A Dynamical Key to the Riemann Hypothesis: Part I

Abstract

We investigate a dynamical basis for the Riemann hypothesis (RH) that the non-trivial zeros of the Riemann zeta function lie on the critical line x = ½. In the process we graphically explore, in as rich a way as possible, the diversity of zeta and L-functions, to look for examples at the boundary between those with zeros on the critical line and otherwise. The approach provides a dynamical basis for why the various forms of zeta and L-function have their non-trivial zeros on the critical line. It suggests RH is an additional unprovable postulate of the number system, similar to the axiom of choice, arising from the asymptotic behavior of the primes as n→∞. Part I of this article includes: Introduction; The Impossible Coincidence; Primes and Mediants - Equivalents of RH; A Mode-Locking View of Dirichlet L-functions and their Counterexamples; Widening the Horizon to other types of Zeta and L-Function; L-functions of Elliptic Curves; and Modular and Automorphic Forms.