A Dynamical Criterion Equivalent to the Riemann Hypothesis

Abstract

We introduce a discrete dynamical system on the integers, defined by moving a composite $m$ forward to $m+\pi(m)$ and a prime $p$ backward to $p-\mathrm{prevprime}(p)$. This map produces trajectories whose contraction properties are closely tied to the distribution of primes. We prove unconditional contraction inequalities for error terms derived from these trajectories, using explicit remainder bounds from the smoothed explicit formula. Building on this, we show that the Riemann Hypothesis is equivalent to a sharp contraction condition: the trajectory error functional satisfies $E(X)\ll X^{1/2}\log X$. The forward implication follows directly from von Koch's classical bound under RH. For the converse, we invoke the Landau--Littlewood $\Omega$-results, which guarantee that any off-critical zero forces oscillations large enough to violate the contraction inequality. This establishes a new dynamical reformulation of RH: the critical-line conjecture is equivalent to the assertion that integer trajectories remain uniformly contracted at the scale $X^{1/2}\log X$, equivalently, RH holds. The perspective is distinct from earlier analytic equivalents, as it arises from stability properties of a simple deterministic system rather than from Hilbert space or Dirichlet polynomial approximations.