On the Riemann hypothesis: A fractal random walk approach

Abstract

In his investigation of the distribution of prime numbers Riemann, in 1859, introduced the zeta function with a complex argument. His analysis led him to hypothesize that all the complex zeros of the zeta function lie on a vertical line in the complex plane. The proof or disproof of this hypothesis has been a famous outstanding problem in mathematics. We are able to recast Riemann's Hypothesis into a probabilistic framework connected to the fractal behavior of a lattice random walk. Fractal random walks were introduced by P. Levy, and in the continuum are called Levy flights. For one particular lattice version of a Levy flight we show the connection to Weierstrass' continuous but nowhere differentiable function. For a different lattice version, using a Mellin transform analysis, we show how the zeroes of the zeta function become the singularities of a complex integrand which governs the behavior of a fractal random walk. The laws of probability place restrictions on the locations of the zeroes of the zeta function. No inconsistencies with probability theory are found if the Riemann Hypothesis is false.