Functional Equations for Higher Logarithms

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Sel. math., New ser. 9 (2003) 361 – 377 c Birkh¨auser Verlag, Basel, 2003 1022–1824/03/030361-17 ° DOI 10.1007/s00029-003-0312-z Selecta Mathematica, New Series

Functional equations for higher logarithms

Herbert Gangl

Abstract. Following earlier work by Abel and others, Kummer gave in 1840 functional equations for the polylogarithm function Lim(z)uptom= 5, but no example for larger m was known until recently. We give the ﬁrst genuine 2-variable functional equation for the 7-logarithm. We investigate and relate identities for the 3-logarithm given by Goncharov and Wojtkowiak and deduce a certain family of functional equations for the 4-logarithm.

Mathematics Subject Classiﬁcation (2000). 33E20 (33B99, 11G55, 19F27).

Key words. Polylogarithms, functional equations, algebraic K-theory.

1. Introduction

1.1. An essential property of the logarithm is its functional equation log(xy) = log(x) + log(y). An essential property of the dilogarithm, which is deﬁned by the power series

∞ zn Li (z)= , z < 1, 2 n2 | | n=1 X and can be analytically continued to C [1, ), is the 5-term relation note the 1 x 1−y ∞ involutary symmetry (x, y) ( 1 −y 1 , 1 −x 1 ) 7→ − − − − ¡ 1 x 1 y Li (xy) Li (x) Li (y) Li ¢ Li = (elementary), 2 2 2 2 − 1 2 − 1 − − − 1 y− − 1 x− ³ − ´ ³ − ´ which was found—in a lot of diﬀerent forms—by Spence, Abel and many others (cf. [6], Chapter I). Here the “elementary” right-hand side consists of a sum of products of logarithms. In 1840, Kummer [5] found functional equations for higher (poly)logarithms

∞ zn Li (z)= , z < 1, m nm | | n=1 X 362 H. Gangl Sel. math., New ser.

(which can be likewise analytically continued) up to m 5. His method does not extend to higher m, though; for a more detailed statement≤ we refer the reader to Wechsung’s paper [10], where more functional equations in Kummer’s spirit are derived and where it is shown that there are no such “Kummer-type” equations for m>5. Other efforts in the direction of finding new equations, even with the help of a computer, seemed also to be restricted to m 5 (cf., e.g., [8], [9], where a number of new 1-variable equations in that range were≤ given). In our thesis [2] we gave the first (non-trivial) functional equations for m =6 (in fact, a whole family of equations in two variables) and for m = 7, as well as many new examples for m 5 (some of those results had already been included in [14] and in Chapter 16 of [7]).≤ The main tool in our computer-aided investigation was Zagier’s criterion for functional equations of the associated one-valued versions of the m-logarithm (cf. [14], Proposition 1 and Proposition 3). The relation of polylogarithms and algebraic K-theory, and in particular Za- gier’s conjecture on special values of Dedekind zeta functions, have made it clear that the question of finding functional equations for higher polylogarithms, as well as to relate them to each other, is a central one (cf., e.g., [15]). For m = 3, Gon- charov [4] found a basic functional equation as the key step in his proof of Zagier’s conjecture for this case, while Wojtkowiak [12] gave a whole family of relations. The relationship between these equations had not been clarified so far. The contents of the paper are as follows: in 2, we recall a general functional equation for the dilogarithm, which is found to§ have some “companion” for the trilogarithm (Proposition 3.1) which in turn specializes to Wojtkowiak’s trilogarithm equations (Corollary 3.2). We restate (in 3.1) Goncharov’s equation in increasingly symmetric ways and relate it (in 3.2 and§ 3.3) to Wojtkowiak’s equations—the combining link being a 34-term equation,§ in§ fact a very special case of Wojtkowiak’s equation which turns out to be equivalent to Goncharov’s one. In 4, we state a family of 4-logarithm equations, a uniform and rather simple proof of§ which is given at the end of the paper ( 6). Finally, in 5, we present our “high- score result” (obtained in 1992): a functional§ equation for§ the 7-logarithm in two variables (with 274 terms).

2. A general dilogarithm equation

2.1. Recall that the function Li (z) is a many-valued function on C 0,1 but m −{ } that one has the one-valued continuous function : C R given by Lm → m r 2 Br r m(z)= m log z Lim r(z) , z 1,z 0, 1 , L < r! | | − | |≤ 6∈ { } r=0 µX ¶ where denotes the real part for m odd and the imaginary part for m even, and

Br the r-th Bernoulli number (B0 =1,B1 = 1/2,B2 =1/6,...). For z > 1, − m 1 | | m(z) is given by the functional equation m(z)=( 1) − m(1/z), while for zL 0,1, one extends the function by continuity.L − L ∈{ ∞} For m = 2, this is the famous Bloch–Wigner dilogarithm (cf., e.g., [1]). For m>2, one-valued versions of Lim were introduced by Ramakrishnan, Wojtkowiak and Zagier. The above function (z) was introduced by Zagier [14] who denoted Lm it Pm(z). It has the advantage that the corresponding functional equations of Lim become “clean” for m, e.g., in the 5-term relation stated above the elementary term on the right-handL side disappears when we replace Li by . 2 L2 Note that we can consider as a function on C[C], the ﬁnite formal C-linear Lm combinations of elements in C, by extending it linearly.

