j u x s for some multi-index s, wehave d dz Li w(z) d dz Li s(z) d z n, dz n s n >n 2> >n l ns2 2 nsl l z n n s n >n 2> >n l n s2 2 nsl l Therefore, if s > (which corresponds to the letter x j x 0 ), then d dz Li x 0u(z) z n >n 2> >n l. z n n s n s2 2 nsl l z Li s,s 2,...,s l (z) z Li u(z) and, if s (which corresponds to the letter x j x ), then d dz Li x u(z) n >n 2> >n l z n 2> >n l z n n s2 2 nsl l z n2 n s2 2 nsl l n 2> >n l n s2 2 nsl l n n 2+ z n (26) z Li s 2,...,s l (z) z Li u(z), and the result follows. Lemma motivates another definition of the generalized polylogarithms, which can now be defined for all elements of the algebra H. As above, it suffices to define the polylogarithms just for the words w H and then extend the definition to the whole algebra by linearity. We set Li (z) and log s z if w x s 0 for some s, s! Li w (z) z (27) ω j (z)li u (z)dz if w x j u contains the letter x. 0 In this case, Lemma remains valid for the extended version (27) of the polylogarithms (this yields the coincidence of the new polylogarithms with old ones (24) for the words w in H ); moreover, lim Li w(z) 0 ifthewordw contains the letter x. z 0+0 An easy verification shows that the generalized polylogarithms are continuous realvalued functions on the interval (0, ).

10 0 V. V. Zudilin [W. Zudilin] Lemma 2. The map w Li w (z) is a homomorphism of the algebra H into C((0, ); R). Proof. We must verify the equalities Li w w 2 (z) Li w (z)li w2 (z) for all w,w 2 H; (28) it suffices to do this for any words w,w 2 H. Let us prove the equality (28) by induction on the quantity w + w 2. If w or w 2, then relation (28) becomes tautological by (5). Otherwise w x j u and w 2 x k v, and hence by Lemma and by the induction hypothesis we have d ( Liw (z)li w2 (z) ) d ( Lixj u(z)li ) xkv(z) dz dz d dz Li d x j u(z) Li xkv(z)+li xj u(z) dz Li x kv(z) ω j (z)li u (z)li xkv(z)+ω k (z)li xj u(z)li v (z) ω j (z)li u xkv(z)+ω k (z)li xju v(z) d ( Lixj (u x dz kv)(z)+li xk(x ju v)(z) ) d dz Li x j u x kv(z) d dz Li w w 2 (z). Thus Li w (z)li w2 (z) Li w w 2 (z)+c, (29) and the passage to the limit as z gives the relation C 0 if at least one of the words w,w 2 contains the letter x, otherwise the substitution z givesthe same result if the representations of w and w 2 involve the letter x 0 only. Therefore, the equality (29) becomes the required relation (28), and the lemma follows. Proof of Theorem 4. Theorem 4 follows from Lemma 2 and the relations (23). An explicit evaluation of the monodromy group for the system (25) of differential equations enabled Minh, Petitot, and van der Hoeven to prove that the homomorphism w Li w (z) of the shuffle algebra H over C is bijective, that is, all C-algebraic relations for the generalized polylogarithms are induced just by the shuffle relations (28); in particular, the generalized polylogarithms are linearly independent over C. The assertion that the functions (22) are linearly independent was obtained in the simplest way (as a consequence of elegant identities for the functions) by Ulanskii [25]; the same assertion follows from Sorokin s result in [24]. 6. Duality theorem By Lemma, the following integral representation holds for the word w x ε x ε2 x εk H : z z zk Li w (z) ω ε (z )dz ω ε2 (z 2 )dz 2 ω εk (z k )dz k ω ε (z )ω ε2 (z 2 ) ω εk (z k )dz dz 2 dz k z>z >z 2> >z k >z k>0 (30)

11 Algebraic relations for multiple zeta values in the domain 0 <z<. If x ε x,thatis,w H 0, then the integral in (30) converges in the domain 0 <z. Thus, in accordance with (23), we obtain a representation [30] for the multiple zeta values in the form of Chen s iterated integrals, ζ(w) ω ε (z ) ω εk (z k )dz dz k. (3) >z > >z k>0 The following result is a simple consequence of the integral representation (3). We denote by τ the anti-automorphism of the algebra H Q x 0,x transposing x 0 and x ; for example, τ(x 2 0 x x 0 x ) x 0 x x 0 x 2. Clearly, τ is an involution preserving the weight. It can readily be seen that τ is also an automorphism of the subalgebra H 0. Theorem 7 (duality theorem [30]). The relation ζ(w) ζ(τw) holds for any word w H 0. Proof. To prove the theorem, it suffices to make the change of variables z z k, z 2 z k,..., z k z, and to apply the relations ω 0 (z) ω ( z) following from (26). As a simple consequence of Theorem 7 we (again) note the identity (5), which corresponds to the word w x 2 0 x, and also the general identity ζ(n +2)ζ(2,,...,), n, 2,... (32) }{{} n times for the words of the form w x n+ 0 x. 7. Identities: the generating-function method Another application of the differential equations proved in Lemma for the generalized polylogarithms is the generating-function method. We first note that for an admissible multi-index s (s,...,s l ) the corresponding set of periodic polylogarithms Li {s}n (z), where {s} n (s, s,...,s), n 0,, 2,... }{{} n times (see, for instance, [4] and [28]) possesses the generating function L s (z, t) : Li {s}n (z)t n s, n0 which satisfies an ordinary differential equation with respect to the variable z. For instance, if l(s), that is, s (s), then, by Lemma, the corresponding differential equation is of the form (( ( z) d dz )( z d dz ) s t s ) L s (z, t) 0, and its solution can be given explicitly in terms of generalized hypergeometric series (see [3], [4], and [28]).

