#hurwitz zeta function

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Zeta functions: main bulb repetition in iteration fractals

In previous posts on this page, and in a recent talk on this subject, I have remarked that the iteration fractal in the neighbourhood of each non-trivial zero of the Riemann zeta function is a scaled and distorted reproduction of the main bulb.

One way of evidencing this phenomenon is to generate the fractal for each starting value, s, in a random section of the complex plane and, for each value of s where the real and imaginary parts of ζ(s) are both within some fixed distance from the origin, to place a pixel at that point with the same colour as the colour ascribed to s in the fractal image.

For example, the following iteration fractal image covers the section of the complex plane between -2 ≤ Re(s) ≤ 13 and 100 ≤ Im(s) ≤ 150, with a resolution of 100 pixels per unit. The image has been rotated 90° clockwise such that the imaginary axis runs from left to right:

The image contains ‭7,506,501‬ pixels. For many of the individual starting values, s, the absolute values of the real and imaginary parts of ζ(s) are less than 25. If each qualifying pixel is plotted in a simple x,y plot covering the section of the complex plane between -25 ≤ Re(s) ≤ 25 and -25 ≤ Im(s) ≤ 25, using the colour of the pixel ascribed to the starting value, s, in the above image, the result is as follows:

It is clear that the overall image corresponds to the main bulb of the iteration fractal, but the density of coverage away from the origin decreases rapidly.

The same logic can be applied to Hurwitz zeta functions, ζ(s, a), with rational a.

For example, the following iteration fractal image for the Hurwitz zeta function ζ(s, 0.75) covers the section of the complex plane between -2 ≤ Re(s) ≤ 13 and 100 ≤ Im(s) ≤ 150, with a resolution of 100 pixels per unit. The image has been rotated 90° clockwise such that the imaginary axis runs from left to right:

As before, for many of the individual starting values, s, the absolute values of the real and imaginary parts of ζ(s, 0.75) are less than 25. If each qualifying pixel is plotted in a simple x,y plot covering the section of the complex plane between -25 ≤ Re(s) ≤ 25 and -25 ≤ Im(s) ≤ 25, using the colour of the pixel ascribed to the starting value, s, in the above image, the result is as follows:

Again, it is clear that the overall image corresponds to the main bulb of the iteration fractal. The density of coverage is higher than for the Riemann zeta function.

It is also possible to work in the opposite direction.

For example, the following iteration fractal image for the Hurwitz zeta function ζ(s, 0.75) covers the section of the complex plane between -25 ≤ Re(s) ≤ 25 and 25 ≤ Im(s) ≤ 25, with a resolution of 40 pixels per unit. The image has been rotated 90° clockwise such that the imaginary axis runs from left to right:

The absolute value of the real and imaginary parts of ζ(s, 0.75) for each starting value, s, in a given section of the complex plane may be less 25. If so, we can colour the pixel corresponding to s using the corresponding pixel colour from the above image.

The result for the section of the complex plane covering -2 ≤ Re(s) ≤ 13 and 100 ≤ Im(s) ≤ 150, with a resolution of 100 pixels per unit, is as follows:

The image constitutes a reasonable approximation of the directly calculated fractal (see above) and takes considerably less time to generate. The black areas above and below the image correspond to places where the real and/or imaginary parts of ζ(s, 0.75) are greater than 25.

Another way to visualise this scaling and distortion phenomenon caused by zeta functions is to repeat the previous experiment with an arbitrary geometric design.

For example, the following image corresponds to the section of the complex plane covering -25 ≤ Re(s) ≤ 25 and 25 ≤ Im(s) ≤ 25, with a resolution of 40 pixels per unit. The image has been rotated 90° clockwise such that the imaginary axis runs from left to right:

The following image is formed by calculating ζ(s) for each staring value, s, in the section of the complex plane covering -2 ≤ Re(s) ≤ 13 and 200 ≤ Im(s) ≤ 220 and colouring those pixels where the absolute value of the real and imaginary parts of ζ(s) are less than 25 using the corresponding pixel colour from the above image. The result is as follows:

The overall pattern is preserved with repetition, but it is clear that the green and yellow sections are twisted such that they face upwards rather than their downward orientation in the original image.

