ISA Function

ISA function describes Riemann sphere and is the true analytic continuation of Riemann zeta function


ISA(S) = Sign(Re(s)) * Sign(Im(s)) * Cos(Re(s) * Pi) * e Power (Sign(Re(s)) * Im(s) * i * Pi)


​ISA(1+1/2i) = Sign(Re(1+1/2i)) * Sign(Im(1+1/2i)) * Cos(Re(1+1/2i) * Pi) * e Power(Sign(Re(1+1/2i)) * Im(1+1/2i) * i * Pi) 

ISA(1+1/2i) = 1 * 1 * Cos(1 * Pi) * e Power(1 * 1/2 * i * Pi) = 1 * - 1 * e Power (i * pi /2) = i (Rotate point -1 to 90 degrees in the direction of i)


​ISA(1-1/2i) = Sign(Re(1-1/2i)) * Sign(Im(1-1/2i)) * Cos(Re(1-1/2i) * Pi) * e Power(Sign(Re(1-1/2i)) * Im(1-1/2i) * i * Pi) 
ISA(1-1/2i) = 1 * -1 * Cos(1 * Pi) * e Power(1 * -1/2 * i * Pi) = -1 * - 1 *  e Power (-i * pi /2) = -i (Rotate point 1 to 90 degrees in the direction of -i)


​ISA(-1+1/2i) = Sign(Re(-1+1/2i)) * Sign(Im(-1+1/2i)) * Cos(Re(-1+1/2i) * Pi) * e Power(Sign(Re(-1+1/2i)) * Im(-1+1/2i) * i * Pi) 
ISA(-1+1/2i) = -1 * 1 * Cos(-1 * Pi) * e Power(1 * 1/2 * i * Pi) = -1 * - 1 * e Power (i * pi /2) = i
 (Rotate point 1 to 90 degrees in the direction of i)


​ISA(-1-1/2i) = Sign(Re(-1-1/2i)) * Sign(Im(-1-1/2i)) * Cos(Re(-1-1/2i) * Pi) * e Power(Sign(Re(-1-1/2i)) * Im(-1-1/2i) * i * Pi) 
ISA(-1-1/2i) = -1 * -1 * Cos(-1 * Pi) * e Power(1 * -1/2 * i * Pi) = - 1 * e Power (-i * pi /2) = -i (Rotate point -1 to 90 degrees in the direction of -i)


ISA(1+i) = Sign(Re(1+i)) * Sign(Im(1+i)) * Cos(Re(1+i) * Pi) * e Power(Sign(Re(1+i)) * Im(1+i) * i * Pi) 

ISA(1+i) = 1 * 1 * Cos(1 * pi) * e Power (1 * 1 * i * Pi) = -1 * e Power (i * Pi) = 1   (Rotate point -1 to 180 degrees in the direction of i)


ISA(1-i) = Sign(Re(1-i)) * Sign(Im(1-i)) * Cos(Re(1-i) * Pi) * e Power(Sign(Re(1-i)) * Im(1-i) * i * Pi)
ISA(1-i) = 1 * -1 * Cos(1 * pi) * e Power (1 * -1 * i * Pi) = -1 * -1 * e Power (-i * Pi) = -1  (Rotate point 1 to 180 degrees in the direction of -i)


ISA(-1+i) = Sign(Re(1+i)) * Sign(Im(1+i)) * Cos(Re(1+i) * Pi) * e Power(Sign(Re(1+i)) * Im(1+i) * i * Pi) 
ISA(-1+i) = -1 * 1 * Cos(-1 * pi) * e Power (-1 * 1 * i * Pi) = -1 * -1 * e Power (-i * Pi) = -1 (Rotate point 1 to 180 degrees in the direction of -i)  


ISA(-1-i) = Sign(Re(1-i)) * Sign(Im(1-i)) * Cos(Re(1-i) * Pi) * e Power(Sign(Re(1-i)) * Im(1-i) * i * Pi)
ISA(-1-i) = -1 * -1 * Cos(-1 * pi) * e Power (-1 * -1 * i * Pi) = 1 * -1 * e Power (i * Pi) = 1 (Rotate point -1 to 180 degrees in the direction of i)  


​ISA(1) = Sign(Re(1)) * Sign(Im(1)) * Cos(Re(1) * Pi) * e Power(Sign(Re(1)) * Im(1) * i * Pi) 
ISA(1) = 1 * 0 * Cos(1 * Pi) * e Power(1 * 0 * i * Pi ) = 0


