# Theory of the Riemann Zeta and Allied Functions

### Martin N. Huxley

University of Wales Cardiff, UK### Matti Jutila

University of Turku, Finland### Yoichi Motohashi

Nihon University of Science and Technology, Tokyo, Japan### Samuel J. Patterson

Georg-August-Universität Göttingen, Germany

## Abstract

This meeting, the second Oberwolfach workshop devoted to zeta functions, was attended by 42 participants representing 16 countries. The scientific program consisted of 32 talks of various lengths and a problem session. In addition, social activities were organised: a hike in the mountains and piano recitals by Peter Elliott and Valentin Blomer. \smallskip Since the times of Dirichlet and Riemann, zeta functions and Dirichlet series have played a central role in analytic number theory, and in recent times connections have been found with other areas of mathematics and its applications, including theoretical physics. The talks represented the various aspects of the theory of zeta functions. In particular, the following topics were discussed, among others: {\small \begin{list}{(\hspkl\roman{anum}\hspkl)}{ \leftmargin4mm\labelsep2mm\listparindent0mm \labelwidth3mm\itemsep1mm\topsep1mm \rightmargin0mm\usecounter{anum}\parsep0mm\partopsep0mm} \item[$\bullet$ ] Connections of classical zeta functions with automorphic functions and spectral theory. \item[$\bullet$ ] Estimates of the size of zeta and $L\!$-functions, both at individual points and in mean value. \item[$\bullet$ ] Problems concerning the zeros of zeta functions (Riemann's hypothesis and other questions such as the Siegel zero and the distribution of zeros of Epstein's zeta functions). \item[$\bullet$ ] Applications of zeta and $L\!$-functions to arithmetic functions, and the duality between arithmetic and analysis. \item[$\bullet$ ] Random Matrix Theory, which shed new light light on mean value estimates and their consequences. \item[$\bullet$ ] Numerical calculations related to the zeros of Riemann's zeta function and other computational projects. \elist }

## Cite this article

Martin N. Huxley, Matti Jutila, Yoichi Motohashi, Samuel J. Patterson, Theory of the Riemann Zeta and Allied Functions. Oberwolfach Rep. 1 (2004), no. 4, pp. 2419–2490

DOI 10.4171/OWR/2004/46