Huyghens, Bohr, Riemann and Galois: Phase-Locking

TL;DR

This paper explores mathematical models of phase-locking, extending classical and quantum approaches, linking them to prime number theory, the Riemann zeta function, and Galois fields, revealing deep connections across physics and mathematics.

Contribution

It generalizes Huyghens' classical phase-locking to include harmonic and subharmonic resonances and introduces quantum phase-locking operators related to prime numbers and Galois fields.

Findings

Classical phase-locking linked to 1/f noise and prime numbers.

Quantum phase-locking operators connected to Riemann zeta function and Gauss sums.

Analysis of phase properties in mathematical structures.

Abstract

Several mathematical views of phase-locking are developed. The classical Huyghens approach is generalized to include all harmonic and subharmonic resonances and is found to be connected to 1/f noise and prime number theory. Two types of quantum phase-locking operators are defined, one acting on the rational numbers, the other on the elements of a Galois field. In both cases we analyse in detail the phase properties and find them related respectively to the Riemann zeta function and to incomplete Gauss sums.