##
**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 \(z\in F\).

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) |

### 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 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.