​ISA(-1) = Sign(Re(-1)) * Sign(Im(-1)) * Cos(Re(-1) * Pi) * e Power(Sign(Re(-1)) * Im(-1) * i * Pi) 
ISA(-1) = -1 * 0 * Cos(-1 * Pi) * e Power(-1 * 0 * i * Pi ) = 0


ISA(i)  = Sign(Re(i)) * Sign(Im(i)) * Cos(Re(i) * Pi) * e Power(Sign(Re(i)) * Im(i) * i * Pi) 
​ISA(i) = 0 * 1 * Cos(0 * Pi) * e Power(0 * 1 * i * Pi) = 0


ISA(-i)  = Sign(Re(-i)) * Sign(Im(-i)) * Cos(Re(-i) * Pi) * e Power(Sign(Re(-i)) * Im(-i) * i * Pi) 

​ISA(-i) = 0 * -1 * Cos(0 * Pi) * e Power(0 * -1 * i * Pi) = 0


​ISA(0) = Sign(Re(0)) * Sign(Im(0)) * Cos(Re(0) * Pi) * e Power(Sign(Re(0)) * Im(0) * i * Pi) 
ISA(0) = 0 * 0 * Cos(0 * Pi) * e Power(0 * 0 * i * Pi) = 0 * 1 * 1 = 0

ISA(Infinity) = Sign(Re(Infinity)) * Sign(Im(Infinity)) * Cos(Re(Infinity) * Pi) * e Power(Sign(Re(Infinity)) * Im(Infinity) * i * Pi) 
ISA(Infinity) = Infinity* Infinity* Cos(Infinity* Pi) * e Power(Infinity* Infinity* i * Pi) = Infinity * [1, -1] * Infinity = Infinity​


Plotting all other values of s other than mentioned above, ISA(s) plots points on a circle of radius absolute(Cos(Re(s) * Pi)), for example

ISA(1/3+ni) = Sign(Re(1/3+ni)) * Sign(Im(1/3+ni)) * Cos(Re(1/3+ni) * Pi) * e Power(Sign(Re(1/3+ni)) * Im(1/3+ni) * i * Pi)  (n is any real number)

ISA(1/3+ni) =1 * 1 * Cos(Pi / 3) * e Power(1 * n * i * Pi) = 1 * 0.5 * e Power (n * i * Pi)  (Describes a circle of radius 0.5 around the origin)

​ISA(1/4+ni) = Sign(Re(1/4+ni)) * Sign(Im(1/4+ni)) * Cos(Re(1/4+ni) * Pi) * e Power(Sign(Re(1/4+ni)) * Im(1/4+ni) * i * Pi)  (n is any real number)
ISA(1/4+ni) =1 * 1 * Cos(Pi / 4) * e Power(1 * n * i * Pi) = 1 * 0.70710678 * e Power (n * i * Pi)  (A circle of radius 0.70710678 around the origin)


Plotting ISA(s) for any Re(s) or any Im(s)  on 1-dimensional X-axis or 1-dimensional Y-axis yields a 0-D point (point at zero or point at infinity)


Plotting ISA(s) for all Re(s) or all Im(s)  on 1-dimensional X-axis or 1-dimensional Y-axis yields a 1-D Z-axis (point at zero and point at infinity)

Plotting ISA(s) for all Im(s) for any Re(s) on 1-D X-axis and 1-D Y-axis yields a 2-D 1-sphere (circular disc of radius abs(Cos(Re(s) * Pi)) )

Plotting ISA(s) for all Im(s) for any Re(s) on 2-D X-axis  and 1-D Y-axis yields a 3-D 2-sphere (hollow sphere of radius abs(Cos(Re(s) * Pi)) )

Plotting ISA(s) for all Im(s) for any Re(s) on 3-D X-axis and 1-D Y-axis yields a 4-D 3-sphere (space ball moving in time, universe)

Plotting ISA(s) for all Im(s) for any Re(s) on 3-D X-axis, 1-D Y-axis and 1-D Z-axis yields a 5-D 4-sphere (Hypersphere of multiple universes)

(
Universe is part of a 5-D Hypersphere (wikipedia links), 4-sphere, a self-illuminated space ball moving in time and is consciously self-aware)


Proof of Riemann Hypothesis using ISA function


​ISA(1/2 + i) = Sign(Re(1/2 + i)) * Sign(Im(1/2 + i)) * Cos(Re(1/2 + i) * Pi) * e Power(Sign(Re(1/2 + i)) * Im(1/2 + i) * i * Pi)
ISA(1/2 + i) = 1 * 1 * Cos(1/2 * Pi) * e Power(1 * 1 * i * Pi) = Cos(Pi/2) * e Power (i * Pi) = 1 * 0 * e Power (i * Pi) = 0


