Christopher Deninger Some Ideas on Dynamical Systems and the Riemann Zeta Function [PS format] Preprint series: Proceedings of the ESI conference on the Riemann Zeta Function MSC: 11M06 $zeta (s)$ and $L(s, chi)$ 58F18 Relations with foliations 58F40 Applications Abstract: In this note we explain how the theory of the Riemann zeta function naturally le ads to the investigation of a class of dynamical systems on foliated spaces. The hope is that finding the right dynamical system will be an important step towards a better understanding of $\zeta (s)$. The entire approach carries over to motivic $L$-series the most general kind of $L$-series coming from arithmetic geometry. This is important for various reasons but for simplicity we will mostly be concerned with $\zeta (s)$. In the first section we recall some arguments from \cite{D1} in favour of a possible cohomological interpretation of the Riemann zeta function. In the second section following \cite{D2}, \cite{D3} we single out a class of foliated dynamical systems whose leafwise reduced cohomology has many of the formal properties desired in section one. We close with a number of further remarks and suggestions. For other approaches to $\zeta (s)$ via dynamical systems the reader may consult the works by Berry \cite{B} and Connes \cite{C}. Keywords: Riemann zeta function, dynamical systems on foliated spaces, leafwise reduced cohomology   archive      tutorial      mystery      new      search      home      contact