In one of my previous posts I showed images of the zeta fractal near two examples of Lehmer’s phenomenon, noting that the pair of non-trivial zeros at approximately 7005.062866175i and 7005.100564674i are located on adjacent, but separate, segments.

In a subsequent post, I showed an enlargement of a tiny segment carrying the non-trivial zero at approximately 7006.739662384i.

I was reminded of these properties by the tiny segment, curled around its own butterfly, shown in my last post. One reason why tiny segments appear to occur is the existence of large butterflies that are so wide that those segments that end at the centres of such butterflies carry non-adjacent non-trivial zeros; tiny major segments appear to be necessary in order to ensure that each segment carries exactly one non-trivial zero.

In my last post, the very large butterfly in question was associated with one of the Im(ζ(s)) = 0 contour lines of the x-ray of ζ(s) that extended into the Re(s) >> 0 half plane. I decided to investigate whether these observations can be generalised.

The following animation comprises two frames. Each frame covers the starting values of s between -2 ≤ Re(s) ≤ 5 and 7004 ≤ Im(s) ≤ 7014, a slightly higher and wider section of the complex plane than was covered in my earlier Lehmer’s phenomenon post.

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 tramlines either side delineate the edges of the critical strip at Re = 0 and Re = 1 respectively.

The second frame shows the Re(ζ(s)) = 0 (black) and Im(ζ(s)) = 0 (blue) components of the x-ray of the region superimposed on the fractal map:















Several observations are worth noting:

The non-trivial zeros comprising the Lehmer’s phenomonen pair are associated with overlapping pairs of black and blue contour lines

The large butterfly is, as noted in my previous post, associated with a Im(ζ(s)) = 0 contour line of the x-ray of ζ(s) that does not loop back, but instead extends into the Re(s) >> 0 half plane

As also noted in my previous post, the two segments that end at the centre of the large butterfly carry non-adjacent non-trivial zeros

The tiny segment carrying the non-trivial zero at approximately 7006.739662384i is located between these two segments

The pair of Im(ζ(s)) = 0 contour lines of the x-ray of ζ(s) between approximately 7011.5i and 7013i has one contour line looped around the other (my last post had two Re(ζ(s)) = 0 contour lines looped around each other)

The outer of these two Im(ζ(s)) = 0 contour lines is associated with an unusual and complex pseudo-finger feature whose centre is located at approximately 1.5 + 7012i.