2.2. There is a general functional equation for the dilogarithm found by Rogers —for a polynomial φ without constant term—and generalized by Wojtkowiak [12] and Zagier [16] to any rational function φ (cf. also Wechsung [10], [11]). For m N put ∈

x z y w (x, y, z, w)= cr(x, y, z, w) = − − , Lm Lm Lm x w y z µ − − ¶ ¡ ¢ g where cr denotes the cross ratio.

1 1 1 Theorem 2.1. Let φ : PC PC be a rational function and α, B, C, D P (C). Then, denoting deg(φ) the degree→ of φ and putting A = φ(α), we obtain ∈

(α, β, γ, δ) = deg(φ) (A, B, C, D), L2 · L2 β,γ,δX f f where β, γ and δ run through the preimages under φ of B, C and D, respectively, with multiplicities.

The theorem contains as a special case the 5-term relation (e.g., if we take φ(z)=z(1 z) and B =1,C=0,D= ) and in fact many other known equations. A functional− equation which is not∞ covered by the theorem (and presumably the only one, essentially) is the one relating z andz ¯, its complex conjugate, via 2(¯z)= 2(z). We expect that the latter, together with the equation in the Labove theorem,−L generates all functional equations for the dilogarithm. In fact, a result by Wojtkowiak [13] states that all functional equations for the dilogarithm with arguments C-rational expressions in ﬁnitely many variables can be written as a sum of 5-term relations. 364 H. Gangl Sel. math., New ser.

3. Trilogarithm equations

The following equation for the trilogarithm was found by symmetrizing a functional equation given by Wojtkowiak [12].

1 1 1 Theorem 3.1. Let φ : PC PC be a rational function, Ai,Bj,Ck,Dl P (C), i, j, k, l 1,2 . Then, denoting→ deg(φ) the degree of φ, we have ∈ ∈{ } ( 1)i+j+k+l (α ,β ,γ ,δ ) deg(φ) (A ,B ,C ,D ) =0, − L3 i j k l − · L3 i j k l i,j,k,lX µ αi,βXj ,γk,δl ¶ f f where αi, βj, γk and δl run through the preimages of Ai, Bj, Ck and Dl, respectively, with multiplicities.

Wojtkowiak’s original equation [12], (10.7.2.2), (a related equation had been found earlier by Wechsung, cf. [11], 4) is obtained by specializing the equation from the above theorem and can be stated§ in a simpler form as follows:

1 1 1 Corollary 3.2. Let A, B, C P (C), φ : PC PC a rational function of degree n ∈ 1 → n 1 n and x an independent variable. Let φ− (A)= αi i=1, φ− (B)= βi i=1 and 1 n { } { } φ− (C)= γ . Then { i}i=1 n (φ(x),C,B,A) (x, γ ,β ,α ) L3 − L3 i j k i,j,kX=1 f n f n (x, α ,α ,γ ) (x, β ,β ,γ ) − L3 i j k − L3 i j k i,j,kX=1 i,j,kX=1 n f n f + (x, α ,α ,β )+ (x, β ,β ,α ) L3 i j k L3 i j k i,j,kX=1 i,j,kX=1 = const, f f i.e., the expression on the left-hand side is independent of x.

Here it is understood that αi, βj and γk run through all preimages (counted with multiplicity) of A, B and C, respectively.

3.1. Around Goncharov’s equation

3.1.1. The original description. Goncharov [4] found a beautiful interpretation of certain functional equations in terms of conﬁguration spaces, and as a crucial by-product he provided an equation for the trilogarithm in three variables αi for which he gave a threefold symmetry. We reproduce it here using the shorthand Vol. 9 (2003) Functional equations for higher logarithms 365

βi =1 αi+αiαi 1 (i=1,2,3), where indices are understood modulo 3: form − − the formal linear combination γ(α ,α ,α ) Z[Q(α ,α ,α )] given by 1 2 3 ∈ 1 2 3 3 1 αiαi 1 βi βiαi+1 γ(α ,α ,α )= +[β]+ − + + 1 2 3 α i β β α − β i=1 i i i+1 i+2 i+1 X³h i h i h i h i´ 1 3 β β + i + i + [1] . (3.1) − α α α − α β α α 1 2 3 i=1 i 1 i+1 i i 1 h i X ³h − i h − i ´ Proposition 3.3. With the above notation, we have γ(α ,α ,α ) =0. L3 1 2 3 ¡ ¢ Since there are 22 non-constant terms occurring in γ(α1,α2,α3), we will refer to it as 22-term (or Goncharov’s) relation.

