1,1

This result is my second try at this type of experiment.

I had this idea of looking at the product elements in the

zeta product as limiting near primes.

pe[n]=1/(1-Prime[n]^(-z)

Limit[Pe[n],x->Prime[n]+Delta1+I*Delta2]=0

where z=-1/2+i*4*Pi*Prime[n]

I solved it down to an equation in n x

and then, I looked a near specific primes.

Two types show up:

Normal Riemannian -1/2 primes on a curve

and second type that are attracted to one instead.

There seems to be a spectral effect in the Delta2 values.

The cure ZDelta2 go to a lower limit

and the second types are all well below that limiting curve.

x start at Prime[n]: when the equation z1 = 1/2 - 4*I*Pi*x; z2 = 1 - 8*I*Pi*x; 1 + 2 x^z1 + x^z2 == 0; has a root with Im[x]<-0.05. the starting prime is reported out.

z1 = 1/2 - 4*I*Pi*x; z2 = 1 - 8*I*Pi*x; a1 = Flatten[Table[If[(Im[x] /. FindRoot[1 + 2 x^z1 + x^z2 == 0, {x, Prime[n]}]) < -0.05, Prime[n], {}], {n, 1, 1000}]]

Sequence in context: A159747 A141866 A028472 this_sequence A142566 A063654 A069764

Adjacent sequences: A136643 A136644 A136645 this_sequence A136647 A136648 A136649

nonn,uned

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 14 2008