OFFSET
1,1
COMMENTS
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 curve ZDelta2 go to a lower limit and the second types are all well below that limiting curve.
FORMULA
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.
MATHEMATICA
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}]]
CROSSREFS
KEYWORD
nonn,uned
AUTHOR
Roger L. Bagula, Apr 14 2008
STATUS
approved