3.1.2. A more symmetric description. Since Goncharov’s equation plays such a central role for the theory, it seems worthwhile to analyze its structure a bit further. There is actually a much bigger symmetry group G (of order 192) than the cyclic group on three letters acting on the set of arguments and dividing the 22 non-constant terms into 2 orbits, one of length 16 (corresponding to the 15 terms in the ﬁrst sum in (3.1) plus the single term 1 ), the other one of length α1α2α3 6 (corresponding to the second sum, with the exclusion− of the constant terms [1]). G is generated by two involutions £ ¤

β1 1 1 1 π1 :(α1,α2,α3) α1,α2, ,π2:(α1,α2,α3) , , , 7→ −α1β3 7→ α1 α3 α2 ³ ´ ³ ´ together with the obvious symmetry of order 3 (shifting the indices mod 3). The set consisting of the arguments in the sixteen terms of the ﬁrst orbit is transformed either into itself (e.g., via π1) or into the set of inverses (e.g., via π2). We give a presentation with more obvious symmetries: let t1,...,t4 be four variables, subject to the constraint i ti = 1. Then the following element in Q(t1,t2,t3) is annihilated 1 by 3 for each (meaningful) evaluation of t1,t2,t3 in C with t4 =(t1t2t3)− : L Q 1 ti 1 1 ¡(1 ti)(1 tj) ¢ [ti]+ − [ti tj] − − 3[1]. 1 t 1 − 4 − 8 (1 t 1)(1 t 1) − i,j j− i,j i,j,k,l = k− l− i · ¸ { } · ¸ i=j − i=j 1,2,3,4 − − X X6 X6 { X }

The obvious 4-action on the set of arguments (where we identify [z] and [1/z]), together withS the involution (cf. 1.1) § 1 ti ti − 1 (i mod 4) 7→ 1 t− − i+2 generates the symmetry group G (of order 192) mentioned above. 366 H. Gangl Sel. math., New ser.

3.1.3. A yet more symmetric description. Consider the ﬁnite group G0 (of order 96) of automorphisms of Q(y1,y2,y3,z1,z2,z3), generated by G0 = g, h , where h i 1 g :(y ,y ,y ,z ,z ,z ) ,z ,z ,z ,y ,y , (3.2) 1 2 3 1 2 3 7→ y 2 3 1 2 3 µ 1 ¶ h :(y ,y ,y ,z ,z ,z ) (y ,y ,y ,z ,z ,z ). (3.3) 1 2 3 1 2 3 7→ 2 3 1 2 3 1 y z y z y z The orbits of y and of 1 − 3 2 − 1 3 − 2 under G have order 12 and 1 1 y z 1 y z 1 y z 0 − 1 2 − 2 3 − 3 1 32, respectively. As in the previous subsection, we introduce “parametrization 4 variables” t1,...,t4, subject to the constraint j=1 tj = 1, and form 6 arguments 1 1 1 t− Qj 1 tk− Ai i=1,2,3 = tit4 i, Bi i=1,2,3 = − − i, j, k = 1, 2, 3 . { } { } { } 1 ti 1 t4 { } { } n − − ¯ o The involutory automorphism induced by ¯ ¯ (t ,t ,t ,t ) ι(t ,t ),ι(t ,t ),ι(t ,t ),ι(t ,t ) , 1 2 3 4 7→ 4 1 3 2 2 3 1 4 1 x where ι(x, y)= , acts on¡ (B ,B ,B ,A ,A ,A ) like the¢ automorphism g − 1 1 2 3 1 2 3 1 y− in (3.2) above while− the automorphism induced by (t ,t ,t ) (t ,t ,t ) acts like 1 2 3 7→ 2 3 1 h in (3.3). If we put yi and zi equal to Ai and Bi, respectively, in the above, then y z y z y z the G -orbit of the expression 1 − 3 2 − 1 3 − 2 contains only 16 diﬀerent 0 1 y z 1 y z 1 y z − 1 2 − 2 3 − 3 1 terms up to inversion of the arguments—obviously, the G0-orbit of y1 contains only 6 terms up to inversion—and the resulting 16+6 arguments coincide precisely with the ones occurring in Goncharov’s equation.

3.1.4. Other descriptions. A diﬀerent way to use the above parametrization with Ai and Bi in order to exhibit the symmetries among the 22 terms is obtained by putting α = √A , β = √B . Then the four elements αε1 αε2 αε3 i − i i i 1 2 3 with ε 1 (j=1,2,3) and 3 ε = 1, together with the twelve elements j ∈{± } j=1 j − αεi βεi+1 βεi+2 (i mod 3) with ε 1 and ε = 1, form the 16 arguments in i i+1 i+2 j Q∈{± } j j the second orbit above in 3.1.3, while the α2 and β2 give the 6 arguments in the iQ i ﬁrst orbit. § Yet another description, in terms of equations instead of parametrizations, is x y given by the following 9 (redundant) equations where we put q(x, y)= − : 1 xy − 1 q(Bi 1,Bi+1) Bi− = − , q(Ai+1,Ai 1) − 1 1 1 Ai− =q Ai+1, q Ai 1, , Bi 1 − Bi+1 µ − ¶ µ ¶ Vol. 9 (2003) Functional equations for higher logarithms 367

A (1 A )2 B (1 B )2 i+1 − i = i+1 − i , A (1 A )2 B (1 B )2 i − i+1 i − i+1 where i =1,2,3 and the indices are taken modulo 3. (Obviously, we can break the symmetry and reduce this to a system of 7 equations in only 4 variables by eliminating A1 and B1,say.)

