Monday, January 29, 2018

14.13i............could be seen as both e and pi.....................14.13i...........14 + 13.....= 27.............e starts out...........2.7..............the 13 is under the dec point............pi starts out........3.14.............the beg to middle........end to middle............14 is 14........0.13i.........is 0.13.......is 13..........from end to middle.....31.........over lap the 1's...................and u have the beg of pi.......fractal patterns.......chaos theory...........



The above plot shows |zeta(1/2+it)| for t between 0 and 60. As can be seen, the first few nontrivial zeros occur at the values given in the following table (Wagon 1991, pp. 361-362 and 367-368; Havil 2003, p. 196; Odlyzko), where the corresponding negative values are also roots. The integers closest to these values are 14, 21, 25, 30, 33, 38, 41, 43, 48, 50, ... (OEIS A002410). The numbers of nontrivial zeros less than 10, 10^2, 10^3, ... are 0, 29, 649, 10142, 138069, 1747146, ... (OEIS A072080; Odlyzko).
nSloanet_n
1A05830314.134725
221.022040
325.010858
430.424876
532.935062
637.586178

The 1st zero is like e and pi......


Riemann Zeta Function Zeros

DOWNLOAD Mathematica Notebook Zeros of the Riemann zeta function zeta(s) come in two different types. So-called "trivial zeros" occur at all negative even integers s=-2, -4, -6, ..., and "nontrivial zeros" occur at certain values of t satisfying
 s=sigma+it
(1)
for s in the "critical strip" 0<sigma<1. In general, a nontrivial zero of zeta(s) is denoted rho, and the nth nontrivial zero with t>0 is commonly denoted rho_n (Brent 1979; Edwards 2001, p. 43), with the corresponding value of t being called t_n.
Wiener (1951) showed that the prime number theorem is literally equivalent to the assertion that zeta(s) has no zeros on sigma=1 (Hardy 1999, p. 34; Havil 2003, p. 195). The Riemann hypothesis asserts that the nontrivial zeros of zeta(s) all have real part sigma=R[s]=1/2, a line called the "critical line." This is known to be true for the first 10^(13) zeros.
Wolfram Riemann Zeta Zeros PosterAn attractive poster plotting zeros of the Riemann zeta function on the critical line together with annotations for relevant historical information, illustrated above, was created by Wolfram Research (1995).
RiemannZetaZerosReImAbs
MinMax
Re
ImPowered by webMathematica
The plots above show the real and imaginary parts of zeta(s) plotted in the complex plane together with the complex modulus of zeta(z). As can be seen, in right half-plane, the function is fairly flat, but with a large number of horizontal ridges. It is precisely along these ridges that the nontrivial zeros of zeta(s) lie.
RiemannZetaZerosContoursReImThe position of the complex zeros can be seen slightly more easily by plotting the contours of zero real (red) and imaginary (blue) parts, as illustrated above. The zeros (indicated as black dots) occur where the curves intersect.
RiemannZetaSurfacesThe figures above highlight the zeros in the complex plane by plotting |zeta(z)| (where the zeros are dips) and 1/|zeta(z)| (where the zeros are peaks).
RiemannZetaAbsThe above plot shows |zeta(1/2+it)| for t between 0 and 60. As can be seen, the first few nontrivial zeros occur at the values given in the following table (Wagon 1991, pp. 361-362 and 367-368; Havil 2003, p. 196; Odlyzko), where the corresponding negative values are also roots. The integers closest to these values are 14, 21, 25, 30, 33, 38, 41, 43, 48, 50, ... (OEIS A002410). The numbers of nontrivial zeros less than 10, 10^2, 10^3, ... are 0, 29, 649, 10142, 138069, 1747146, ... (OEIS A072080; Odlyzko).
nSloanet_n
1A05830314.134725
221.022040
325.010858
430.424876
532.935062
637.586178

Everyone knows..........but no one does anything except to stare at my penis......and testicles......which is sexual harassment........but it is ok in the USA.........b/c women are involved....



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