A Random Walk From the Landau to the Riemann Hypothesis

Thomas J. Osler, Marcus Wright

Mathematics Department, Rowan University, Glassboro, NJ 08028.

Abstract: The Riemann hypothesis is arguably the most important unsolved problem in mathematics. It is even difficult to state to beginning students since it requires a knowledge of the zeta function for complex values of the argument. However, there are at least 23 equivalent statements of this hypothesis that are much easier to state. In this paper we examine one such idea called the Landau hypothesis. This Landau hypothesis has a very simple interpretation in terms of  the prime factorization of integers. Surprisingly this hypothesis also has a useful description in terms of a one dimensional random walk. We show by means of a known, but not often seen, integral representation for any Dirichlet series that the Landau hypothesis implies the Riemann  hypothesis.

Keywords: Riemann hypothesis, Landau hypothesis, random walk, number theory.

Pages: 252 – 262 | Full PDF Paper