The Riemann hypothesis and holomorphic index in complex dynamics

TL;DR

This paper links the Riemann hypothesis to complex dynamics by showing that the hypothesis holds if a specific meromorphic function lacks attracting fixed points, using holomorphic index to analyze fixed point properties.

Contribution

It introduces a novel interpretation of the Riemann hypothesis through the lens of complex and topological dynamics, employing holomorphic index as a key tool.

Findings

Riemann hypothesis is equivalent to a meromorphic function having no attracting fixed points.

All zeros of the Riemann zeta function are simple if the fixed point condition is met.

Holomorphic index characterizes local fixed point properties in complex dynamics.

Abstract

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