This blog is an illustration of how it becomes possible to model the most dizzyingly complicated complex functions known in rich detail highlighting the zeros of Riemann's zeta and Xi functions.
Newton's method finds the zeros of a function by iteratively running up to the graph of the function (green) and then down the slope of the function (red) and repeating the process, converging towards a zero of the function.
Left: 1(a) is the zeta function showing trivial x-axis zeros, non-trivial on x = 1/2 zeros and the red infinity at z = 1. In (b) is the derivative of zeta showing the critical points. (c) shows the additive parameter plane f(z) = zeta(z) +c for the critical plateau z ~ 1000 showing sample Julia sets with the periods illustrated filling the right half plane. In (d) is the multiplicative parameter plane is portrayed. Right Julia sets of Newton's method on the Dirichlet L-function (5,2) and Riemann xi function.
This follows the formula below defining the Newton function
Nf of a function
f. For example, if we take the complex function
f(
z)=
z3-1, whose zeros are the complex cube roots of unity, we have the following iterations.
When we carry out this iteration in complex numbers, we find that some points don't tend to the zeros and instead enter a chaotic state, becoming the fractal
Julia set of the iteration illustrated below.
The three red, brown and green fractal basins are the points iterating to each of the three zeros, but the points on the black skeleton in the inset are on the
chaotic Julia set of the iteration.
Notice that every point on the Julia set bounds all three basins, so this is a fractal landscape involving three regions in which every point on any of their boundaries bounds all three regions, a seeming topological impossibility unless the set is fractal.
What then happens if we try to investigate the most complex and enigmatic function of all, the
Riemann zeta function, whose zeros seem to form a Fourier transform, or hologram, of the prime numbers?
The following set of equations show how we have to work a 'Houdini act' to define zeta, and extend it firstly from Re(
z)>1 down to Re(
z)>0 and then over the entire complex plane except the singularity at
z=1, using the gamma function and how we can in turn approximate the gamma function to calculate zeta.
But to get the Newton function of Riemann's zeta, we have to first differentiate it, which requires even more equation subtlety and a lot more processing power.
Thus we finally arrive above at a workable formula for Newton zeta and the related Newton Xi which also has the same enigmatic set of zeros.
Above we see at left the zeta function with its zeros, next the Newton function of zeta, attracting the non-trivial zeros on the line x=1/2 in green and the trivial zeros on the negative real axis at x=-2n in red. Notice the fractal way in which satellites of the basins are entangled.
Finally above center we have the Newton function of Xi with the basin of the third non-trivial zeta zero in red and the basins of the others in green, showing how the basins of the zeros are fractally reflected in one another, as shown in the two exploded images at right of fractal neighbourhoods of ensuing boundaries between the zeros immediately above and below the third one.
This has all been made possible thanks to the GCC complex.h math library, and the XCode integrated development environment for Mac in generating the applications downloadable in the previous blog.
Riemann zeta function and a series of Dirichlet L-functions