12 2 V. V. Zudilin [W. Zudilin] Lemma 3 ([4], Theorem 2). The following equality holds: L (3,) (z, t) F ( 2 ( + i)t, 2 ( + i)t;;z) F ( 2 ( i)t, 2 ( i)t;;z), (33) where F (a, b; c; z) stands for the Gauss hypergeometric function. Proof. A routine verification (using Lemma for the left-hand side) shows that both sides of the required equality are annihilated by the differential operator ( ( z) d ) 2 ( z d ) 2 t 4 ; dz dz moreover, the first terms in the expansions of both sides of (33) in powers of z coincide, + t4 8 z2 + t4 8 z3 + t8 +44t 4 z This implies the assertion of the lemma. Theorem 8 ([4], [28]). The identity ζ({3, } n ) 2π4n (4n +2)! holds for any integer n. Proof. By the Gauss summation formula ([29], Ch. 4) we have sin πa F (a, a;;) Γ( a)γ( + a) πa, (35) and, substituting z into the equality (33), we obtain ζ({3, } n )t 4n L (3,) (,t) sin ( + i)πt 2 n0 2 ( + i)πt sin ( i)πt 2 ( i)πt 2 2π 2 t (e (+i)πt/2 e (+i)πt/2)( e ( i)πt/2 e ( i)πt/2) 2 2π 2 t 2 (e πt + e πt e iπt e iπt) 2π 2 t 2 ( + ( ) m i m ( i) m ) (πt)m 2π 4n t 4n m! (4n +2)!. m0 n0 Comparing the coefficients of like powers of t gives the desired identity. The assertion of Theorem 8 was conjectured in [30]. The identity (34) is far from being the only example using the generating-function method. Let us present some other identities of [3] similar to (34) for which the above method is also effective: ( ) 2n+ ( ) 2n+ ζ({2} n ) 2(2π)2n, ζ({4} n ) 4(2π)4n, (2n +)! 2 (4n +2)! 2 (( + ) 4n+2 + ( ) 4n+2 ) 2, 2 ζ({6} n ) 6(2π)6n (6n +3)!, ζ({8} n) 8(2π)8n (8n +4)! ( + 5 ζ({0} n ) 0(2π)0n (0n +5)! ( + 2 ) 0n+5 ( ) 0n+5 ), (34) (36)

13 where n, 2,... The identities Algebraic relations for multiple zeta values 3 ζ(m +2, {} n )ζ(n +2, {} m ), m,n 0,, 2,..., can be obtained both by the generating-function method [0] and by applying Theorem 7 proved above. A somewhat different example of generating functions is related to generalizations of Apéry s identity [] ζ(3) 5 2 ( ) k k 3( ) ; 2k k k namely, the following expansions are valid [6], [2]: ζ(2n +3)t 2n k 3 ( t 2 /k 2 ) n0 k ( ) k ( k 3( ) 2k t 2 /k 2 k n0 k ζ(4n +3)t 4n k k 3 ( t 4 /k 4 ) 5 2 k ) k ) ( t2, l ( ) k k 3( ) 2k k l 2 t 4 /k 4 k l +4t 4 /l 4 t 4 /l 4. (37) The proofs of these identities and of some other ones are based on the use of transformation and summation formulae for generalized hypergeometric functions similar to the way in which the formula (35) was used in the proof of Theorem 8. The identities (37) are very useful for the fast evaluation of values of the Riemann zeta function at odd integers. We also note the relations ζ({2} n, ) 2ζ(2n +), n, 2,..., (38) obtained by successive application of the results in [26] (or [33]) and [32]. The equalities (38) also generalize Euler s identity (5) and are closely related to one of the ways to prove Apéry s theorem [] and Rivoal s theorem [22], which were mentioned in. However, it is still not known how to derive the relations (38) from Theorems 4 6 for an arbitrary integer n. 8. Quasi-shuffle products A construction suggested by Hoffman [2] enables one to view each of the algebras H and H as a special case of some general algebraic structure. The present section is devoted to the description of that structure. We consider a non-commutative polynomial algebra A K A graded by degree over a field K C; herea stands for a locally finite set of generators (that is, the set of generators of any given positive degree is finite). As usual, we refer to the elements of the set A as letters, and to the monomials in these letters as words.

14 4 V. V. Zudilin [W. Zudilin] To any word w we assign its length l(w) (the number of letters in the representation) and its weight w (the sum of the degrees of the letters). The unique word of length 0 and weight 0 is the empty word, which is denoted by ; thiswordisthe unit of the algebra A. The neutral (zero) element of the algebra A is denoted by 0. Let us now define the product (by extending it additively to the whole algebra A) by the following rules: for any word w, and w w w (39) a j u a k v a j (u a k v)+a k (a j u v)+[a j,a k ](u v) (40) for any words u, v and letters a j,a k A, where the functional [, ]: Ā Ā Ā (4) (Ā : A {0}) satisfies the following properties: (S0) [a, 0] 0 for any a Ā; (S) [[a j,a k ],a l ][a j, [a k,a l ]] for any a j,a k,a l Ā; (S2) either [a j,a k ]0 or [a k,a j ] a j + a k for any a j,a k A. Then A : (A, ) becomes an associative graded K-algebra and, if the additional property (S3) [a j,a k ][a k,a j ] for any a j,a k Ā holds, then A is a commutative K-algebra ([2], Theorem 2.). If [a j,a k ] 0 for all letters a j,a k A, then(a, ) is the standard shuffle algebra; in the special case A {x 0,x } we obtain the shuffle algebra A H of the multiple zeta values (or of the polylogarithms). The stuffle algebra H corresponds to the choice of the generators A {y j } j and the functional [y j,y k ]y j+k for any integers j andk. Let us equip the algebra A on which a functional (4) is given with the dual product by the rules w w w, ua j va k (u va k )a j +(ua j v)a k +(u v)[a j,a k ] instead of (39) and (40), respectively. Then A : (A, ) is a graded K-algebra as well (which is commutative if the property (S3) holds). Theorem 9. The algebras A and A coincide. Proof. It suffices to prove the relation w w 2 w w 2 (42)

