## The Bloch-Wigner-Ramakrishnan polylogarithm function.(English)Zbl 0698.33001

The dilogarithm function $$Li_ 2(z)=\sum^{\infty}_{n=1}z^ n/n^ 2$$ $$(| z| <1)$$ has many beautiful properties and plays a role in connection with problems in many parts of mathematics, most recently $$K$$-theory. This function cannot be extended in a one-valued way to the entire complex plane, but the modified function $D(z)={\mathcal I}(Li_ 2(z)+\log (1-z)\log | z|)$ introduced by D. Wigner and S. Bloch can be extended as a continuous real-valued function on $${\mathbb C}$$ which is real-analytic except at 0 and 1. For the higher polylogarithm functions $$Li_ m(z)=\sum^{\infty}_{n=1}z^ n/n^ m$$ a similar one-valued modification $$D_ m(z)$$ was defined in principle, but not written down in closed form or studied in detail, by D. Ramakrishnan.
In the present paper the function $$D_ m(z)$$ is given explicitly and some of its properties are studied. It turns out that values of certain Kronecker double series related to special values of Hecke $$L$$-series for imaginary quadratic fields can be given in terms of the functions $$D_ m(z)$$ (generalizing a result of S. Bloch involving $$D_ 2=D)$$ and that the function $$D_ m$$ is related to a certain Green’s function for the quotient of the upper half-plane by the group of translations $$\tau \mapsto \tau +n$$, $$n\in {\mathbb Z}$$. Most interestingly, a conjecture is presented according to which the value of the Dedekind zeta-function $$\zeta_ F(s)$$ as $$s=m$$ for an arbitrary number field $$F$$ and integer $$m>1$$ can be expressed in closed form in terms of finitely many values of $$D_ m(z)$$ at arguments $$z\in F$$.
Reviewer: D.Zagier

### MSC:

 11G55 Polylogarithms and relations with $$K$$-theory 11R42 Zeta functions and $$L$$-functions of number fields 11R70 $$K$$-theory of global fields 19F27 Étale cohomology, higher regulators, zeta and $$L$$-functions ($$K$$-theoretic aspects)
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### References:

 [1] Bloch, S.: Higher regulators, algebraicK-theory, and zeta-functions of elliptic curves. Lecture Notes, U.C. Irvine, 1977 [2] Lewin, L.: Polylogarithms and associated functions. New York: North-Holland 1981 · Zbl 0465.33001 [3] Ramakrishnan, D.: Analogs of the Bloch-Wigner function for higher polylogarithms. Contemp. Math.55, 371-376 (1986) · Zbl 0655.12005 [4] Suslin, A.A.: AlgebraicK-theory of fields, in: Proceedings of the International Congress of Mathematicians 1986. Am. Math. Soc., pp. 222-244 (1987) [5] Weil, A.: Elliptic functions according to Eisenstein and Kronecker. Ergebnisse der Mathematik 88. Berlin Heidelberg New York: Springer 1977 · Zbl 0318.33004 [6] Zagier, D.: Hyperbolic manifolds and special values of Dedekind zeta functions. Invent. Math.83, 285-302 (1986) · Zbl 0591.12014 [7] Zagier, D.: The remarkable dilogarithm. In: Number theory and related topics. Papers presented at the Ramanujan Colloquium, Bombay 1988, TIFR and Oxford University Press, pp. 231-249 (1989) and J. Math. Phys. Sci.22, 131-145 (1988) [8] Zagier, D.: Green’s functions for quotients of the upper half-plane. In preparation [9] Zagier, D.: Polylogarithms, Dedekind zeta functions, and the algebraicK-theory of fields. In preparation · Zbl 0728.11062
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