The Riemann Hypothesis and Holomorphic Index in Complex Dynamics

ABSTRACT

We present an interpretation of the Riemann hypothesis in terms of complex and topological dynamics. For example, the Riemann hypothesis is true and all zeros of the Riemann zeta function are simple if and only if a meromorphic function that is explicitly given in this note has no attracting fixed point. To obtain this, we use the holomorphic index (residue fixed point index) that characterizes the local properties of the fixed points in complex dynamics.

Keywords:

• Riemann hypothesis
• holomorphic index
• complex dynamics

2010 AMS Subject Classification:

• 11M26
• 37F99

Acknowledgments

The author would like to thank Masatoshi Suzuki and the referee for helpful, interesting, and stimulating comments.

Additional information

Funding

Research partially supported by JSPS KAKENHI Grant.