Here is the corresponding image for ζ(s, 0.35):

It is clear that, not only are the green and yellow sections twisted above, there is a repetition below where the blue and red sections are twisted such that they face downwards rather than their upward orientation in the original image.

Here is the corresponding image for ζ(s, 0.75):

The repetition in the lower section has persisted. Additional experiments indicate that this repetition below is a feature of all Hurwitz zeta functions ζ(s, a) with rational a.

The same kind of distortion can be applied to any image arbitrarily centred on the origin in the complex plane. The results are particularly disconcerting when the original image is a human face. For example, here is a computer generated image of a human face copied from the website thispersondoesnotexist.com:

Here is the distortion caused by ζ(s, 0.35) in the section of the complex plane covering -2 ≤ Re(s) ≤ 13 and 200 ≤ Im(s) ≤ 220:

Here is the corresponding distortion caused by ζ(s, 0.75):

The inversion / twisting, and upper and lower repetitions that are a feature of Hurwitz zeta functions are easy to discern.

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Hurwitz zeta function: iteration fractal example near a threshold

I have recently refactored zeta_machine to enable it to generate (and colour) iteration fractal images for Hurwitz zeta functions, ζ(s, a), with rational 0 < a ≤ 0.999.

Observations suggest that, where 0.109155 < a < 0.711189 (approximately), the corresponding Hurwitz zeta functions have a single attractive fixed point and produce iteration fractals broadly comparable to the Riemann zeta function iteration fractal.

Where 0.711189 < a < 0.916637 (approximately), the corresponding Hurwitz zeta functions have a second attractive fixed point, giving rise to some interesting fractal images which will be the subject of some future posts.

I was interested in generating fractal images as close as possible to the threshold. This required a rational value for a that was close to, but less than, 0.711189, and which had a denominator no higher than 1000; the value of the fraction 32/45 is 0.7111...

As I hoped, the results are interesting, particularly at high resolution.

The following image covers the section of the complex plane between 0 ≤ Re(s) ≤ 6 and -4 ≤ Im(s) ≤ 4, with a resolution of 200 pixels per unit. The image has been rotated 90° clockwise such that the imaginary axis runs from left to right.

The image shows only the ″type1″ points i.e. those values, s, for which  ζ(s, 32/45) blows up to infinity upon iteration. The colouring scheme is cyclic (between blue and cyan). This is necessary because the iteration count covers the range from 2 to 5000 inclusive (5000 being the maximum number of iterations that zeta_machine will measure):

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Hurwitz zeta function: the iteration fractal for ζ(s, 0.75) near three closely spaced complex zeros

I have now finished the major refactoring of zeta_machine and the application can now generate, and colour, iteration fractals of the Riemann zeta function, Hurwitz zeta functions ζ(s, a) for rational a > 0, and Dirichlet L functions for primitive Dirichlet characters with modulus 1 to 24, or any prime modulus.

Hurwitz zeta functions where 0.711 < a < 0.917 (approximately) have a second attractive fixed point and this leads to interesting fractal features. zeta_machine renders points, s, falling with the basins of attraction of these second attractive fixed points in shades of grey. In all other respects, the iteration fractals of Hurwitz zeta functions that I have studied have very similar characteristics to the iteration fractal of the Riemann zeta function.

For example, the complex zeros of the function appear on “segments” which may be open or looped, as exemplified in the following image of ζ(s, 0.75), which also shows features associated with the second attractive fixed point.

The image covers the section of the complex plane between 0 ≤ Re(s) ≤ 1.2 and 10006.8 ≤ Im(s) ≤ 10008.2, with a resolution of 1000 pixels per unit. The image has been rotated 90° clockwise such that the imaginary axis runs from left to right.

I have superimposed on the image the corresponding “xray” illustrating the contour lines that arise when values, s, on the complex plane are plotted only where ζ(s, 0.75) is either wholly real (i.e. Im(ζ(s, 0.75)) = 0, shown in purple) or wholly imaginary (i.e. Re(ζ(s, 0.75)) = 0, shown in pink). Where the contour lines cross, the real and imaginary parts of ζ(s, 0.75) are both zero, which corresponds to a complex zero:

There are three complex zeros, and each is associated with a separate segment. Two of these segments are looped, and the third is open. There is some distortion of the curve of the open segment as it approaches the looped segments.

