A Dynamical Systems Framework for Generating the Riemann Zeta Function and Dirichlet L-functions

TL;DR

This paper introduces a dynamical systems approach to analytically continue and study the zeros of the Riemann zeta and Dirichlet L-functions, providing evidence that the Riemann Hypothesis and its generalization are almost surely true.

Contribution

It develops a novel dynamical systems framework for analyzing the zeros of these functions, offering an alternative perspective to traditional methods.

Findings

Framework suggests the Riemann Hypothesis is almost surely true.

Extends to Dirichlet L-functions and supports the generalized Riemann Hypothesis.

Provides a new tool for investigating the distribution of non-trivial zeros.

Abstract

We first construct a dynamical systems model which in its steady-state serves as an analytic continuation of the completed Riemann zeta function over the entire critical strip. The resulting mathematical construct involves a linear interpolation of two symmetric generator functions which can be used to infer the global properties of the non-trivial zeros of the Riemann zeta function using concentration bounds. The proposed dynamical systems framework thus provides an alternative method for investigating the celebrated Riemann Hypothesis which is shown in this paper to be almost surely true. We also show that the framework is general enough to study the non-trivial zeros of the Dirichlet L-functions and in this paper we show that under specific conditions, the generalized Riemann Hypothesis is also almost surely true.