15 Algebraic relations for multiple zeta values 5 just for all words w,w 2 K A. Let us proceed by induction on the quantity l(w )+l(w 2 ). If l(w )0orl(w 2 ) 0, then the relation (42) becomes an obvious identity. If l(w )l(w 2 ), that is, if w a and w 2 a 2 are letters, then a a 2 a a 2 + a 2 a +[a,a 2 ]a a 2. If l(w ) > andl(w 2 ), then, writing w a ua 2 and w 2 a 3 applying the induction hypothesis, we obtain A and a ua 2 a 3 a (ua 2 a 3 )+a 3 a ua 2 +[a,a 3 ]ua 2 a (ua 2 a 3 )+a 3 a ua 2 +[a,a 3 ]ua 2 a ((u a 3 )a 2 + ua 2 a 3 + u[a 2,a 3 ]) + a 3 a ua 2 +[a,a 3 ]ua 2 a ((u a 3 )a 2 + ua 2 a 3 + u[a 2,a 3 ]) + a 3 a ua 2 +[a,a 3 ]ua 2 (a (u a 3 )+a 3 a u +[a,a 3 ]u)a 2 + a ua 2 a 3 + a u[a 2,a 3 ] (a u a 3 )a 2 + a ua 2 a 3 + a u[a 2,a 3 ] (a u a 3 )a 2 + a ua 2 a 3 + a u[a 2,a 3 ] a ua 2 a 3. We can proceed similarly (with more cumbersome manipulations) with the remaining case l(w ) > andl(w 2 ) >. Namely, writing w a ua 2 and w 2 a 3 va 4 and applying the induction hypothesis, we see that a ua 2 a 3 va 4 a (ua 2 a 3 va 4 )+a 3 (a ua 2 va 4 )+[a,a 3 ](ua 2 va 4 ) a (ua 2 a 3 va 4 )+a 3 (a ua 2 va 4 )+[a,a 3 ](ua 2 va 4 ) a ((u a 3 va 4 )a 2 +(ua 2 a 3 v)a 4 +(u a 3 v)[a 2,a 4 ]) + a 3 ((a u va 4 )a 2 +(a ua 2 v)a 4 +(a u v)[a 2,a 4 ]) +[a,a 3 ]((u va 4 )a 2 +(ua 2 v)a 4 +(u v)[a 2,a 4 ]) a ((u a 3 va 4 )a 2 +(ua 2 a 3 v)a 4 +(u a 3 v)[a 2,a 4 ]) + a 3 ((a u va 4 )a 2 +(a ua 2 v)a 4 +(a u v)[a 2,a 4 ]) +[a,a 3 ]((u va 4 )a 2 +(ua 2 v)a 4 +(u v)[a 2,a 4 ]) (a (u a 3 va 4 )+a 3 (a u va 4 )+[a,a 3 ](u va 4 ))a 2 +(a (ua 2 a 3 v)+a 3 (a ua 2 v)+[a,a 3 ](ua 2 v))a 4 +(a (u a 3 v)+a 3 (a u v)+[a,a 3 ](u v))[a 2,a 4 ] (a u a 3 va 4 )a 2 +(a ua 2 a 3 v)a 4 +(a u a 3 v)[a 2,a 4 ] (a u a 3 va 4 )a 2 +(a ua 2 a 3 v)a 4 +(a u a 3 v)[a 2,a 4 ] a ua 2 a 3 va 4. This completes the proof of the theorem. Remark. If the property (S3) holds, then the above proof can be simplified significantly. However, in our opinion it is of importance that the algebras A and A coincide in the most general situation, that is, when the functional (4) satisfies the conditions (S0) (S2). To conclude this section, we prove an auxiliary statement.

16 6 V. V. Zudilin [W. Zudilin] Lemma 4. The following identity holds for each letter a A and any words u, v A: a uv (a u)v u(a v av). (43) Proof. Let us prove the statement by induction on the number of letters in the word u. If the word u is empty, then the identity (43) is evident. Otherwise, let us write u in the form u a u,wherea A and the word u consists of fewer letters, and hence satisfies the identity Then as was to be proved. a u v (a u )v u (a v av). a uv (a u)v a a u v (a a u )v aa u v + a (a u v)+[a, a ]u v (aa u + a (a u )+[a, a ]u )v a (a u v (a u )v) a u (a v av) u(a v av), 9. Functional model of the stuffle algebra A functional model of the stuffle algebra H cannot be described by a perfect analogy with the polylogarithmic model of the shuffle algebra H, because the rule (7) has no differential interpretation, in contrast to (6). Therefore we use a difference interpretation of the rule (7), namely, consider the (simplest) difference operator Df(t) f(t ) f(t). It can readily be seen that D ( f (t)f 2 (t) ) Df (t) f 2 (t)+f (t) Df 2 (t)+df (t) Df 2 (t) (44) and that the inverse map Ig(t) g(t + n) n (such that D(Ig(t)) g(t)) is defined up to an additive constant provided that some additional restrictions are imposed on the function g(t) ast +, for instance, g(t) O(t 2 ). Remark. By [6], 3., the operator D is related to the differential operator d/dt as follows: D e d/dt ( ) n d n n! dt. n n

