Unlocking the Riemann Hypothesis: Novel Approaches from Dynamical Systems

Exploring Dynamical Systems and the Riemann Hypothesis

This article investigates potential connections between the mathematical frameworks presented in arXiv:XXXX.XXXXX and the Riemann Hypothesis (RH). The paper focuses on dynamical systems, and while seemingly distant from the RH, we aim to identify and leverage potential connections.

Framework 1: Chaotic Systems and Zeta Function Zeros

The paper presents a chaotic system described by the following equations:

dx/dt = -(y+z); dy/dt = x+ay; dz/dt = b+xz-cz

We propose to construct an analogous dynamical system whose stability properties are directly linked to the location of the zeros of the Riemann zeta function. This could involve relating the system's parameters (a, b, c) to the properties of the zeta function.

• Formulation: Relate coefficients to the local maxima and minima of |ζ(s)| on critical lines.
• 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)|.

Framework 2: Functional Transformations and Gamma-Zeta Relations

Explore the transformation f(x) = g(u), a functional equation involving gamma functions and the zeta function. This framework builds upon known functional equations related to the Riemann zeta function, exploring potential new symmetries.

• Formulation: Analyze singularities of g(u) to understand their relationship to zeta function zeros.
• 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).

Framework 3: Infinite Product Representations

Investigate the series and product representation given by ∏(1 - λ_nz) and its connections to sums involving φ(t). This approach explores alternative representations of the zeta function, potentially revealing hidden structures.

• Formulation: Establish a direct link between the coefficients λ_n and the zeros of the Riemann zeta function.
• Potential Theorem: Prove that the convergence properties of this infinite product are directly related to the distribution of zeta zeros.
• Connection: Connect the convergence behavior of the product to the distribution of zeros.

Computational Experiments

Computational experiments could be designed to test the proposed connections. For example, numerical simulations of the dynamical system could be performed, with parameters adjusted to reflect properties of the zeta function. The results could then be analyzed to see if they provide any insights into the distribution of zeros.