In my recent posts I have: (a) described a fractal pattern that arises from the iteration of the Riemann zeta function; (b) identified various fractal features that appear at all levels of magnification; and © demonstrated a correlation between the positions of some of these fractal features (specifically the “segments”) and the positions of the non-trivial zeros of the Riemann zeta function.

In this and my next few posts I will provide evidence of some further correlations, this time between the local maxima of the Riemann-Siegel Z function and some other features of the fractal (specifically the “butterflies”).

(See my earlier posts for an explanation of the terms, “segment” and “butterfly”. In the following text, references to “major” segments and butterflies mean those segments and butterflies that are the most prominent on or near the critical strip at 1 x magnification; obviously there are smaller segments and butterflies at every level of magnification.)

It is clear from the images set out in my earlier posts that the major segments and butterflies are rarely wholly contained within the critical strip. Indeed, many of these segments, and almost all of the butterflies, extended over the edges of the critical strip.

I have therefore generated some additional data covering the regions to the immediate left and right of the critical strip, such that my dataset now covers all starting values of s between -1 ≤ Re(s) ≤ 4 and 0 ≤ Im(s) ≤ 500 at a resolution of 0.01 and 20 significant figures, i.e. a vertical strip five times the width of the critical strip.

Among the many interesting features of the fractal that are visible on this expanded strip is a remarkable correlation between the number, position and relative size of the major butterflies and the number, position and relative amplitude of the maxima of the Riemann-Siegel Z function.

The following series of images together constitute the surface map for the section of the complex plane between -1 ≤ Re(s) ≤ 4 and 0 ≤ Im(s) ≤ 500 with a resolution of 0.01 and a precision of 20 significant figures.



The surface map has been divided into 25 equal sections, each spanning 20i, with each section being rotated 90° clockwise in order to maximise the resolution on this page. In each section, therefore, Re(-1) is at the top of the map, Re(4) at the bottom and Im(s) increases from left to right. The surface map of each section has 1,002,501 plotted points in around 72 different colours.

The white line running through each image from left to right in the top third of 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. Each non-trivial zero of the Riemann zeta function is marked by a black dot on the critical line, although these will be difficult to see at this scale (see earlier posts for clearer images with guidance marks).

Each surface map is shown together with a graph of the corresponding section of the Riemann-Siegel Z function plotted at the same horizontal scale, with t increasing from left to right. The graph data was generated using the siegelz function of the Python mpmath library. The graphs were plotted using Microsoft Excel 2003.

The evidence in these images, and those that will appear in my next few posts, supports the following propositions:

There is a 1:1 ratio between the number of major butterflies and the number of maxima of the Riemann-Siegel Z function. This is not surprising if, as suggested in my earlier posts, every major segment has exactly one non-trivial zero, because there is a major segment between each pair of major butterflies in the same way that there is a zero of the Riemann-Siegel Z function between each pair of maxima.

The centres of each major butterfly and each corresponding maximum of the Riemann-Siegel Z function appear at around the same height above the origin. In other words, the value of t that corresponds to the position of each maximum of the Riemann-Siegel Z function, Z(t), is approximately the same as the value of the imaginary part of the value of s at the centre of each major butterfly.

The size of each major butterfly is proportional to the size of the corresponding maximum. Further investigations will be required to establish the precise nature of the correlation, particularly whether the size correlation is based on height / width or area.

Here are the images:



0i to 20i

20i to 40i

40i to 60i

60i to 80i

80i to 100i



100i to 120i

120i to 140i

140i to 160i

160i to 180i



180i to 200i

200i to 220i

220i to 240i

240i to 260i

260i to 280i

280i to 300i

300i to 320i

320i to 340i

340i to 360i

360i to 380i

380i to 400i

400i to 420i

420i to 440i



440i to 460i

460i to 480i

480i to 500i

In the next post I will provide further evidence in support of the above propositions by showing a map of a region of the expanded strip at a height of around 2447i where the fractal exhibits a particularly large butterfly sandwiched between two particularly tiny butterflies. As anticipated, the corresponding section of the graph of the Riemann-Siegel Z function features a relatively large maximum, with an amplitude of around 12, between two relatively tiny maxima.