The Riemann hypothesis and dynamics of Backtracking New Q-Newton's method

Abstract

A new variant of Newton's method - named Backtracking New Q-Newton's method (BNQN) - was recently introduced by the second author. This method has good global convergence guarantees, specially concerning finding roots of meromorphic functions. This paper explores using BNQN for the Riemann xi function. We show in particular that the Riemann hypothesis is equivalent to that all attractors of BNQN lie on the critical line. We also explain how an apparent relation between the basins of attraction of BNQN and Voronoi's diagram can be helpful for verifying the Riemann hypothesis or finding a counterexample to it. Some illustrating experimental results are included, which convey some interesting phenomena. The experiments show that BNQN works very stably with highly transcendental functions like the Riemann xi function and its derivatives. Based on insights from the experiments, we discuss some concrete steps on using BNQN towards the Riemann hypothesis, by combining with de Bruijn -Newman's constant. Ideas and results from this paper can be extended to other zeta functions.