3.2. Relating Goncharov’s and Wojtkowiak’s equations

In order to compare the equations resulting from Goncharov’s and Wojtkowiak’s approach, we propose to study an “intermediate” relation, with 34 terms, which allows an interpretation for both situations. 1. In Wojtkowiak’s equation, the “intermediate” relation occurs when we consider the rational function x a x b φ(x)= − − =cr(x, xbc, b, abc) x c 1 ·x abc − − − (for x = 1 this is a Bi-type term above) and (A, B, C)=( ,0,1), and subtracting two different specializations for x of the ensuing terms from∞ Corollary 3.2—each specialization providing 17 terms—we are left with 34 terms which form a functional equation in three variables. The generic 17 terms are given, in factored form, as the arguments of the following formal linear combination in Z[F ], F = Q(a, b, c, t): (1 ct)a (1 ct)b 1 ct 1 ct f(a, b, c, t)= − + − + − + − a t b t c(a t) c(b t) − − − − h abc it h abc it h abc it h abc ti + − + − + − + − (a t)bc (b t)ac a t b t − − − − (a t)b(ac 1)h (t ia)(1h bc) i(tc h 1)b(aci 1)h (tci 1)a(bc 1) + − − + − − + − − + − − (b t)a(bc 1) (t b)(1 ac) (abc t)(bc 1) (abc t)(ac 1) − − − − − − − − h (1 ct)abc i h (1 ct) it ha b(a t) i h(b t)(a t)c i − − − − − − . − abc t − (abc t)c − t b − a(b t) − (abc t)(1 ct) h − i h − i h − i h − i h − − i Definition 3.4. The 34-term relation is given by the difference

f(a, b, c, t) f(a, b, c, u) Z[F ],F=Q(a, b, c, t, u). − ∈ Remark 3.5. From Corollary 3.2 it results that vanishes on 34-term relations. L3 2. Let us recall that Goncharov [4] has deﬁned a projective invariant for 6 points 2 in the projective plane PF over a ﬁeld F , their triple ratio with values in a quotient of Z[F ], in such a way that it respects a 7-term relation: for any 7 distinct points 368 H. Gangl Sel. math., New ser.

2 P1,...,P7 in PF one puts 7 ( 1)i triple ratio(P ,...,P ,...,P )=0. − · 1 i 7 i=1 X In the setting of configurations, the 34-term relationb mentioned above implies the well-definedness of the triple ratio associated to a configuration of six points Pi (i =1,...,6) in general position in the plane as follows. Goncharov reduces such a configuration with the help of the 7-term relation above to more degenerate configurations by introducing the intersection point Q of the line through the points P1,P2 with the line passing through P3 and P4. He uses the fact that he has already defined the triple ratio for more degenerate configurations which result by leaving out any of the other 6 original points Pi. But there are different possibilities for choosing Q, depending on the ordering of the Pi, e.g., one could switch the roles of P2 and P3, and one verifies (this is not explicitly done in [4]) that the difference of two such possibilities, if non-zero, essentially results in two sets of 17 terms which correspond precisely to the 34 terms in question. (Here “essentially” alludes to the fact that we argue modulo two formal linear combinations arising from simple equations for the trilogarithm, the inversion relation [x]=[1/x] and the 3-term relation [x]+[1 1/x]+[1/(1 x)] = [1].) − −

3.3. Relating the 34-term and 22-term equation

In this subsection, we will indicate a proof of the following fact:

Proposition 3.6. The 22-term relation and the 34-term relation are equivalent in the sense that each one, together with its specializations, implies the other.

If we take the diﬀerence of the specializations of f(a, b, c, t)tot= 1 and t =0, respectively, then a number of terms degenerate and we are left with the sum of a Kummer–Spence relation and a 22-term relation. (Again, we work up to inversion and 3-term relation, the latter being a specialization of both 22-term and 34-term equation.) Similarly, the diﬀerence of the specializations to t = a and t = 0 gives a version of the Kummer–Spence relation itself. Thus the 34-term relation implies the 22-term relation. Conversely, the substitution

1 1 (t ,t ,t ) cr(t, 0,c− ,a),cr(t, 0,b,c− ),cr(t, 0,a,abc) 1 2 3 7→ maps the terms of the 22-term¡ relation above in the form given in¢ 3.1.2 to the 17 terms of f(a, b, c, t) together with ﬁve terms (plus a constant term)§ which are independent of t. This shows immediately that the 22-term relation implies the 34-term relation. Combining the proposition with the previous subsection, we conclude: Vol. 9 (2003) Functional equations for higher logarithms 369

Proposition 3.7. Goncharov’s 22-term relation is subsumed in Wojtkowiak’s family of equations in Corollary 3.2.

