Cellular Automata and Signals: A Novel Approach to the Riemann Hypothesis

Unlocking the Riemann Hypothesis with Cellular Automata and Signal Processing

The Riemann Hypothesis, one of the most important unsolved problems in mathematics, continues to inspire new approaches. A recent paper (arXiv:hal-02101868) explores the potential of cellular automata (CA) and signal processing techniques to shed light on the distribution of prime numbers and the zeros of the Riemann zeta function. This article delves into potential research pathways inspired by this work.

Mathematical Frameworks: Signals, Automata, and Increasing Functions

The paper introduces several intriguing mathematical frameworks:

• Signal Propagation: Signals propagating on "sites" defined by number-theoretic functions. These signals, denoted as T, F', G, R, H, could represent the distribution of primes or the behavior of divisor sums.
• Cellular Automata: A CA model where states correspond to the input word aⁿ. The evolution of this CA might encode properties of prime numbers, with its asymptotic behavior reflecting properties of the zeta function's zeros.
• Increasing Functions: The properties of increasing functions are used to define relationships between signals, potentially linking them to the behavior of the zeta function.

Novel Approaches: Combining CA with Zeta Function Theory

Hybrid CA and Explicit Formulas

One novel approach combines the CA model with explicit formulas relating to the zeta function. By defining CA states based on contributions of individual zeros and simulating interactions between terms, we might uncover new symmetries or regularities in the distribution of zeros.

Slope Analysis in Complex Plane Dynamics

Another approach uses the paper's intersection and slope analysis framework to study dynamics in the complex plane. By modeling paths traced by zeros under various transformations, we could explore new geometric or topological properties of the zeta function's zeros.

Tangential Connections: Quantum Systems and Dynamical Systems

Quantum States and Zeta Zeros

Connections between quantum mechanical systems and the distribution of zeros can be explored, inspired by the propagation of signals in complex systems. Analogies between quantum probability amplitudes and signal strengths can lead to conjectures about the statistical distribution of zeros.

Discrete Dynamics and Zeta Properties

The evolution of a cellular automaton is a discrete-time dynamical system. The Riemann Hypothesis can be related to the long-term behavior of certain dynamical systems, potentially linking the chaotic nature of prime distribution to chaos in a dynamical system.

Research Agenda: Conjectures and Tools

A detailed research agenda would involve:

• Formulating precise conjectures about the relationship between CA stability and zero distributions.
• Developing advanced numerical methods for CA simulations.
• Utilizing complex analysis techniques for studying dynamics in the complex plane.
• Establishing basic properties of CA models in the context of zeta zeros.

For example, one could simulate a simplified CA model where only a few zeros are considered, checking for predicted stability or periodicity. Another approach involves analyzing a specific transformation in the complex plane for a simplified version of the zeta function, studying the resulting dynamics and intersections.

Conclusion

By combining rigorous mathematical analysis with innovative models, these approaches offer new pathways toward understanding and potentially proving the Riemann Hypothesis. The key lies in establishing concrete connections between the discrete world of cellular automata and the continuous world of complex analysis and number theory.