The Proof of the Riemann Hypothesis and an Application to Physics

Rensley Meulens

doi:10.4236/am.2019.1011068

Abstract

In this manuscript, a proof for the age-old Riemann hypothesis is delivered, interpreting the Riemann Zeta function as an analytical signal, and using a signal analyzing affine model used in radar technology to match the warped Riemann Zeta function on the time domain with its conjugate pair on the warped frequency domain (a Dirichlet series), through a scale invariant composite Mellin transform. As an application of above, since the Navier Stokes system solution’s Dirichlet transforms are also Dirichlet series, a minimal general solution of the 3d Navier Stokes differential equation for viscid incompressible flows is constructed through a fractional derivative Fourier transform of the found begin-solutions preserving the geometric properties of the 2d version assuming that the solution is an analytic solution that suffices the Laplace equation in cylindrical coordinates, which is the momentum equation for both the 2d and the 3d Navier Stokes systems of differential equations.

Highlights

The age-old Riemann Hypothesis is a conjecture that the Riemann Zeta function has its zeros only at the negative even integers and complex numbers with real part 1 and is firstly stated in the essay by Bernard Riemann [1]. 2The fact that the Riemann Zeta function ζ +iv under certain conditions will arise as the solution of a second order ordinary differential equation on the time domain while the Riemann Zeta function ζ (1− v) may arise as the same solution but on the warped frequency domain
Since the extended form of the Riemann Zeta function is found to be an eigenfunction of the Fourier Transform operator, a composite Mellin Transform is found that projects ζ into ζ (1− v) preserving scale properties and is de facto a line-invariant transform between ζ and ζ (v), since there exists a well-known functional relationship between ζ (1− v) and ζ (v), the so-called functional equation of the extended
The discrete super-position of the mentioned eigenfunctions are used to find a new eigenfunction of the Fourier Transform and to build solutions of the known and studied second order Ordinary Differential Equations (ODEs) on both time and warped frequency domain

Summary

Introduction

The age-old Riemann Hypothesis is a conjecture that the Riemann Zeta function has its zeros only at the negative even integers and complex numbers with real part 1 and is firstly stated in the essay by Bernard Riemann [1]. 2. The discrete and collapsed forms of these hyperbolic (eigen)functions are used as analytical input signals in [3] [4] [5] [6] [7] for an analytical signal analyzing affine model In this manuscript, the discrete super-position of the mentioned (warped) eigenfunctions are used to find a new eigenfunction of the Fourier Transform and to build solutions of the known and studied second order Ordinary Differential Equations (ODEs) on both time and warped frequency domain. A general solution of the Navier Stokes 3d equation for the viscid incompressible flows is constructed from the begin conditions in the warped frequency domain, using geometric properties of the 2d version. Please do see Appendix A for a study of the Navier Stokes d.e

The Euler Differential Equation and a Think Experiment

The Proof of the Age-Old Riemann Hypothesis

Conclusions and Future Work