Quantum and classical study of prime numbers, prime gaps and their dynamics

Abstract

A wave function constructed from prime counting function is employed to study the properties of primes using quantum dynamics. The prime gaps are calculated from the expectation values of position and a formula for maximal gaps is proposed. In an analogous nonlinear system, the trajectories, associated nodes with their stability condition and the bifurcation dynamics are studied using classical dynamics. It is interesting to note that the Lambert W functions appear as a natural solution for the fixed points as functions of energy. The derived potential with the divergence resembles the effective potential experienced by a particle near a massive spherical object in general theory of relativity. The coordinate time and proper time corresponding to a black hole serendipitously find their analogy in the solution of the nonlinear dynamics representing primes. The stereographic projection obtained from quantum dynamics on unit circle in the $(\theta ,p_{\theta })$ phase space of the real numbers present along x-axis in general and prime numbers in particular provides a simple way to calculate a formula for upper bounds on the prime gaps. The estimated prime gaps is found to be significantly better than that of Cramer’s predicted values.

Appendices

6 Appendix A

1.1 A. Potentials

In order to compare the characteristics of prime potential with the effective potential associated with General Theory of Relativity we write the effective prime potential in 3-D as [22]

$$\begin{aligned} V^\textrm{Prime}_\textrm{eff}(r) = {\frac{1}{4r^2}} {\bigg (}{\frac{1}{ln (r)} + \frac{3}{2ln^2(r)}}{\bigg )} - \frac{L^2}{2r^2}\ \ \ \ \end{aligned}$$

(27)

In the above equation we have taken in the last term $-L^2$ instead of $+L^2$, since it has been conjectured [7] that centrifugal force reversal [39] happens for $r<3M$ black holes. Also the negative sign is due to quantum anti-centrifugal force being metric related [40] and the same negative sign appear [41] in case of an inverted harmonic half-oscillator with negative fractional phase shift associated with an attractive centrifugal barrier(i.e. well) in relation to prime numbers.

The effective potential in General Theory of Relativity is given as

$$\begin{aligned} V^\textrm{GTR}_\textrm{eff}=-\frac{\epsilon \ M}{r} +\frac{L^2}{2r^2} -\frac{GML^2}{m^2c^2r^3} \end{aligned}$$

(28)

It is to be noted that the mathematical dependence on r though appears to be different in both cases, the graphically they appear to be same.

7. Appendix B

1.1 A. Four acceleration for GTR and prime potential

The magnitude of four acceleration defined as $a_0=(-a_{\mu } a^{\mu })^{1/2}$ is given with $c=1$ as [19]

$$\begin{aligned} a_0=(-a_{\mu } a^{\mu })^{1/2}=\frac{GM}{r^2(1-\frac{2GM}{r})^{1/2}} \end{aligned}$$

(29)

So we see that at $r=r_{sch}=2GM/c^2$ for a black hole the magnitude of the four acceleration diverges. But since the four acceleration has to be divergent in GTR for a black hole , our potential having a divergence is a boon since it mimics the four acceleration for a black hole and defines the Schwarzschild radius at $r=1$.