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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 √Čtale 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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