In 1981 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 J. Elstrodt, F. Grunewald and 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.
Moreover the results by 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.