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


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)
Full Text: DOI EuDML


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