3.4. A 21-term equation

Yet another related equation of possible interest is a sum of four 22-term relations which combines to a functional equation in 3 variables with less, viz. only 21, different non-constant arguments (up to inversion), but with coeﬃcients in 1, 2 : with γ as in (3.1), consider {± ± } 1 1 x 1 1 xy 1 Γ(x, y, z)=γ , , 1 z + γ 1 , , − − 1 1 x 1 xy − − x y(1 x) 1 z− ³ − − ´ ³ − − ´ and note that its symmetrization in the ﬁrst 2 arguments, which of course also constitutes a functional equation for the trilogarithm, has the following form if we 1 1 1 u− introduce a “symmetrizing variable” z2 =(x1x2z1)− and put j(t, u)= −1 t : − Γ(x ,x ,z )+Γ(x ,x ,z )= 2[x x ] 2[1] 1 2 1 2 1 1 − 1 2 − 2

+2 [xi]+[j(xi,x3 i)] + [zi]+[j(zi,z3 i)] − − i=1 X ³ [xi j(z1,z2)] [j(xi,x3 i)j(z1,z2)] [xiz1] [j(xi,x3 i)z1] − − − − − − 2 ´ + [xi z1 j(xi,x3 i)j(z1,z2)] + [j(xi,x3 i)j(z1,z2)/(xi z1)] . − − i=1 X³ ´

4. 4-logarithm equations

It is in general a tedious job to verify even a single functional equation for the polylogarithms. Therefore it seems worthwhile to find a rather short way to verify a whole family of them. We propose to do this for a family of equations for the 4-logarithm, using essentially only one (and actually readily checked) polynomial equation. One can proceed similarly (albeit in a somewhat more complicated fashion) with other families (cf. [3], Thm. 4.4 and Thm. 4.9), even up to n =6. Let us emphasize that not only had there been no example for n 6 previously known (apart from the trivial distribution relations) but Wechsung [10]≥ even proved that the type of equation (called “Kummer-type”) which gave the only examples for n =4,5 known at the time is not “good enough” for n 6. The following example, which is proved in 6, is not of Kummer-type. ≥ § n 1 Theorem 4.1. Let F be a field of characteristic 0, φ(x)=x − (x 1) for some n 1 n 1 − n N. For some t, u F , let x = φ− (t) resp. y = φ− (u) be the sets ∈ ∈ { i}i=1 { i}i=1 370 H. Gangl Sel. math., New ser. of preimages in some finite extension field F 0 of F . Then the following element in Z[F 0] is annihilated by : L4 n 1 n x 1 x− 1 x n(n 2) i i (n 1)2 − i + n2 − i − y − − 1 y 1 1 y j j i,j=1 j− i,j=1 j · Q ¸ X · − ¸ X · − ¸ Q n x n 1 1 n2(n 1)2 i + n(n 1)2 1 1 . (4.1) − − y − − x − − y i,j=1 · j ¸ i=1 i i X X ³£ ¤ £ ¤´

5. A 7-logarithm equation

In this section we give a 2-variable functional equation for the 7-logarithm. This functional equation consists of 274 terms, but by using an appropriately sym- metrized notation we can write it in a more digestible form.

5.1. Let F be the rational function field in two variables Q(t, u). Put z z 1 z(1 z) f (z)= − ,f(z)= − ,f(z)= − 1 1 z+z2 2 1 z+z2 3 1 z + z2 − − − z2(1 z)2 and f(z)= f (z)f (z)f (z)= − . − 1 2 3 (1 z + z2)3 − Notice that the group G = generated by z 1/z and z 1 z permutes the ∼ S3 → → − f and leaves f invariant. For a, b, c, d Z define elements a, b; c, d (t, u) and i ∈ { }0 a, b; c, d in C[C]by { } 3 f(t)af (t)b a a, b; c, d (t, u)= i − and { }0 f(u)cf (u)d c i,j=1 j − X· ¸ a, b; c, d = a, b; c, d (t, u)+ c, d; a, b (t, u) { } { }0 { }0 = a, b; c, d (t, u)+ a, b; c, d (u, t), { }0 { }0 where the latter equation holds only up to inversions of the arguments (which, of course, does not affect the functional equation to be presented below). Finally, set (3) ξ7 = 1 609 1, 1; 1, 1 1 35 1, 1; 2, 1 + 1 105 1, 1; 3, 5 − 18 · 4 {− − − − }− 3 · {− − − } 3 · 8 {− − − } 1 21 1, 1; 1, 4 1 15 1, 1; 2, 5 + 1 15 1, 1; 3, 4 , − 3 · {− − − }− 3 · {− − − } 3 · {− − − } (2) ξ7 = + 1 700 1, 0; 1, 0 + 1 175 1, 3; 1, 3 + 1 28 2, 3; 2, 3 2 · { } 2 · 4 { − − } 2 · {− − } Vol. 9 (2003) Functional equations for higher logarithms 371