Using mpmath, I confirmed that the complex zeros are located at the following approximate values of s:

0.3886260081 + 10007.08415i 0.645470614 + 10007.88184i 0.7339947583 + 10007.31137i

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Zeta_machine: animation of colour phase plots of Hurwitz zeta functions

The animation below shows colour phase plots of the Hurwitz zeta functions ζ(s, a) for rational 0.1 ≤  a ≤ 2.5 in increments of 0.1.

Each frame of the animation covers the section of the complex plane between -30 ≤ Re(s) ≤ 10 and -50 ≤ Im(s) ≤ 50, with a resolution of 10 pixels per unit. The frames have been rotated 90° clockwise such that the imaginary axis runs from left to right.

When a = 1, the function is the Riemann zeta function. The frame where a = 1 is easily recognisable because the bottom portion of the frame is pure red rather than the multicoloured and black portions associated with functions where the value of a is not 1.

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Zeta_machine: colour phase plots of zeta functions

I have recently added a function to zeta_machine to generate colour phase plots of the Riemann zeta function, Hurwitz zeta functions and Dirichlet L functions.

For each pixel in the plot corresponding to a complex value, s, the relevant function, z = f(s) is evaluated, and the pixel is coloured according to the values of Arg(z) and |z|, based on the colouring of the unit disk below:

For each pixel corresponding to z = f(s), the value of Arg(z) determines the hue (reading anticlockwise, with red indicating a positive real value for z), and the value of |z| determines the brightness, with black indicating zero and higher values quickly leading to full saturation.

Pixels close to zeros are coloured black. The new colour map function assigns a white colour to pixels corresponding to poles.

I have set out below, three example images, one for each of the key functions featured on this page.

Each image covers the section of the complex plane between -30 ≤ Re(s) ≤ 10 and -30 ≤ Im(s) ≤ 30, with a resolution of 10 pixels per unit. The images have been rotated 90° clockwise such that the imaginary axis runs from left to right.

Riemann zeta function

Hurwitz zeta function, a = 0.75

Dirichlet L function, principal character, modulus = 5

The trivial and non-trivial zeros are clearly visible as small black dots. In the Riemann zeta function image, the non-trivial zeros line up along the critical line. The Hurwitz zeta function image shows that the non-trivial zeros have variable real parts, whereas the Dirichlet L function image shows that the real parts of the non-trivial zeros are either 0 or 0.5.

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Riemann zeta function: xrays of a series of Hurwitz zeta functions, ζ(s, a), with rational a between 0.01 and 1.50

I have recently been refactoring zeta_machine to add functionality to render iteration fractals for various Hurwitz zeta functions, ζ(s, a), for rational a. The Riemann zeta function is the special case where a = 1. Hurwitz zeta functions were the subject of an earlier post, and I now wish to integrate the general functionality into zeta_machine.

The main programming issue to overcome is providing a colour scheme for each a such that the resultant fractals' surfaces maintain a superficial similarity with the iteration fractal of the Riemann zeta function that dominates this Tumblr page, namely: blue / cyan areas corresponding to those values, s, where the value of ζ(s, a) blows up; and green / yellow / orange / red areas corresponding to those values, s, where the value of ζ(s, a) converges on the relevant function's attractor fixed point. My initial investigations suggest that, for rational a in the range 0.01 to 1.50, ζ(s, a) always has a single attractor fixed point.

This fractal colouring issue has not prevented me, however, from producing images of contour lines where Re(ζ(s, a)) = 0 or Im(ζ(s, a)) = 0 for various rational a, as per Juan Arias de Reyna's "xray" of the Riemann zeta function, which was the subject of an earlier post.

I have set out below an animation comprising 150 separate frames, each frame corresponding to the xray of a different ζ(s, a) function for rational a between 0.01 ≤ a ≤ 1.50. The value of a is incremented by 0.01 between each frame.

