Quantum and classical study of prime numbers, prime gaps and their dynamics

A wave function constructed from prime counting is employed to study the properties of primes using quantum dynamics. The gaps are calculated expectation values position and a formula for maximal proposed. In an analogous nonlinear system, trajectories, associated nodes with their stability condition bifurcation dynamics studied classical It interesting note that Lambert W functions appear as natural solution fixed points energy. derived potential divergence resembles effective experienced by particle near massive spherical object in general theory relativity. coordinate time proper corresponding black hole serendipitously find analogy representing primes. stereographic projection obtained on unit circle $$(\theta ,p_{\theta })$$ phase space real numbers present along x-axis particular provides simple way calculate upper bounds gaps. estimated found be significantly better than Cramer’s predicted values.