Eisensteinreihen auf dem dreidimensionalen hyperbolischen Raum und Zetafunktionen

Greefrath G

Zusammenfassung

In 1981 {ıt D. Zagier} [Automorphic forms, representation theory and arithmetic, Pap. Colloq. Bombay 1979, 275-301 (1981; Zbl 0484.10019)] derived interesting relations between the real-analytic Eisenstein series on the complex upper half-plane and nontrivial zeros of the Riemann and Dedekind zeta-functions. In his thesis under review the author generalizes these results to the three-dimensional hyperbolic space. The analytic properties of the Eisenstein series associated with an imaginary quadratic number field $K$ were already investigated by {ıt J. Elstrodt, F. Grunewald} and {ıt J. Mennicke} [J. Reine Angew. Math. 360, 160-213 (1985; Zbl 0555.10012)]. The author considers zeta-functions associated with positive definite and indefinite binary Hermitian forms over the ring $O_K$ of integers in $K$. This leads to relations for special values of the Eisenstein series and nontrivial zeros of the Riemann zeta-function. As an application a Kronecker limit formula is derived in a particular case. \par Moreover the results by {ıt P. Bauer} [Proc. Lond. Math. Soc. 69, 250-276 (1994; Zbl 0801.11022)] on zeta-functions of binary quadratic forms over $K$ lead to an analogous identity between special values of the Eisenstein series on the three-dimensional hyperbolic space and the nontrivial zeros of the Dedekind zeta-function of $K$. The final chapter refers to the Rankin-Selberg method. It is shown that the Rankin zeta-function is divisible by the Dedekind zeta-function whenever the class number of $K$ is 1. Moreover an Euler product expansion of the Rankin zeta-function is derived in this case.