I have been working in complex dynamics for years and

have done recent work in minimal surfaces

and constant mean curvature surfaces.

It never occurred to me to use that knowledge

on the Zeta zeros connection to the distribution of the primes.

The key seems to be something called the Gauss map:

{x0,y0,z0}={ 2*Re[Zeta[z]]/(1+Abs[Zeta[z]]^2),

2*Im[Zeta[z]]/(1+Abs[Zeta[z]]^2), (1-Abs[Zeta[z]]^2)/(1+Abs[Zeta[z]]^2)}

where

x0^2+y0^2+z0^2=1

Suppose we have a nontrivial zeta zero:

f[n]=Zeta[1/2+I*b[n]]=0

Then we get for the Gauss Map:

{x0,y0,z0}={ 2*Re[Zeta[1/2+I*b[n]]], 2*Im[Zeta[1/2+I*b[n]]], 1}

This gives

1+2*Re[Zeta[1/2+i*b[n]]]+I*2*Im[Zeta[1/2+i*b[n]]]=Exp[I*2*Pi*C[n]]

My hypothesis/ conjecture is that in order for the result to be on a

unit circle as the Gauss Map puts it:

C[n]=Arg[1+2*Re[Zeta[1/2+I*b[n]]],2*Im[Zeta[1/2+I*b[n]]]]/(2*Pi)

That

C[n]=Mod[Prime[n]*Irr,1]

That is for the primes to be connected to the distribution of the zeta zeros

as Riemann thought, there must be some irrational number "Irr"

that forms an irrational rotation that connects the Primes to the zeta

zeros.

Another way to put is is that

when the 1/2+I*b[n] is projected

onto a unit sphere in the Riemannian manner

the result is a circle with points

at the b[n] points that correspond to the irrational rotation

determined by the primes themselves.

The alternative to this conjecture is

a sequence of irrational numbers with no determining

equation. My quantum energy equation for the primes shows that there

is an ordering principle involved.

I further conjecture that "Irr" is an transcendental number

and that Riemann's conjecture as a result can

never be proved as it is a Turing problem with no stopping point/

transfinite.

So this idea gives an inverse problem between the primes and the Zeta

zeros.

And it isn't an easy one: either the constant "Irr" exists or there is a

sequence,

but in both cases it is still an inverse problem. And the problem still

involves

a circle and irrational numbers....

Respectfully, Roger L. Bagula

tftn@..., 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :

URL :

http://home.earthlink.net/~tftn
URL :

http://victorian.fortunecity.com/carmelita/435/