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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. 0
89, 619, 877, 1193, 1319, 1481, 1747, 2437, 2447, 2521, 2647, 3163, 3371, 3449, 3469, 5711, 6007, 6089, 6659, 7001, 7121 (list; graph; refs; listen; history; text; internal format)
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

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Last modified July 31 14:49 EDT 2021. Contains 346374 sequences. (Running on oeis4.)