Riemann Zeroes from a Parametric Oscillator analyzed with Adiabatic Invariance, Hill Equation and the Least Action Principle

TL;DR

This paper explores a novel approach to locating Riemann zeta function zeroes using a parametric oscillator model analyzed through adiabatic invariance, Hill equations, and the least action principle, revealing potential connections between physics and number theory.

Contribution

It introduces a new physical model employing a parametric oscillator with a variable frequency linked to RZF zeroes, applying three different formalisms to identify zeroes on the critical line.

Findings

Oscillator states correspond to RZF zeroes within finite intervals.

The formalism remains effective even with stochastic parameter variations.

Multiple mathematical tools converge to locate zeroes on the critical line.

Abstract

Adiabatic Invariance (AdI), Hill Equation formalism (HEF), and the Least Action Principle (LAP), three relevant tools of theoretical physics are here separately applied to a one-dimensional parametric oscillator of time-variable frequency that depends on an integer parameter Lambda. This oscillator is subjected to a perturbation which is a functional of the modulus of Riemann Zeta Function (RZF) times an oscillatory function, expecting that nontrivial zeroes could be obtained, in the critical strip, either (i) by optimizing the oscillator AdI, (ii) verifying the Magnus-Winkler equation associated to the oscillator Hill equation, (iii) from evaluating the Action integral of the perturbed oscillator. The optimum value of parameter Lambda is firstly obtained when applying the AdI formalism, and we find that the three formalisms do lead to parametric oscillator states giving the RZF zeroes in arbitrary finite intervals on the critical line. The parameter Lambda is later replaced by a random integer-valued function Lambda(sigma) written in terms of twin primes, actually defining an infinite set of perturbed oscillators of different Lagrangian. Nonetheless, when applying the LAP to these oscillators we still get those Riemann zeroes at the critical line, in spite of both complex plane coordinates varying simultaneously in the critical strip.