A136646 Primes that give find root imaginary results for: z1 = 1/2 - 4*I*Pi*x; z2 = 1 - 8*I*Pi*x; 1 + 2 x^z1 + x^z2 == 0; with Im[x]<-0.05. The resulting root share value of the Zeta[root] being near one.

89, 619, 877, 1193, 1319, 1481, 1747, 2437, 2447, 2521, 2647, 3163, 3371, 3449, 3469, 5711, 6007, 6089, 6659, 7001, 7121

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.

Links

Table of n, a(n) for n=1..21.

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

Sequence in context: A176634 A224740 A028472 * A142566 A063654 A308801

Adjacent sequences: A136643 A136644 A136645 * A136647 A136648 A136649

Keyword

nonn,uned

Author

Roger L. Bagula, Apr 14 2008

Status

approved