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

TL;DR

This paper investigates the application of the Backtracking New Q-Newton's method (BNQN) to the Riemann xi function, proposing a novel approach to explore the Riemann hypothesis through dynamical systems and basin of attraction analysis.

Contribution

It introduces the use of BNQN for the Riemann xi function and establishes a new equivalence between the Riemann hypothesis and the attractors of BNQN lying on the critical line.

Findings

BNQN demonstrates stable convergence with the Riemann xi function.

The attractors of BNQN are related to the critical line, supporting the Riemann hypothesis.

Experimental results reveal interesting basin of attraction phenomena.

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.