17 Algebraic relations for multiple zeta values 7 The above equality is justified by a formal application of the Taylor expansion, ( ) n d n f(t ) f(t)+ n! dt n f(t); n in fact, the last formula is valid for any entire function. The exponentiation of derivations (on algebras of words) is discussed in 2 below in connection with a generalization of Theorem. According to (7) and (44), the natural analogy with Lemmas and 2 assumes the existence of functions ω j (t) such that ω j (t)ω k (t) ω j+k (t) for the integers j andk. The simplest choice is given by the formulae ω j (t), j, 2,..., tj and this leads to the functions ( Ri s (t) Ri s,...,s l,s l (t) :I t sl ) Ri s,...,s l (t), Ri (t) :, defined by induction on the length of the multi-index. By definition, we have D Ri uyj (t) t j Ri u(t), (45) which is in a sense a discrete analogue of formula (25). Lemma 5. The following identity holds: Ri s (t) ; (46) (t + n ) s (t + n l ) sl sl (t + n l ) in particular, n > >n l >n l Ri s (0) ζ(s), s Z l, s 2, s 2,..., s l, (47) lim s(t) 0, t + s Z l, s 2, s 2,..., s l. (48) Proof. By definition, ( ) Ri s (t) I Ri s,...,s t sl l (t) ( ) I t sl (t + n ) s (t + n l ) sl n n > >n l (t + n) sl n > >n l (t + n + n) s (t + n l + n) sl (t + n n > >n l >n )s (t + n l )sl (t + n), sl and this implies the required formula (46).

18 8 V. V. Zudilin [W. Zudilin] Let us now define the multiplication on the algebra H (and, in particular, on the subalgebra H 0 )bytherules w w w, (49) uy j vy k (u vy k )y j +(uy j v)y k +(u v)y j+k, instead of (5) and (7). Lemma 6. The map w Ri w (z) is a homomorphism of the algebra (H 0, ) into C([0, + ); R). Proof. We must verify the relation Ri w w 2 (z) Ri w (z)ri w2 (z) for all w,w 2 H 0. (50) We assume without loss of generality that w and w 2 are words of the algebra H 0. Let us prove the relation (50) by induction on the quantity l(w )+l(w 2 ); if w or w 2, then the validity of (50) is evident by (49). Otherwise we write w uy j, w 2 vy k and apply the formulae (44) and (45) and the induction hypothesis, D ( Ri w (t)ri w2 (t) ) D ( Ri uyj (t)ri vyk (t) ) D Ri uyj (t) Ri vyk (t)+ri uyj (t) D Ri vyk (t) + D Ri uyj (t) D Ri vyk (t) t j Ri u(t)ri vyk (t)+ t k Ri uy j (t)ri v (t)+ t j+k Ri u(t)ri v (t) t j Ri u vy k (t)+ t k Ri uy j v(t)+ t j+k Ri u v(t) D ( Ri (u vyk)y j (t)+ri (uyj v)y k (t)+ri (u v)yj+k (t) ) D Ri uyj vy k (t) D Ri w w 2 (t). Therefore Ri w (t)ri w2 (t) Ri w w 2 (t)+c, (5) and, passing to the limit as t tends to +, weobtainc 0 by (48). Thus, relation (5) becomes the required equality (50), and the lemma follows. Proof of Theorem 5. By (47), Theorem 5 follows from Lemma 6 and Theorem Hoffman s homomorphism for the stuffle algebra Another way to prove Theorem 5 (and Lemma 6 as well) uses Hoffman s homomorphism φ: H Q[[t,t 2,...]], where Q[[t,t 2,...]] is the Q-algebra of formal power series in countably many (commuting) variables t,t 2,... (see [] and [3]). Namely, the Q-linear map φ is defined by setting φ() : and φ(y s y s2 y sl ): n >n 2> >n l t s n t s2 n 2 t sl n l, s Z l, s,..., s l.

