Christopher Deninger
Some Ideas on Dynamical Systems and the Riemann Zeta Function
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 towar
ds a better understanding of $\zeta (s)$. The entire approach carries over to mo
tivic $L$-series the most general kind of $L$-series coming from arithmetic geom
etry. 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 poss
ible cohomological interpretation of the Riemann zeta function. In the second se
ction following \cite{D2}, \cite{D3} we single out a class of foliated dynamical
systems whose leafwise reduced cohomology has many of the formal properties des
ired 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