The Classical Trilogarithm, Algebraic ^-theory of Fields, and Dedekind Zeta Functions

In this paper we show how to express the values of fF(3) for arbitrary number field F in terms of the trilogarithms (D. Zagier's conjecture) and how to relate this result to algebraic K-theory. 1. THE CLASSICAL POLYLOGARITHM FUNCTION The classical polylogarithm function n (1.1) Up(z) := ^ ^ ( 2 € C , | 2 | < l , p € N ) n-\ during the last 200 years was the subject of much research—see [L]. Using the inductive formula Li (z) = J^lAp_{(t)t~ dt, Lij(z) = -log(l z), the /7-logarithm can be analytically continued to a multivalued function on C\{0, 1}. However, D. Wigner and S. Bloch introduced [Bl] the single-valued cousin of the dilogarithm, namely (1.2) D2{z) := Im(Li2(z)) + arg(l z) • log|z|. Of course, for Lit such function is l og | z | . Analogous functions D (z) for p > 3 were introduced in [R] and computed explicitly in [Z]. Let us consider the slightly modified function (1.3) ^ 3 (z ) := Re [Li3(z) log|z| -Li2(z) + ±log 2 \z\ -Li^z)] . Such modified functions were considered also for all p by D. Zagier, A. A. Beilinson and P. Deligne [Z3, Bel]. ~S^(z) is real-analytic on CP\{0, 1, oo} and continuous on CP. Let F be a field. Let PF be the projective line over F, and let Z[Pp\0, 1, oo] be the free abelian group generated by symbols {x} , where x G PF{Q, 1, oo} . Received by the editors February 9, 1990 and, in revised form, June 15, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 19F27,11F67. The proofs for this paper were reviewed by the editor. © 1991 American Mathematical Society 0273-0979/91 $1.00+ $.25 per page 155