19 Algebraic relations for multiple zeta values 9 The image of the homomorphism φ (which is in fact a monomorphism) is the algebra QSym of quasi-symmetric functions. Here by a quasi-symmetric function we mean a formal power series (of bounded degree) in t,t 2,... in which the coefficients of n l and t s n t s2 n t sl 2 n coincide whenever n >n 2 > >n l and n >n 2 > l (our definition slightly differs from the corresponding definition in [3] but leads to the same algebra QSym of quasi-symmetric functions). In these terms, the homomorphism w Ri w (t) in Lemma 6 is defined as the restriction of the homomorphism φ to H 0 given by the substitution t n /(t + n), n, 2,... Another approach to the proof of the stuffle relations for multiple zeta values was recently suggested by Cartier (see [28]). Slightly modifying the original scheme of Cartier, we show the main ideas of the approach by the example of proving Euler s identity t s n t s2 n 2 t sl >n l ζ(s )ζ(s 2 )ζ(s + s 2 )+ζ(s,s 2 )+ζ(s 2,s ), s 2, s 2 2. (52) To this end, we need another integral representation (as compared with (3)) for the admissible multi-indices s: l t t 2 t s+ +s ζ(s) j dt dt 2 dt s, l l(s), (53) t t 2 t s+ +s j t t 2 t s+s 2+ +s l [0,] s j which was kindly pointed out to us by Nesterenko and can be proved by straightforward integration of the series t t n. n0 Substituting u t t s, v t s+ t s2 into the elementary identity ( u)( v) uv + u ( u)( uv) + v ( v)( uv) and integrating over the hypercube [0, ] s+s2 in accordance with (53), we arrive at the identity (52).. Derivations As in 8, let us consider a graded non-commutative polynomial algebra A K A over a field K of characteristic 0 with a locally finite set of generators A. By a derivation of the algebra A we mean a linear map δ : A A (of the graded K-vector spaces) satisfying the Leibniz rule δ(uv) δ(u)v + uδ(v) for all u, v A. (54) The commutator of two derivations [δ,δ 2 ]:δ δ 2 δ 2 δ is a derivation, and thus the set of all derivations of the algebra A forms a Lie algebra Der(A) (naturally graded by degree). It can readily be seen that it suffices to define a derivation δ Der(A) juston the generators of A and then to extend it to the whole algebra by linearity and by using (54). The next assertion gives examples of derivations of the algebra A equipped with an additional multiplication having the properties (39) and (40).

20 20 V. V. Zudilin [W. Zudilin] Theorem 0. The map δ a : w aw a w (55) is a derivation for any letter a A. Proof. It is clear that δ a is linear. By Lemma 4, for any words u, v A we have δ a (uv) auv a uv auv (a u)v u(a v av) (δ a u)v + u(δ a v), and thus (55) is really a derivation. By Theorem 0, the maps δ : H H and δ : H H defined by the formulae δ : w x w x w, δ : w y w y w x w x w, (56) are derivations; according to the rule (8), the map δ is a derivation on the whole algebra H. We note that the derivations (56) act on the generators of the algebra according to the rules (5) (8) as follows: δ x 0 x 0 x, δ x x 2, δ x 0 0, δ x x 2 x 0x. (57) For any derivation δ of the algebra H (or of the subalgebra H 0 ) we define the dual derivation δ τδτ,whereτ is the anti-automorphism of the algebra H (and H 0 ) introduced in 6. A derivation δ is said to be symmetric if δ δ and anti-symmetric if δ δ. Sinceτx 0 x, an (anti-)symmetric derivation δ is uniquely determined by the image of one of the generators x 0 or x, whereas an arbitrary derivation can be reconstructed just from the images of both generators. We now define the derivation D of the algebra H by setting Dx 0 0andDx x 0 x (that is, Dy s y s+ for the generators y s of the algebra H ) and represent the statement of Theorem in the following form. Theorem (derivation theorem, [3], Theorem 2.). The identity ζ(dw) ζ(dw) (58) holds for any word w H 0. Proof. Expressing any word w H 0 in the form w y s y s2 y sl (with s > ), we note that the left-hand side of the equality (7) corresponds to the element Dw D(y s y s2 y sl )y s+y s2 y sl +y s y s2+y s3 y sl + +y s y sl y sl+ (59) of the algebra H 0. On the other hand, Dw τd ( x 0 x sl x 0 x sl x 0 x s2 x 0 x s l s k 2 τ x 0 x sl x 0 x sk+ x 0 x j x 0x sk j x 0 x sk x 0 x s k s k 2 l k s k 2 j0 s k 2 j0 ) x s 0 x x sk 0 x x sk j 0 x x j 0 x x sk+ 0 x x sl 0 x, (60)

21 Algebraic relations for multiple zeta values 2 which corresponds to the right-hand side of (7). Applying the map ζ to the resulting equalities (59) and (60), we obtain the required identity (58). Remark. The condition w H 0 in Theorem cannot be weakened. The equality (58) fails for the word w x : ζ(dx )ζ(x 0 x ) 0ζ(Dx ). Proof of Theorem 6. Comparing the action (57) of the derivations (56) with the action of D and D on the generators of the algebra H, Dx 0 0, Dx x 0 x, Dx 0 x 0 x, Dx 0, we see that δ δ D D. Therefore, application of Theorem to the word w H 0 leads to the required equality ζ(x w x w) ζ ( (δ δ )w ) ζ ( (D D)w ) ζ(dw) ζ(dw) 0. This completes the proof. Remark. Another proof of Theorem 6, based on the shuffle and stuffle relations for so-called coloured polylogarithms Li s (z) Li (s,s 2,...,s l)(z,z 2,...,z l ): n >n 2> >n l z n zn2 2 znl l, (6) n s ns2 2 nsl l can be found in [28]. (It is clear that the specialization z 2 z l transforms the functions (6) to the generalized polylogarithms (22).) We do not intend to present the properties of the functional model (6) in this survey and refer the interested reader to [4], [7], and [28]. 2. Ihara Kaneko derivations and Ohno s relations Theorem has a natural generalization. For any n we define an antisymmetric derivation n Der(H) bytherule n x 0 x 0 (x 0 + x ) n x. As was mentioned in the proof of Theorem 6, we have D D δ δ. The following assertion holds. Theorem 2 [4] (see also [3]). The identity holds for any n and any word w H 0. ζ( n w) 0 (62) Below we sketch the proof of Theorem 2 presented in the preprint [4] (the proof in [3] uses other ideas). The following result, which was proved in [2] by the generating-function method, contains Theorems, 3, and 7 as special cases (the corresponding implications are also given in [2]).

