We discuss computation of the special values of partial zeta functions associated to totally real... more We discuss computation of the special values of partial zeta functions associated to totally real number fields. The main tool is the Eisenstein cocycle Ψ , a group cocycle for GLn(Z); the special values are computed as periods of Ψ , and are expressed in terms of generalized Dedekind sums. We conclude with some numerical examples for cubic and quartic fields of small discriminant.
We define certain higher-dimensional Dedekind sums that generalize the classical Dedekind-Rademac... more We define certain higher-dimensional Dedekind sums that generalize the classical Dedekind-Rademacher sums, and show how to compute them effectively using a generalization of the continued-fraction algorithm. We present two applications. First, we show how to express special values of partial zeta functions associated to totally real number fields in terms of these sums via the Eisenstein cocycle introduced by the second author. Hence we obtain a polynomial-time algorithm for computing these special values. Second, we show how to use our techniques to compute certain special values of the Witten zetafunction, and compute some explicit examples.
Newsletter of the European Mathematical Society, 2016
This article introduces a set of recently discovered lecture notes from the last course of Leopol... more This article introduces a set of recently discovered lecture notes from the last course of Leopold Kronecker, delivered a few weeks before his death in December 1891. The notes, written by F. von Dalwigk, elaborate on the late recognition by Kronecker of the importance of the "Eisenstein summation process", invented by the "companion of his youth" in order to deal with conditionally convergent series that are known today as Eisenstein series. We take this opportunity to give a brief update of the well known book by André Weil (1976) that brought these results of Eisenstein and Kronecker back to light. We believe that Eisenstein's approach to the theory of elliptic functions was in fact a very important part of Kronecker's planned proof of his visionary "Jugendtraum".
We propose a novel method to guide THz radiation with low losses along thin layers of water. This... more We propose a novel method to guide THz radiation with low losses along thin layers of water. This approach is based on the coupling of evanescent surface fields at the opposite sides of the thin water layer surrounded by a dielectric material, which leads to a maximum field amplitude at the interfaces and a reduction of the energy density inside the water film. In spite of the strong absorption of water in this frequency range, calculations show that the field distribution can lead to propagation lengths of several centimeters. By means of attenuated total reflection measurements we demonstrate the coupling of incident THz radiation to the long-range surface guided modes across a layer of water with a thickness of 24 μm. This first demonstration paves the way for THz sensing in aqueous environments.
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Papers by Robert Sczech