Lyapunov exponents for the map that passes through the non-trivial zeros of Riemann zeta-function

TL;DR

This paper investigates the dynamics of a non-linear map related to the Riemann zeta-function's zeros by calculating Lyapunov exponents to optimize parameter estimation, contributing to understanding the Riemann Hypothesis.

Contribution

It introduces a method to optimize the estimation of non-trivial zeros of the Riemann zeta-function using Lyapunov exponents for a specific non-linear map.

Findings

Lyapunov exponents calculated for the map

Identification of bifurcation parameter values

Analytical results on map dynamics

Abstract

The Riemann Hypothesis is the main open problem of Number Theory and several scientists are trying to solve this problem. In this regard, in a recent work [8], a difference equation has been proposed that calculates the nth non-trivial zero in the critical range. In this work, we seek to optimize this estimation by calculating Lyapunov numbers for this non-linear map in order to seek the best value for the bifurcation parameter. Analytical results are presented.