22 22 V. V. Zudilin [W. Zudilin] Theorem 3 (Ohno s relations). Let a word w H 0 and its dual w τw H 0 have the following representations in terms of the generators of the algebra H : w y s y s2 y sl, w y s y s 2 y s k. Then the identity ζ(y s+e y s2+e 2 y sl+e l ) e,e 2,...,e l 0 e +e 2+ +e ln e,e 2,...,e k 0 e +e 2+ +e kn ζ(y s +e y s 2 +e 2 y s k +e k ) holds for any integer n 0. Following [4], we define the derivation D n Der(H) for each integer n by setting D n x 0 0andD n x x n 0 x. One can readily see that the derivations D,D 2,... commute; this fact holds for the dual derivations D, D 2,... as well. Let us consider the completion of H, namely, the algebra Ĥ Q x 0,x of formal power series in non-commuting variables x 0,x over the field Q. The action of the anti-automorphism τ and of the derivations δ Der(H) can be extended naturally to the whole algebra Ĥ. For simplicity, let us write w ker ζ if all homogeneous components of the element w Ĥ belong to ker ζ. The maps D n D n n, D n D n n are derivations of the algebra Ĥ, and it follows from the standard relation between the derivations and homomorphisms that the maps σ exp(d), σ τστ exp(d) are automorphisms of the algebra Ĥ. In these terms, Ohno s relations can be stated as follows. Theorem 4 [4]. The inclusion holds for any word w H 0. Proof. SinceDx 0 0and Dx (σ σ)w ker ζ (63) ( ) x 0 + x x x ( log( x 0 ))x, it follows that D n x 0 0andD n x ( log( x 0 )) n x, and hence σx 0 x 0 and σx n0 n! ( log( x 0)) n x ( x 0 ) x (+x 0 + x x3 0 + )x.

23 Algebraic relations for multiple zeta values 23 Therefore, for the word w y s y s2 y sl H 0 we have σw σ(x s 0 x x s2 0 x x sl 0 x ) x0 s ( + x 0 + x )x x0 s2 ( + x 0 + x )x x sl 0 ( + x 0 + x )x n0 e,e 2,...,e l 0 e +e 2+ +e ln x s +e 0 x x s2 +e2 0 x x sl +el 0 x ; thus, σw στw ker ζ by Theorem 3. Applying now Theorem 7, we obtain the desired inclusion (63). Let us return to the derivations, 2,... and consider the derivation n Lemma 7. The following equality holds: n n Der(Ĥ). exp( ) σ σ. (64) Proof. We first note that the operators n, n, 2,..., commute. Indeed, since n (x 0 + x ) 0 for any n, it suffices to prove the equality n m x 0 m n x 0 for n, m. Since n (x 0 + x ) k 0foranyn andk 0, we see that n m x 0 n (x 0 (x 0 + x ) m x ) x 0 (x 0 + x ) n x (x 0 + x ) m x x 0 (x 0 + x ) m x 0 (x 0 + x ) n x x 0 (x 0 + x ) n (x 0 + x x 0 )(x 0 + x ) m x x 0 (x 0 + x ) m (x 0 + x x )(x 0 + x ) n x x 0 (x 0 + x ) n x 0 (x 0 + x ) m x + x 0 (x 0 + x ) m x (x 0 + x ) n x m n x 0, as was to be proved. Let us consider the family ϕ(t), t R, of automorphisms of the algebra ĤR R x 0,x that are defined on the generators x 0 x 0 + x and x by the rules ( ϕ(t): x 0 x 0, ϕ(t): x ( x 0 )t x ( ) x 0 )t x, t R. A routine verification [4] shows that d ϕ(t )ϕ(t 2 )ϕ(t + t 2 ), ϕ(0) id, dt ϕ(t) t0, ϕ() σ σ ; hence ϕ(t) exp(t ), and the substitution t leads to the required result (64). x 0

24 24 V. V. Zudilin [W. Zudilin] ProofofTheorem2. Let us show how Theorem 2 follows from Theorem 4 and Lemma 7. On the one hand, we have log(σ σ ) log( (σ σ)σ ) (σ σ) and on the other hand ((σ σ)σ ) n σ, n n σ σ ( σ σ )σ ( exp( ))σ n n σ; n! hence, H 0 (σ σ)h 0, and Theorem 4 yields the required identities (62). Does there exist a simpler way to prove relations (62)? The explicit computations in [4] show that δ δ, 2 [δ, δ ], 3 2 [δ, [, δ ]] 2 [δ, 2 ] 2 [δ, 2 ], 4 6 [δ, [, [, δ ]]] 6 [δ, [δ, [, δ ]]] + 6 [, [ 2, δ ]] + 3 [ 3,δ ]+ 3 [ 3, δ ], and, moreover, δ + δ δ + δ ; therefore, the cases n, 2, 3, 4 in Theorem 2 can be processed by induction (with Theorem as the base of induction). This motivates the following conjecture. Conjecture 3 [4]. For any n the above anti-symmetric derivation n is contained in the Lie subalgebra of Der(H) generated by the derivations δ, δ, δ, and δ. We also note that the preprint [4] contains some other ideas (as compared with Conjecture 2) about a complete description of the identities for multiple zeta values in terms of regularized shuffle-stuffle relations. 3. Open questions Along with the above Conjectures 3, we also mention some other important conjectures concerning the structure of the subspace ker ζ H. We denote by Z k the Q-vector subspace in R spanned by the multiple zeta values of weight k and we set Z 0 Q and Z {0}. Then the Q-subspace Z R spanned by all multiple zeta values is a subalgebra of R over Q graded by weight. Conjecture 4 ([8], [28]). When regarded as a Q-algebra, the algebra Z is the direct sum of the subspaces Z k, k 0,, 2,... We can readily see that the relations (9) (2) for multiple zeta values are homogeneous with respect to weight, and hence Conjecture 4 follows from Conjecture 2. Let d k be the dimension of the Q-space Z k, k 0,, 2,... Wenotethatd 0, d 0,d 2 (sinceζ(2) 0),d 3 (sinceζ(3) ζ(2, ) 0),andd 4 (since Z 4 Qπ 4 by (32), (34), and (36)). For k 5 the above identities enable one to write down upper bounds; for instance, d 5 2, d 6 2, and so on.