35 1, 3; 2, 3 140 2, 3; 1, 0 + 175 1, 0; 1, 3 , − { − − }− {− } { − } (1) ξ7 = + 1 700 1, 2; 1, 2 + 3150 0, 1; 1, 1 + 1 1575 1, 1; 1, 1 2 · { − − } { − } 2 · {− − } 2100 1, 2; 0, 1 + 1 6300 0, 1; 0, 1 1050 1, 2; 1, 1 − { − − } 2 · { − }− {− − } 1 700 1, 2; 1, 2 1 1575 1, 1; 1, 1 1 6300 0, 1; 0, 1 − 2 · {− − }− 2 · {− − }− 2 · { } + 1050 1, 2; 1, 1 + 2100 0, 1; 1, 2 3150 1, 1; 0, 1 . {− − } { − − }− { − − } (1) (2) (3) Theorem 5.1. The element ξ7 = ξ7 + ξ7 + ξ7 Q[F ], where F = Q(t, u),is a functional equation for , i.e., ∈ L7 ξ (t, u) =0 L7 7 for any t, u C where fj(t),fj(u) /¡ 0, ,¢j =1,2,3. ∈ ∈{ ∞} Proof. Functional equations for polylogarithms can be characterized by an algebraic criterion (cf. [14], Proposition 1), so the proof can be reduced to checking this criterion for ξ7 which was done with a computer program. The statement applies, of course, to any functional equation; the diﬃculty lies in ﬁnding functional equations, not in checking their validity. ¤

5.2. Remark. In ξ7 the denominator of the first factor of a coefficient (which we suppress if it is 1) gives the multiplicity of an argument in the sum a, b; c, d (where we identify [z] and [1/z]). We associate a weight to each a, b{; c, d by} wt( a, b; c, d ) = wt(a, b) in the following way: setting wt(a, b)= 1(a{ + a+}b + { } 2 | | | | 2a+b) we find that in all terms a, b; c, d of ξ7 we have wt(a, b) = wt(c, d). This | | (i) { } provides each ξ7 (i =1,2,3) with weight i. There is a weight-preserving operation on these blocks which also includes the coefficients. We explain this operation in more detail in the following subsection.

5.3. More symmetric form of ξ7

3 3 3 3 Let (Z )0 = (a, b, c) Z a + b + c =0 and (Z )1 = (a, a, b) Z .We consider the map{ ∈ | } { ∈ } Θ:Z3 Z3, (5.1) → (a, b, c) (a, b c, b a) (5.2) 7→ − − − 3 3 which sends (Z )0 to (Z )1, and a second one (the fi are taken as in 5.1) Φ:Z3 φ:P1 P1 φrational , (5.3) →{ C → C | } α α =(α ,α ,α ) φ : z f (z) 1 f (z)α2 f (z)α3 . (5.4) 1 2 3 7→ α 7→ − 1 2 3 ³ ¡ ¢ ´ 372 H. Gangl Sel. math., New ser.

The symmetric group operates on (Z3) and Z3 via permutation. For each k Z S3 0 ∈ we have the element (k, 1, 1 k) (Z3) . We set − − ∈ 0 A =Θ (k, 1, 1 k) ,k=1,2,3. k S3· − − Then we have ¡ ¢ A = (1, 1, 2), ( 1, 1, 1), (0, 0, 1) , 1 {± − ± − − ± } A = (2, 2, 3), ( 1, 1, 3), ( 1, 1, 0) , 2 { − − − − − } A = ( 1, 1, 1), ( 1, 1, 4), ( 2, 2, 5), ( 2, 2, 1), (3, 3, 4), (3, 3, 5) . 3 { − − − − − − − − − − − } We will denote by δ the 3-invariant element ( 1, 1, 1) of A3 which will play a special role. Finally we deﬁneS for α =(α ,α ,α− )=− 0− a “weight” 1 2 3 6 1 ω(α)= . α α 1− 3 Then Theorem 5.1 can be restated more concisely by saying that 3 (i) 60 ξ7 = ξ i=1 X with 29 φ (t) φ (t) φ (t) ξ(3) = δ + ω(α) σα + δ , − 20 φδ(u) φδ(u) φσα(u) α A δ σ / h i ∈X3\{ } ∈SX3 S2 ³h i h i´ 20 φ (t) ξ(2) = ω(α)ω(β) σα , 3 φτβ(u) α,β A2 σ,τ / X∈ ∈SX3 S2 h i φ (t) ξ(1) = 30 ω(α)ω(β) σα . − φτβ(u) α,β A1 σ,τ / X∈ ∈SX3 S2 h i Altogether we get 1 + 30 + 81 + 162 = 274 diﬀerent arguments (up to inverses) since in the last sum (over A1) the arguments associated to (α, β) and ( α, β) are inverse to each other. − −

6. A proof of Theorem 4.1

Proof. Using Zagier’s criterion (cf. [14], Proposition 1) alluded to earlier, we need only verify that the above combination (4.1) lies in the kernel of the map

