Riemann Hypothesis and Random Walks: The Zeta Case

Abstract

In previous work, it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its L-function is valid to the right of the critical line ℜ(s)>12, and the Riemann hypothesis for this class of L-functions follows. Building on this work, here we propose how to extend this line of reasoning to the Riemann zeta function and other principal Dirichlet L-functions. We apply these results to the study of the argument of the zeta function. In another application, we define and study a one-point correlation function of the Riemann zeros, which leads to the construction of a probabilistic model for them. Based on these results we describe a new algorithm for computing very high Riemann zeros, and we calculate the googol-th zero, namely 10100-th zero to over 100 digits, far beyond what is currently known. Of course, use is made of the symmetry of the zeta function about the critical line.

Keywords

• zeta functions
• prime numbers
• probabilistic number theory