25 Algebraic relations for multiple zeta values 25 Conjecture 5 [30]. For k 3 we have the recurrence relations in other words, k0 d k d k 2 + d k 3 ; d k t k t 2 t. 3 Even if the answers to Conjectures 4 and 5 are positive, the question of choosing a transcendence basis of the algebra Z and (or) a rational basis of the Q-spaces Z k, k 0,, 2,..., would be still open. In this connection, the following conjecture of Hoffman is of interest. Conjecture 6 []. For any k 0,, 2,... the number set { ζ(s) : s k, sj {2, 3}, j,...,l(s) } (65) is a basis of the Q-space Z k. Not only the experimental confirmation for k 6 (under the assumption that Conjecture 2 is true) but also the coincidence of the dimension of the Q-space spanned by the numbers (65) with the dimension d k of the space Z k in Conjecture 5 (the last fact was proved by Hoffman in []) is an argument in favour of Conjecture q-analogues of multiple zeta values Thirty-three years after Gauss s work on hypergeometric series, Heine [9] considered series depending on an additional parameter q and possessing properties similar to those of Gauss s series. Moreover, as q tends to (at least termwise), the Heine q-series become hypergeometric series, and thus Gauss s results can be obtained from the corresponding results for q-series by this passage to the limit and the theorem on analytic continuation. Similar q-extensions of classical objects are possible not just in analysis; we refer the interested reader to Hoffman s paper [2], in which a possible q-deformation of the stuffle algebra H is discussed. The objective of the present section is to discuss problems of q-extension for multiple zeta values. The simplest (and rather obvious) way is as follows: for positive integers s, s 2,..., s l we set ζq (x s )ζq (s) ζq (s,s 2,...,s l ) q ns+n2s2+ +nlsl :, q <, (66) ( q n ) s ( q n2 ) s2 ( q nl ) sl n >n 2> >n l and additively extend the Q-linear map ζq to the whole algebra H. An easy verification shows that, if s >, then lim ( q) s ζq (s) ζ(s), q 0<q<

26 26 V. V. Zudilin [W. Zudilin] that is, the series in (66) are really q-extensions of the series in (4). Moreover, ζq is a (q-parametric) homomorphism of the stuffle algebra H ; to prove this fact, it suffices to consider the specialization t n q n /( q n ) of the Hoffman homomorphism φ defined in 0. Hence, ζq (w w 2 )ζq (w )ζq (w 2 ) for all w,w 2 H. This model of multiple q-zeta values (and also of generalized q-polylogarithms) is described in [23]; the main defect of the model is the absence of any description of other linear and polynomial relations over Q, in other words, the absence of a suitable q-shuffle product. Another way to q-extend (non-multiple) zeta values was suggested simultaneously and independently in [5] and [34], ζ q (s) σ s (n)q n n n n n s q n, s, 2,..., (67) qn where σ s (n) d n ds stands for the sum of powers of the divisors; the limit relations lim ( q) s ζ q (s) (s )! ζ(s), s 2, 3,..., q 0<q< are also proved in these papers. The q-zeta values (67) can readily be recalculated in terms of (66) with l,namely, q n ζ q () q n, ζ q n q(2) ( q n ) 2, ζ q n ( + q n ) q(3) ( q n ) 3, ζ q (4) n n q n ( + 4q n + q 2n ) ( q n ) 4, ζ q (5) n n q n ( + q n +q 2n + q 3n ) ( q n ) 5, and, generally, q n ρ k (q n ) ζ q (k) ( q n, ) k n k, 2, 3,..., where the polynomials ρ k (x) Z[x] are determined recursively by the formulae ρ, ρ k+ (+(k )x)ρ k + x( x)ρ k for k, 2,... (see [34]). If s 2 is even, then the series E s (q) 2sζ q (s)/b s, where the Bernoulli numbers B s Q are already defined in (3), are known as the Eisenstein series. This fact enables one to prove the coincidence of the rings Q[q, ζ q (2),ζ q (4),ζ q (6), ζ q (8),ζ q (0),...] and Q[q, ζ q (2),ζ q (4),ζ q (6)] (cf. the corresponding result in for ordinary zeta values). However, the problem of constructing a model of multiple q-zeta values that includes the ordinary multiplicity-free model (67) remains open. The natural requirement concerning such a model is the existence of q- analogues of the shuffle and stuffle product relations. In conclusion we present a possible q-extension of Euler s formula (5) for the quantity ζ q (2, ) n >n 2 q n ( q n ) 2 ( q n2 ).

