Sel. math., New ser. 9 (2003) 361 { 377 c BirkhÄauser 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 ¯rst 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 Classi¯cation (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 de¯ned by the power series 1 zn Li (z)= ; z < 1; 2 n2 j j n=1 X and can be analytically continued to C [1; ), is the 5-term relation note the 1 x 1¡y 1 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 di®erent 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 1 zn Li (z)= ; z < 1; m nm j j 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 e®orts in the direction of ¯nding 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 ¯rst (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 ¯nding 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 clari¯ed so far. The contents of the paper are as follows: in 2, we recall a general functional equation for the dilogarithm, which is found tox have some \companion" for the trilogarithm (Proposition 3.1) which in turn specializes to Wojtkowiak's triloga- rithm equations (Corollary 3.2). We restate (in 3.1) Goncharov's equation in increasingly symmetric ways and relate it (in 3.2 andx 3.3) to Wojtkowiak's equa- tions|the combining link being a 34-term equation,x inx 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 ofx which is given at the end of the paper ( 6). Finally, in 5, we present our \high- score result" (obtained in 1992): a functionalx equation forx 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 ¡f g 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! j j ¡ j j· 62 f g r=0 µX ¶ where denotes the real part for m odd and the imaginary part for m even, and <m Vol. 9 (2003) Functional equations for higher logarithms 363 Br the r-th Bernoulli number (B0 =1;B1 = 1=2;B2 =1=6;:::). For z > 1, ¡ m 1 j j 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 2f 1g 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 ¯nite 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 Á without constant term|and generalized by Wojtkowiak [12] and Zagier [16] to any rational function Á (cf. also Wechsung [10], [11]). For m N put 2 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 Á : PC PC be a rational function and ®; B; C; D P (C). Then, denoting deg(Á) the degree! of Á and putting A = Á(®); we obtain 2 (®; ¯; γ; ±) = deg(Á) (A; B; C; D); L2 ¢ L2 ¯;γ;±X f f where ¯, γ and ± run through the preimages under Á of B, C and D, respectively, with multiplicities. The theorem contains as a special case the 5-term relation (e.g., if we take Á(z)=z(1 z) and B =1,C=0,D= ) and in fact many other known equa- tions. A functional¡ equation which is not1 covered by the theorem (and presumably the only one, essentially) is the one relating z andz ¹, its complex conjugate, via 2(¹z)= 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 ¯nitely 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 Á : PC PC be a rational function, Ai;Bj;Ck;Dl P (C), i; j; k; l 1;2 . Then, denoting! deg(Á) the degree of Á, we have 2 2f g ( 1)i+j+k+l (® ;¯ ,γ ;± ) deg(Á) (A ;B ;C ;D ) =0; ¡ L3 i j k l ¡ ¢ L3 i j k l i;j;k;lX µ ®i;¯Xj ,γk;±l ¶ f f where ®i, ¯j, γk and ±l run through the preimages of Ai, Bj, Ck and Dl, respec- tively, 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 statedx in a simpler form as follows: 1 1 1 Corollary 3.2. Let A; B; C P (C), Á : PC PC a rational function of degree n 2 1 ! n 1 n and x an independent variable. Let Á¡ (A)= ®i i=1, Á¡ (B)= ¯i i=1 and 1 n f g f g Á¡ (C)= γ . Then f igi=1 n (Á(x);C;B;A) (x; γ ;¯ ;® ) L3 ¡ L3 i j k i;j;kX=1 f n f n (x; ® ;® ,γ ) (x; ¯ ;¯ ,γ ) ¡ L3 i j k ¡ L3 i j k i;j;kX=1 i;j;kX=1 n f n f + (x; ® ;® ;¯ )+ (x; ¯ ;¯ ;® ) 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 ®i, ¯j 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 con¯guration spaces, and as a crucial by-product he provided an equation for the trilogarithm in three variables ®i for which he gave a threefold symmetry.