2 2 β : Z[F 0] Sym F 0× F 0×, 4 −→ ⊗ 2 [x] x¯ x (1^ x) . 7−→ ⊗ ∧ − Here the 2 denotes the second exterior product¡ (i.e., two-fold¢ tensors subject to the relations x x = 0). Note that xy z = x z + y z. V ∧ ∧ ∧ ∧ Vol. 9 (2003) Functional equations for higher logarithms 373

Our strategy is as follows: a few preliminary considerations (Steps 0–2) allow one to rewrite the β4-images in a more convenient way, so that the theorem is essentially reduced to Claim 6.1 (in Step 3). We work modulo 2-torsion. Part I: Reformulations. 0. Note that t φ(z) is a polynomial in (the variable) z with roots equal to x and therefore − { i}i t φ(z)=φ(x) φ(z)=λ (x z) − i − i− i Y for some constant λ. Thus, for any fixed l and m, t u = φ(x ) φ(y )=λ (x y )=µ (x y ) − l − m i− m l− j i j Y Y for some constants λ and µ (which actually turn out to be equal to 1 and therefore can be neglected in the following). ± 1. As a first preliminary step, we express α and 1 α in terms of the factors x − i and yj for each of the arguments α in (4.1). In order to save indices, we put x = xl and y = y for some fixed l, m 1,...,n . We obtain m ∈{ } n 1 1 n 1 x y − X 1 x− y X − = and − = , 1 y xn 1Y 1 y 1 xnY − − − − where we have set X = i xi, Y = j yj. A further decomposition we need is n 1 1 n 1 1 x y −Q Q 1 x− y − 1 − = (x y) and 1 − = (x y). − 1 y ± Y − − 1 y 1 ∓ xY − − − − With these preparations, we can write the corresponding images under β4 as 2 1 x yn 1X ⊗ yn 1X yn 1 β = − − − (x y) (6.1) 4 − n 1 n 1 1 y x − Y ⊗ x − Y ∧ Y − µh − i¶ µ ¶ and 2 1 x 1 ynX ⊗ ynX yn 1 β − = − (x y). (6.2) 4 − 1 n n 1 y− x Y ⊗ x Y ∧ xY − µh − i¶ µ ¶ These two expressions decompose “naturally” into two parts, one of which contains 2 the factor (x y) in the last tensor factor. E.g., the part in F 0× in the first expression (6.1)− factors as V yn 1X yn 1 X yn 1 yn 1X − − (x y)= − + − (x y) (6.3) n 1 n 1 n 1 x − Y ∧ Y − x − ∧ Y x − Y ∧ − where the first summand on the right has been reduced using the defining property of . ∧2. As a second preliminary step, we “switch” to an additive notation—formally, we put ξi = “ log xi”, ηj = “ log yj”, and introduce ζlm = “ log(xl ym)” as well as the shorthand ξ = ξ and η = η (note also that the − now satisfies i i j j ∧ P P 374 H. Gangl Sel. math., New ser.

(a + b) c = a c + b c); then the term (6.1) resp. (6.2) is written, using the decomposition∧ (6.3),∧ as ∧

(n 1)(η ξ )+ξ η 3 ζ − m − l − ∧ lm 2 ¡ + (n 1)(η ξ )+¢ξ η ξ (n 1)ξ η +(n 1)η (6.4) − m − l − · − − l ∧ − − m resp. ¡ ¢ ³ ´ ³ ´

n(η ξ )+ξ η 3 ζ m − l − ∧ lm 2 ¡+ n(ηm ξl)+ξ ¢η ξ (n 1)ξl + ηm η +(n 1)ηm ξl . (6.5) − − · − − ∧ − − − Introducing¡ the shorthand¢ S¡= ξ η and slm =¢ξl ¡ ηm, we rewrite the latter¢ two equations as − −

S (n 1)s 3 ζ + S (n 1)s 2 ξ (n 1)ξ η +(n 1)η (6.6) − − lm ∧ lm − − lm · − − l ∧ − − m resp.¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ S ns 3 ζ + S ns 2 ξ (n 1)ξ +η η +(n 1)η ξ . (6.7) − lm ∧ lm − lm · − − l m ∧ − − m − l ¡ Part II:¢ Calculations.¡ 3.¢ So¡ far, we have only¢ reformulated¡ the objects¢ under consideration. Now we can proceed to the two actual calculations involved, the ﬁrst of which follows easily from the readily veriﬁed identity

ξ (n 1)ξ η (n 1)η = (2 n)δil+δjm ξ η . (6.8) − − l ∧ − − m − i ∧ j i,j ¡ ¢ ¡ ¢ X Here δij(= 1 if i = j, and = 0 otherwise) denotes the usual “Kronecker-δ”. Using relation (6.8) together with the presentations (6.6) and (6.7), the β4-image of the ﬁrst two terms in (4.1) becomes

