The Proof of the Riemann Hypothesis and an Application to Physics

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.


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 2 and is firstly stated in the essay by Bernard Riemann [1].
The fact that the (warped) Riemann Zeta function A fact that confirms the age-old conjecture of Bernard Riemann.
The C ∞ -form of the eigenfunctions of the Fourier transform and the Dilation operator are thoroughly studied in [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.

The Euler Differential Equation and a Think Experiment
Consider the solutions for the Euler differential equation 2 0 at t y y by ′ ′ + + = ′ (2) that may be deducted from the second order polynomial characteristic equation The solutions of the Euler differential equation [8] is given by and D the discriminant equivalent with If we think the variable t as restricted to the set of natural numbers » , and 1 c and 2 c as constants and equal to one and take improper super-position of all identified solutions, then the Riemann Zeta Function will arise as the family bundle of solutions of (2) as follows: with the zeros of the characteristic polynomial (3) defined as 1,2 r α β = ± for 0 D ≥ (8) and as 1,2 r i α β = ± when 0 D < (9) In our think experiment we have got both the Riemann Zeta function The Riemann Zeta function arises also as a special case of the super-position of the general solutions (see Table B1 for examples in Appendix B) of other second order ordinary differential equations like the confluent hyper geometric differential Eigenfunctions of the Fourier Transform and the Dilation operators are thoroughly discussed in the thesis of Garas in [2] and the denomination "warped frequency domain" that is used in this manuscript is adapted from [2].
The Mellin transform operator [10] [11] does consist of the warping operator x U , [2] p. 122 and further, followed by a Fourier transform. Its image will therefore reside on the warped frequency domain, when the initial function does reside on the time domain.

The Extended Riemann Zeta Function as an Eigenfunction of the Fourier Transform
Consider the monomial represented by x α and its defined unitary Fourier transform (with ordinary frequency) If we take the improper sum of both x α and its Fourier Transform, a procedure also known as the Poisson Summation formula, then will the extended form of the Riemann Zeta function in the complex plane emerge as follows, with 2 i r µ β = π + using the fact that the fractional derivative of ( ) ( ) ( ) for 0 α ≥ and 0 ξ ≥ [7]. The identified scaling and dilative effect (or Doppler effect) makes this transform ideal for analytical signal reconstruction and signal analysis purposes applied in radar technology.

The Proof of the Age-Old Riemann Hypothesis
For normalization purposes and to be coherent with the used Fourier transform we will use An affine or a line-preserving map between Q.e.d. The "quod erat demonstrandum" abbreviation refers to the confirma- and with cos x x = π , with 2v a defined as in the On-Line Encyclopedia of Integer Sequences as OAEIS A053117 [13], that is comparable with a characteristic polynomial of a higher order ODE for some v → ∞ . Here are v T and respectively v U the Chebyshev polynomials of resp. the first and second kinds [14].
Since the index v has to be taken even in the cumulative v U polynomial gives the opportunity to form repeating pairs of conjugate complex roots as the resulting characteristic polynomial may then be decomposed in product of lower order parabolic expressions which is a conditio sine qua non for the warped Riemann Zeta function to exist as a solution of the discussed ODE.
The expression ( ) is equivalent with the Fourier series of the zero-th Bernoulli polynomials, The last definition can be found in the public accessible on-line Wikipedia.
If we do interchange the variable x and n with each other, a logic way to calculate the Dirichlet Transform of an Ordinary Generating Function resulting in a Dirichlet series, yields: [15] p. 177.

Conclusions and Future Work
Using the fact that the extended Riemann Zeta function is an eigenfunction under the Fourier Transform, it is proved that there exists a line-invariant composite Mellin transform operator that projects what confirms the validation of the age-old Riemann Hypothesis, knowing that there exists a well-known functional equation between ( ) De facto they are conjugate pairs of each other using repetitively a fractional derivative Fourier Transform operator or the so-called Mellin Transform operator f ξ  (12). The Mellin transform will expand the transform variable for n  from the set of natural numbers  to the set of real numbers » and x  will expand the transform variable from the set of real numbers to the set of (negative) complex numbers  .
Our future work will entail the study of the full spectrum of the eigenspaces belonging to the (fractional derivative) Fourier Transform operator and the behavior of the so-called conjugate pairs on their respectively domains with regard to known solution procedures for partial differential equations.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.
If we discretize n θ = ∈ » , then take the superposition of all discrete solutions, is still a general solution of above equation, namely is invariant under the Fourier Transform or if we would have taken summation over the variable x. if we would have used a general term n a [15]. Please note that the Heaviside step function [16].
Above, the so-called functional equation of the Riemann Zeta function (1) The Mellin Transform of (32), for ( ) 0 taylor series expansion:  When the begin conditions differ per independent solution legs, the same method may be applied but now instead of using e i n x − , its real part (with regard to the term with ( ) 1 C s ) must be used and respectively its imaginary part (with regard to the term with ( ) 2 C s ) must be used to calculate (38). The time domain equivalent of (38) may be calculated by the inverse Dirichlet transform and is defined by The Dirichlet Transform of a function f is defined as The above solution (38) With the begin function f equivalent with 1 does Equation (46) yield the Dirichlet series    Figure A1 confirms the volatile characteristics of the Dirichlet series. Figure   A2 is the inverse Dirichlet transform of Figure A1, using [15].       On the edge of the domain the following is true for the normal vectors Please do see Figure A3. Figure A3 does show the contour graph of the warped frequency distribution in 3 dimensions.
The total energy of each wave system is equal on both the time and frequency domain [6] [7]. In symbols, is the analytical signal on the frequency domain and its Fourier Transform inverse ( ) x t on the time domain and 2  the total (kinetic) energy of the system and r is a warp parameter that may be set 1 2 − to yield a factor The Wigner-Ville time-frequency distribution function is a unitary affine function that suffices the following equation makes it also unbounded. The last maximum value of the total energy is also supported by the fact that the Wigner-Ville Distribution function definition also does have a factor of the hyperbolic cosecant with half argument in its integrand (defined on what may lead to variants of the improper Riemann contour integral [1] ( ) ( ) ( ) ( ) ( ) Below schematically, the application of the presented formulas for solving up the Laplace equation in cylindrical coordinates in an analytic signal analyzing model used by [6]. The constructed transformation is scale invariance: An analytic signal analyzing model : The result is equivalent with only for ( ) a n is totally multiplicative [17]. Solving up the zeros of the denominator in the last expression Interesting to note is that the so-called blow-up time within the numeric schemes may manifest whenever the numerical pivoting method strikes a known singularity of the general solution.
When the analytical signal condition is dropped or when negative discrete so- converts the weber equation to