Mobius Function - page 2

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The Mobius Function

4. When the Mobius Function is equal to +1 the number n is composite. This suggests a position in 3D space or, as I suggest, onto the interior face of the complex plane. When n = 1 we have a unique situation as 1 is both prime and a square number. While it is not composite this function would actually allow the number to lie upon the interior face of the complex plane too. This allows for a unit circle at n = 0 to possibly translate onto the complex plane.

The Geometry of Complex Space.
1. I have used logarithms to the base 10 just for illustrative purposes although I expect natural logarithms to be the proper base.

2. Now if we look at Figure A1-1 we see how distances change in complex space. Note first how the axes are not linear and the origin is equivalent to the translation of the first prime = 1. It is a crude drawing of the ztar's arms as I want to show how I arrive at the shape of these arms.
3. Of course we have three dimensions to consider so we end up with the rectangular hyperbolic star diagrammatically shown in figure A1 - 2. This represents a new 'konic' in UNDISTORTED complex space.
 
4. Remember this new 'konic' (let's call it a 'ztar') represents the geometry of space and distance but not force. in fact it represents the shape of UNDISTORTED Complex Space.

5. A prime in 3D can thus map into 4 complex numbers in the complex plane and then remapped to a single point at the exterior 4D surface. However through Analytic Number Theory we see that this distribution in the complex plane takes wave forms. When the Ztar’s arms are equal, as shown, their terminal points can represent positions for a plane section through my zunnel ( a complex manifold connecting a 3D circle and 4D square).
6. How exactly does a force from the complex plane behave as it moves towards 3D or vice versa? The equation i^2 = -1 and the Mobius Function hint at this. If we have a number squared it tends to lie on the Real Plane (say 3D for convenience). So a number undergoing a parabolic increase of y = x^2 can translate from the Complex Plane into 3D. To reverse the operation we do not need the inverse of y but it’s square root implying a logarithmic property.
7. The arms of the Ztar could be argued to have an exponential curvature and has we will see later they are stepped. In fact the cross section of the ztar’s arm follows the rectangular hyperbola of y = 1/x corresponding to the work done.
8. It is the force needed to propagate a wormhole or zunnel in complex space that needs to be exponential. This indicates why wormholes do not open up so easily.
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