1 2 1 xl 2 1 xl− lm lm lm lm β4 n − (n 1) − 1 = T1 + T2 + T3 + T4 (6.9) 1 ym − − 1 ym− µ h − i h − i¶ where T lm = (2n 1)S3 3n(n 1)S2s + n2(n 1)2s3 ζ , 1 − − − lm − lm ∧ lm 2 T lm = ¡ n2 S (n 1)s (n 1)2(S ns )2 ¢ (2 n)δil+δjm ξ η , 2 − − − lm − − − lm − i ∧ j i,j ³ ¡ ¢ ´ X Tlm =(n 1)2(S ns )2 η ξ , 3 − − lm · m ∧ l T lm =(n 1)2(S ns )2 ξ ξ + η η . 4 − − lm · ¡ ∧ l ¢ m ∧ We will sum the latter expression¡ over all¢l and m (unless explicitly indicated otherwise, all sums in the following run from 1 to n). Vol. 9 (2003) Functional equations for higher logarithms 375

Claim 6.1. With notations as above, we have 1 2 1 xl 2 1 xl− β4 n − (n 1) − 1 1 ym − − 1 ym− Xl,m µ h − i h − i¶ 3 1 2 = n(n 2) S ζlm + S ξl ηm − Ã − ∧ n ∧ ! ³ Xl,m ´ Xl,m 2 2 3 2 + n (n 1) slm ζlm slm(ξl ηm) − Ã ∧ − ∧ ! Xl,m Xl,m + f1(ξi)+ f2(ηj) i j X X for some functions fk (k =1,2) depending on only one of the variables t and u.

Proof. We will need the following immediately veriﬁed identities: 1 Z := ζ = ζ = ζ , l, m, (6.10) n ij im lj ∀ ∀ i,j i j X X X 1 S = ξ η = s , (6.11) i − j n ij i j i,j X X X (ξ nξ )= (η nη )=0, (6.12) − m − l m X Xl and the “distribution properties” (for ﬁxed l, m):

(2 n)δil+δjm = (2 n)δil =1, (6.13) − − i,j i X X as well as some simple consequence of the above

s (2 n)δil+δjm = S (n 1)s . (6.14) lm − − − ij Xl,m (Proof: write slm = ξl ηm and use successively the equations in (6.13).) − lm We will first treat the sum l,m T4 which can be easily reduced to the form i f1(ξi)+ jf2(ηj) since each of its terms which is dependent on both ξ’s and η’s is cancelled in the sum by (6.12).P P P 3 lm Thus it will suffice to identify the sum i=1 l,m Ti with the first two lines on the right-hand side of the claim. The first two terms of P P T lm =(2n 1)S3 ζ 3n(n 1)S2 s ζ 1 − ∧ lm − − lm ∧ lm Xl,m Xl,m Xl,m + n2(n 1)2 s3 ζ (6.15) − lm ∧ lm Xl,m 376 H. Gangl Sel. math., New ser. combine to n(n 2)S3 Z − − ∧

note that l,m slm ζlm = S Z by using (6.10) and (6.11) , while the third term on the right of (6.15)∧ is already∧ of the desired form. ¡ The termP ¢

T lm = (2n 1)S2 2n(n 1)Ss (2 n)δil+δjm ξ η 2 − − − − lm − i ∧ j i,j Xl,m Xl,m µ ¶ X decomposes into two simpler sums, the ﬁrst of which is given as

(2n 1)S2 (2 n)δil+δjm ξ η = (2n 1)S2 ξ η , (6.16) − − − i ∧ j − − i ∧ j i,j i,j X Xl,m X where we have used (6.13), while the second one equals

2n(n 1)S s (2 n)δil+δjm ξ η − lm − i ∧ j i,j Xl,m X =2n(n 1)S2 ξ η 2n(n 1)2S s ξ η (6.17) − i ∧ j − − ij i ∧ j i,j i,j X X ¡ ¢ lm by (6.14). Finally, expanding l,m T3 as a “polynomial” in S gives us three further sums, one being quadratic, one linear and one constant in S; while the quadratic one combines with (6.16)P and with the first term on the right in (6.17) to n(n 2)S2 ξ η , the linear one cancels with the second term in (6.17), − i,j i ∧ j and the constant one gives n2(n 1)2 s2 (ξ η ). This proves Claim 6.1. P − − l,m lm l ∧ m 4. We finish by observing that theP first two terms on the right-hand side of Claim 6.1 correspond precisely to the images under β of n(n 2) x / y 4 − − i j and n2(n 1)2 x /y , respectively, while T lm, which corresponds to i,j i j l,m 4 £ Q ¤ the last two− terms of the claim, can be recognized readily—after moreQ careful P £ ¤ P investigation of the terms “purely in ξi or ηj”— as the image under β4 of the expression 2 1 1 n(n 1) 1 x− 1 y− . − − − i − − j ¤ i j ¡ X £ ¤ X £ ¤¢

Acknowledgements. We would like to heartily thank Don Zagier for not only suggesting the original problem, but even more for his steady support, encourage- ment and enlightenment, and constant ﬂow of ideas surrounding this work. It was made possible through the generosity and hospitality of the Max-Planck-Institut f¨ur Mathematik in Bonn with its unsurpassed working conditions. Vol. 9 (2003) Functional equations for higher logarithms 377

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