Exploring Chaotic Systems and the Riemann Hypothesis
This article proposes a novel approach to the Riemann Hypothesis by exploring the potential connections between the distribution of prime numbers and the behavior of chaotic dynamical systems. The central idea is to construct a dynamical system whose stability properties are directly related to the location of the zeros of the Riemann zeta function.
A Chaotic Dynamical System
Consider the following system of ordinary differential equations, which exhibits chaotic behavior:
dx/dt = -(y + z)
dy/dt = x + ay
dz/dt = b + xz - cz
where a, b, and c are parameters. We propose to modify this system to create an analogous system whose stability properties are intrinsically linked to the zeros of the Riemann zeta function.
Relating the System to the Zeta Function
- Formulation: Relate the coefficients a, b, and c to the local maxima and minima of |ζ(s)| on the critical line (Re(s) = 1/2).
- Potential Theorem: Prove that these coefficients can model or approximate the behavior of the Riemann zeta function ζ(s) near potential zeros.
- Connection: Establish criteria or bounds for the location of zeros on the critical line by examining the transformations’ impact on |ζ(s)|.
Functional Transformations and Gamma-Zeta Relations
Another avenue of investigation involves exploring functional transformations involving gamma functions and the Riemann zeta function. We can explore a transformation of the form f(x) = g(u), where f(x) involves the zeta function and g(u) involves gamma functions.
- Formulation: The equation f(x) = g(u) is the core. Analyze singularities of g(u).
- Potential Theorem: Demonstrate how these transformations maintain or reveal symmetries of the zeta function, particularly related to its zeros.
- Connection: Relate the zeros of Ξ(z) directly to the zeros of g(u).
Infinite Product Representations
Investigate the series and product representation given by Π(1 - λnz) and its connections to sums involving φ(t), where λn represents the non-trivial zeros of the Riemann zeta function and φ(t) is a suitable function. Analyzing the convergence properties of this product representation might offer insights into the distribution of zeros.