In my previous posts, I have noted that the non-trivial zeros of the Riemann zeta function appear exclusively on the “segment” features of the iteration fractal, and that each such segment carries exactly one non-trivial zero.

There are two types of segment. In the first type, the segment spans two spatially separated segment endpoints. In the second type, the segment is looped around a single segment endpoint. I will refer to these segment types as “open” and “looped” respectively.

In my latest posts, I have provided some evidence that, at every segment endpoint, s, the value of ζ(s) is close to 1.83377. I noted that this is also the value of s (and ζ(s)) at the repeller fixed point within the main bulb of the iteration fractal.

I have also recently noted that each segment endpoint is surrounded by a cluster of similarly valued points, which I will refer to as an “endpoint cluster”.

In some cases, the endpoint cluster is large and diffuse. In other cases, the endpoint cluster is small and tightly packed. The variation in local gradients that is suggested by the relative sizes of the endpoint clusters is consistent with the variation in the gradient of the real part of the value of the Riemann zeta function in the neighbourhood of each endpoint cluster; shallow gradients and large endpoint clusters are a feature of the half plane where Re(s) >> 1 where the value of ζ(s) is known to converge to 1.

The relative sizes and positions of these endpoint clusters appears to have a profound effect on the configuration of the iteration fractal and, in particular, the configuration of the segments.

Looped segments, which are very much in the minority (certainly for those non-trivial zeros that have an imaginary part less than 10000), appear in two configurations:

In the first configuration, which is typically found adjacent to large and diffuse endpoint clusters spanned by other segments, the looped segment is located within the main fractal surface.

In the second configuration, the looped segment is located on an “island” fragment of the fractal surface, separated from the main clumps of the fractal surface that carry multiple non-trivial zeros.



In this next series of two posts I will set out the results of my investigations into the origins of looped segments. I will show examples of each configuration and provide evidence in support of my hypothesis about their origins.

The following images show plots of the section of the complex plane between -2 ≤ Re(s) ≤ 5 and 28976 ≤ Im(s) ≤ 28983 at resolutions of 0.01 and 0.001. In each case, the plots have been been rotated 90° clockwise so that the imaginary axis runs from left to right.

The first image shows the iteration fractal at a resolution of 0.01. The critical line at Re(s) = 0.5 is shown as a white line, with a single black pixel at each non-trivial zero:









There are nine non-trivial zeros in the above image. Each of the first three and the last three non-trivial zeros appears on an open segment. Each of the middle three non-trivial zeros, however, appears on a looped segment. Of these three looped segments, the outer two are located within the main fractal surface and the middle one is located on a fractal island.

Note that the open segments immediately adjacent to the outer pair of looped segments have to be considerably longer than their neighbours in order to span their respective pairs of segment endpoints.

Here are close-up images of the three looped segments comprising unit square plots of the iteration fractal at a resolution of 0.001 centred on the relevant non-trivial zero and spanning the critical strip:





First looped segment, carrying the non-trivial zero at 0.5 + 28978.1521i:





Second (island) looped segment, carrying the non-trivial zero at 0.5 + 28979.4096i:





Third looped segment, carrying the non-trivial zero at 0.5 + 28980.6594i:





The next image is the same section of the complex plane between -2 ≤ Re(s) ≤ 5 and 28976 ≤ Im(s) ≤ 28983, also at a resolution of 0.01, but this time with each point that satisfies the following set description coloured using a heat map colour scheme where the hue value at each qualifying point, s, is proportional to the absolute difference between ζ(s) and 1.83377:









Colours at the blue end of the spectrum indicate points, s, where ζ(s) is closest to 1.83377, with cyan, green, yellow and orange indicating increasing distance. Points, s, where |ζ(s) - 1.83377| > 1 are coloured red, regardless of the actual value of ζ(s).

The critical line at Re(s) = 0.5 is shown as a white line with a single black pixel at each of the nine non-trivial zeros:









One can see immediately that the endpoint clusters associated with each of the three looped segments share the following properties:

the distances between these endpoint clusters and their associated non-trivial zeros are smaller than the distances between the other endpoint clusters (which are spanned by open segments) and their associated non-trivial zeros

these are the closest endpoint clusters to the large and diffuse endpoint cluster that dominates the lower half of the image

these endpoint clusters are relatively small and tightly packed

There appears to be a one-to-one ratio between the total number of endpoint clusters and the total number of non-trivial zeros. This is to be expected if each segment carries exactly one non-trivial zero.

However, there does not appear to be any obvious direct correlation between the absolute sizes and positions of the endpoint clusters (being simply points, s, where ζ(s) happens to be close to 1.83377) and the positions of the non-trivial zeros (being points, s, where ζ(s) = 0), aside from the fact that the absolute value of ζ(s) is known to undulate between the non-trivial zeros.

It is clear, however, that the configuration of the segments appears to be at least partly dependent on the properties and relative positions of the endpoint clusters and their nearest non-trivial zeros.



At this stage, I believe that the probability of an endpoint cluster being associated with a looped segment, rather than an open segment, is at least partly dependent on its proximity to its nearest non-trivial zero; the closer the proximity, the more likely the corresponding segment will be looped. It also appears likely that looped segments are associated only with small and tightly packed endpoint clusters, rather than with large and diffuse endpoint clusters.

To test my hypothesis, I carried out a systematic search of the immediate neighbourhood of each non-trivial zero with an imaginary part less than 7000, looking for points, s, that had the characteristics of segment endpoints.*

For those non-trivial zeros that had at least one qualifying point in their immediate neighbourhood, I calculated and plotted a unit square of the corresponding section of the iteration fractal at a resolution of 0.01, centred on the relevant non-trivial zero and spanning the critical strip.

I found that, in every case, such a plot enabled me to determine instantly whether the segment carrying that non-trivial zero was open or looped and, if looped, whether it was located within the main fractal surface or on a fractal island. Overall, the test was highly selective for looped segments over open segments, but tended to identify looped segments located on fractal islands rather than those located within the main fractal surface.

In the next post I will present the results of this test.











*In this description of the test, the phrase “immediate neighbourhood” means, in relation to any given non-trivial zero, “within a 0.4 x 0.4i square centred on that non-trivial zero”, and the phrase “characteristics of segment endpoints” means, for any point, s, that |ζ(s) - 1.83377| < 0.35.