27 Algebraic relations for multiple zeta values 27 Theorem 5. The following identity holds: 2ζ q (2, ) ζ q (3). Proof. As in the proof of Theorem, we use the method of partial fractions, namely, the expansion s ( u)( uv) s ( v) s ( u) v, s, 2, 3,... ( v) j+ ( uv) s j j0 (68) This identity can be proved in the same way as (9), by summing the geometric progression on the right-hand side. For s 2 we multiply the identity (68) by u( + v), u( + v) ( u)( uv) 2 u( + v) ( v) 2 ( u) uv( + v) ( v)( uv) 2 uv( + v) ( v) 2 ( uv), set u q m and v q n, and sum over all positive integers m and n. This results in an equality with left-hand side the double sum m n q m ( + q n ) ( q m )( q n+m ) 2 and right-hand side the double sum ( n m n m n m q n ( + q m ) ( q n )( q n+m ) 2 q m ( + q n ) ( q n ) 2 ( q m ) q n+m ( + q n ) ( q n )( q n+m ) 2 q n+m ( + q n ) ( q n ) 2 ( q n+m ) +q n ( ) q m ( q n ) 2 q m qn+m q n+m n n m Carrying the last sum to the left-hand side, we obtain q n ( + q m )+q n+m ( + q n ) ( q n )( q n+m ) 2 n m +q n ( ) q m ( q n ) 2 q m qn+m +q n q n+m ( q n ) 2 n +q n ( q n ) 2 m ( q n n q n + m q m q m n ) ζ q (3) + n>m ) q n+m ( + q n ) ( q n )( q n+m ) 2. n m q m q m ( + q n )q m ( q n ) 2 ( q m ). (69) On the other hand, the left-hand side of the last equality can be represented in the form (n + m l) n ln+ q n +2q l + q l+n ( q n )( q l ) 2 l>n q n +2q l + q l+n ( q l ) 2 ( q n ), (70)

28 28 V. V. Zudilin [W. Zudilin] and hence, setting n n and n 2 m on the right-hand side of (69) and n l and n 2 n in (70), we finally obtain the desired identity ζ q (3) n >n 2 n >n 2 q n2 +2q n + q n+n2 ( q n ) 2 ( q n2 ) 2q n ( q n ) 2 ( q n2 ). n >n 2 ( + q n )q n2 ( q n ) 2 ( q n2 ) Bibliography [] R. Apéry, Irrationalitédeζ(2) et ζ(3), Astérisque 6 (979), 3. [2] J. Borwein and D. Bradley, Empirically determined Apéry-like formulae for ζ(4n +3), Experiment. Math. 6:3 (997), [3] J. M. Borwein, D. M. Bradley, and D. J. Broadhurst, Evaluations of k-fold Euler/Zagier sums: A compendium of results for arbitrary k, Electron. J. Combin. 4:2 (997), #R5; printed version in J. Combin. 4:2 (997), [4] J. M. Borwein, D. M. Bradley, D. J. Broadhurst, and P. Lisoněk, Special values of multiple polylogarithms, Trans. Amer. Math. Soc. 353 (200), [5] L. Euler, Meditationes circa singulare serierum genus, Novi Comm. Acad. Sci. Petropol. 20 (775), 40 86; reprinted in Opera Omnia Ser. I, vol. 5, Teubner, Berlin 927, pp [6] A. O. Gel fond, Calculus of finite differences, Nauka, Moscow 967; English transl., (Intern. Monographs Adv. Math. Phys.) Hindustan Publ. Corp., Delhi 97. [7] A. B. Goncharov, Polylogarithms in arithmetic and geometry, Proceedings of the International Congress of Mathematicians (ICM 94, Zürich, August 3, 994), vol. I (S. D. Chatterji, ed.), Birkhäuser, Basel 995, pp [8] A. B. Goncharov, The double logarithm and Manin s complex for modular curves, Math. Res. Lett. 4 (997), [9] E.Heine, Über die Reihe..., J. Reine Angew. Math. 32 (846), [0] M. E. Hoffman, Multiple harmonic series, Pacific J. Math. 52 (992), [] M. E. Hoffman, The algebra of multiple harmonic series, J. Algebra 94 (997), [2] M. E. Hoffman, Quasi-shuffle products, J. Algebraic Combin. (2000), [3] M. E. Hoffman and Y. Ohno, Relations of multiple zeta values and their algebraic expression, E-print math.qa/ [4] K. Ihara and M. Kaneko, Derivation relations and regularized double shuffle relations of multiple zeta values, preprint [5] M. Kaneko, N. Kurokawa, and M. Wakayama, A variation of Euler s approach to values of the Riemann zeta function, E-print math.qa/ [6] M. Koecher, Letter to the editor, Math. Intelligencer 2:2 (979/980), [7] F. Lindemann, Über die Zahl π, Math. Ann. 20 (882), [8] H. M. Minh, G. Jacob, M. Petitot, and N. E. Oussous, Aspects combinatoires des polylogarithmes et des sommes d Euler Zagier, Sém. Lothar. Combin. 43: Art. B43e (electronic) (999); slc/wpapers/s43minh.html. [9] H. M. Minh and M. Petitot, Lyndon words, polylogarithms and the Riemann ζ function, Formal Power Series and Algebraic Combinatorics (Vienna, 997), Discrete Math. 27 (2000), [20] H. M. Minh, M. Petitot, and J. van der Hoeven, Shuffle algebra and polylogarithms, Formal Power Series and Algebraic Combinatorics (Toronto, ON, June 998), Discrete Math. 225 (2000), [2] Y. Ohno, A generalization of the duality and sum formulas on the multiple zeta values, J. Number Theory 74 (999), [22] T. Rivoal, La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs, C. R. Acad. Sci. Paris Sér. IMath. 33 (2000),