  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.