​ISA(1/2 + 2i) = Sign(Re(1/2 + 2i)) * Sign(Im(1/2 + 2i)) * Cos(Re(1/2 + 2i) * Pi) * e Power(Sign(Re(1/2 + 2i)) * Im(1/2 + 2i) * i * Pi)
ISA(1/2 + 2i) = 1 * 2 * Cos(1/2 * Pi) * e Power(1 * 2 * i * Pi) = 2 * Cos(Pi/2) * e Power (2 * i * Pi) = 2 * 0 * e Power (2 * i * Pi) = 0


​ISA(1/2 + ni) = Sign(Re(1/2 + ni)) * Sign(Im(1/2 + ni)) * Cos(Re(1/2 + ni) * Pi) * e Power(Sign(Re(1/2 + ni)) * Im(1/2 + ni) * i * Pi)
ISA(1/2 + ni) = 1 * 1 * Cos(1/2 * Pi) * e Power(1 * n * i * Pi) = 1 * Cos(Pi/2) * e Power (n * i * Pi) = 1 * 0 * e Power (n * i * Pi) = 0


​ISA(-1/2 - i) = Sign(Re(-1/2 - i)) * Sign(Im(-1/2 - i)) * Cos(Re(-1/2 - i) * Pi) * e Power(Sign(Re(-1/2 - i)) * Im(-1/2 - i) * i * Pi)
ISA(-1/2 - i) = -1 * -1 * Cos(-1/2 * Pi) * e Power(-1 * -1 * i * Pi) = 1 * Cos(-Pi/2) * e Power (i * Pi) = 1 * 0 * e Power (i * Pi) = 0


​ISA(-1/2 - 2i) = Sign(Re(-1/2 - 2i)) * Sign(Im(-1/2 - 2i)) * Cos(Re(-1/2 - 2i) * Pi) * e Power(Sign(Re(-1/2 - 2i)) * Im(-1/2 - 2i) * i * Pi)
ISA(-1/2 - 2i) = -1 * -2 * Cos(-1/2 * Pi) * e Power(-1 * -2 * i * Pi) = 2 * Cos(-Pi/2) * e Power (2 * i * Pi) = 2 * 0 * e Power (2 * i * Pi) = 0


​ISA(-1/2 - ni) = Sign(Re(-1/2 - ni)) * Sign(Im(-1/2 - ni)) * Cos(Re(-1/2 - ni) * Pi) * e Power(Sign(Re(-1/2 - ni)) * Im(-1/2 - ni) * i * Pi)
ISA(-1/2 - ni) = -1 * -1 * Cos(-1/2 * Pi) * e Power(-1 * -n * i * Pi) = 1 * Cos(-Pi/2) * e Power (n * i * Pi) = 1 * 0 * e Power (n * i * Pi) = 0


The reason for ISA(s) being 0 at all points of abs(Re(s)) = 1/2 is the effect of simultaneous rotation of invisible Z-axis when a point on X-axis is rotated.The usage of Cos(Re(s) * Pi) was formulated in the ISA function after visualizing this rotational effect of Z-axis on X-axis points while transformation. ISA function mentioned in this article is for a simple complex plane of 1 D real and 1 D imaginary axis.

For study of complicated complex plane involving 3 D real and 1 D imaginary axis we need to use mathematics of Complex coordinate space and Spherical coordinate systems.​​ 


Implications of the proof : In the most abstract sense the universe is purely mathematical. Also the proof of ISA function describing Riemann sphere shows the holographic nature of our universe. Any point on the 2-dimensional complex plane can be plotted on to Riemann sphere using the ISA function. Surface of the Riemann sphere is the event horizon, I consider the universe as a Light Whole Noumenon instead of the currently popular black hole phenomenon, as what lies with in the sphere totally encompasses what lies outside of it. The most fundamental aspect of our universe is the Source Of Universal Light (SOUL or Consciousness). It is the consciousness that brings forth the light, space and time and molds them in to relativistic matter and moves them as it wishes, like a kid throwing stones in a pond and creating water droplets and ripples just for fun. In this analogy the kid is the consciousness, stone is the light, water is the space-time fabric. As we can see matter is just space-time contortion, it is just a temporary construct and is continually trying to resolve itself to the original state and this is the pull of gravity that all materials experience. So answer to Yang-Mills least massive particle possible is infinitesimal, from our earlier analogy, it is like asking how small of a droplet can the kid create with a stone's throw.