One way to visualise the values of a function, f, that takes a complex argument, s, is to plot the “x-ray of f”. This entails plotting contour lines joining adjacent starting values of s where either the real part or the imaginary part of the value of f(s) is 0.

In this post I will show how the x-ray of the Riemann zeta function correlates to features of the iteration fractal described in my earlier posts.

In order to illustrate the correlation, I have chosen a region of the complex plane referred to in J. Brian Conrey‘s famous paper entitled The Riemann Hypothesis, which was published in the March 2003 edition of Notices of the AMS and won the 2008 Levi L. Conant Prize for expository writing in the field of mathematics.

The following image is taken from the second page of Professor Conrey’s paper:









The image on the left is a 3D plot of |Re(ζ(s))| and the image on the right is the x-ray of the same region, with Re(ζ(s)) = 0 (solid) and Im(ζ(s)) = 0 (dotted). The paper states that this region, “may be the first place in the critical strip where the curves Reζ(s) = 0 loop around each other”.

I have set out below an animation made up of nine separate frames. Each frame covers the starting values of s between -2 ≤ Re(s) ≤ 5 and 121410 ≤ Im(s) ≤ 121417, corresponding to a slightly enlarged version of the region referred to in Professor Conrey’s paper.

The first frame is the fractal map for this region with a resolution of 0.01 and a precision of 20 significant figures. The white line running vertically through the image is the “critical line” at Re = 0.5 and the black tramlines delineate the edges of the critical strip at Re = 0 and Re = 1 respectively.

The second to fourth frames comprise, respectively, the Re(ζ(s)) = 0 (black) and Im(ζ(s)) = 0 (blue) components of the x-ray of the region, and their combination, plotted using Wolfram Mathematica 10. The fifth to seventh frames show the x-ray components superimposed on the fractal map, separately and together.

The final two frames include a box highlighting a region where there is a tiny segment on which the non-trivial zero at approximately 121412.990421458… is located. An enlargement of which is shown below the animation:

















It is clear that the edges / centres of the finger features of the fractal map coincide closely with the contour lines of the x-ray components. The very large butterfly in the centre of the plot corresponds to one of the contour lines of Im(ζ(s)) = 0 that does not loop back, but continues into the region Re(s) >> 0.

The large butterfly is so wide that the two segments that end at its centre carry non-adjacent non-trivial zeros. The non-trivial zero at approximately 121412.990421458… is located between this pair of non-adjacent non-trivial zeros, and it appears on a very tiny segment that begins and ends on its own small butterfly. This tiny segment is almost invisible at the centre of the red box.

The following image is that same highlighted section, this time with a resolution of 0.001, the non-trivial zero is marked by a black dot in the centre of the image:









Finally, the following image shows the section of the Riemann-Siegel Z function, Z(t), for 121410 ≤ t ≤ 121417:









As expected, there is a very large maximum close to the position of the large butterfly.

I will continue to investigate correlations between the fractal map and other aspects of the Riemann zeta and Rieman-Siegel Z functions and report my results on this page.