If our chaotic operator is derived correctly, then the Riemann hypothesis holds true (2404.00583v1)

Abstract: In this paper, we explore novel chaotic dynamics derived from the Riemann and von Mangoldt function formula regarding the distribution of nontrivial zeros of the Riemann zeta function. By computing Lyapunov exponents, we demonstrate that the derived dynamics exhibit chaotic behavior when the gaps between zeros are within a certain bound, specifically up to 2.4. Beyond this threshold, the dynamics do not display chaotic behavior. Furthermore, we derive a chaotic operator for the Riemann zeta function within the critical strip, utilizing the correction term from the Riemann-von Mangoldt formula. We establish the chaotic nature and Hermiticity of this operator, and discuss its diagonalization properties. Moreover, our study reveals a remarkable compatibility between our derived chaotic operator and the quantum hydrogen model, as evidenced by the analysis of its eigenvalues resembling the energy levels of hydrogen. Numerical evidence, including Lyapunov exponents, bifurcation analysis, and entropy computation, underscores the unpredictability of the system. Additionally, we establish a connection between our chaotic operator and the prime number theorem regarding the density of primes. Furthermore, our investigation suggests that this chaotic operator strongly supports the validity of the Riemann hypothesis, as proposed by Hilbert and Polya. These findings shed light on the intricate relationship between chaotic dynamics, number theory, and quantum mechanics, offering new perspectives on the behavior of the Riemann zeta function and its zeros. Finally, we demonstrate the Hermiticity and diagonalization properties of our operator using the spectral theorem, further elucidating its mathematical properties and unboundedness.