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
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


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