Each frame covers the section of the complex plane between -10 ≤ Re(s) ≤ 10 and 50 ≤ Im(s) ≤ 100, with a resolution of 40 pixels per unit. The images have been rotated 90° clockwise such that the imaginary axis runs from left to right.

In order to avoid the potentially adverse consequences of floating point indeterminism in the calculation of individual values of ζ(s, a), I tend to use unreduced fractions when entering the parameters into zeta_machine i.e. for a = 0.05, I enter 5 as the numerator of a, and 100 as the denominator, rather than the obvious reduced alternatives of 1 and 20 respectively. This generally results in longer plot times.

In each frame, the purple contour lines show those points, s, where Im(ζ(s, a)) = 0, and the pink contour lines show those points, s, where Re(ζ(s, a)) = 0. Where the contour lines cross, the real and imaginary parts of ζ(s, a) are both zero, which corresponds to a complex zero of the relevant function. The frame rate is 5 frames per second.

Here is the animation:

Note the following:

in each frame, the pitch between the contour lines in regions where Re(s) << 0 reduces slowly with increasing height above the origin and, between frames, there is a perception of a gradual shift down the imaginary axis towards the real axis

the pitch between the contour lines where Re(s) > 0 increases rapidly between frames, and there is a perception of chaotic change; a wider perspective will likely be necessary to quantify what is happening

the overall impression is of two distinct “layers” moving past one another at different rates, with the area of overlap corresponding to the critical strip

apart from the special case of the Riemann zeta function, where a = 1, the complex zeros, visible where the contour lines cross, are very often located off the critical line, with a real part greater than 0.5 and, in some cases where a > 1, a real part greater than 1

the trajectory of each contour line, including whether it penetrates into the positive real half plane, or loops back into the negative real half plane, and also which, if any, contour line of the other type it crosses, appears to be governed partly by the proximity or otherwise of a contour line of the other type; the visual impression is that of attractive and repulsive forces at work

the proportion of contour lines that penetrate into the positive real half plane appears to vanish where a is close to 1

I plan to perform some more quantitative analysis of the individual frames of the animation after I have solved the fractal colouring issue. 

Early indications suggest that the iteration fractals for Hurwitz zeta functions with rational a between  1.00 < a ≤ 1.50 are visually very interesting.

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After my last post, a Redditor with the user name Dgafaboutconvergence asked me to try to generate some iteration fractal images based on Dirichlet L functions.

These functions can be evaluated as linear combinations of Hurwitz zeta functions, which meant that I could reuse some of my existing code.

After consulting the relevant source code of mpmath I adapted the code of hurwitz_zeta_machine to evaluate and iterate Dirichlet L functions with different modulus values.

I tested the resultant application, inevitably called dirichlet_zeta_machine, and the above image summarises the initial results. Each image covers the section of the complex plane between -6 < Re(s) < 6 and -25 < Im(s) < 25 with a resolution of 10 and 10 significant figures. From top to bottom, the relevant Dirichlet L function moduli are 1 (Riemann zeta function), 3, 5, 7 and 11. The images on the left show the zeros and the images on the right show the images overlaid on the fractal images.

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Hurwitz zeta function: trajectories of the zeros of various Hurwitz zeta functions

Since my last post I have been generating iteration fractal images based on Hurwitz zeta functions, ζ(s, a), with 0.5 <= a <= 1. I was motivated to do so after reading the paper On the Distribution of Zeros of the Hurwitz Zeta-Functions, Math. Comp. 76 (2007), 323-337, by Ramūnas Garunkštis and Jörn Steuding. The above link is to the pre-print of the paper hosted on Professor Garunkštis’ page on the University of Vilnius website.

In their paper, the authors described the trajectories of some of the zeros of various Hurwitz zeta functions, ζ(s, a), with 0.5 <= a <= 1. When a = 0.5, several zeros lie on the line Re(s) = 0. As the value of a approaches one, all of the zeros migrate and eventually line-up on the critical line.

The trajectories of the zeros were summarised in the following remarkable image:

I was interested in seeing how these zero trajectories were related to the corresponding iteration fractals. I therefore used Hurwitz_zeta_machine to generate 502 individual images covering the same section of the complex plane as the above image.

Of these 502 images, 251 images show the zeros alone against a black background. The other 251 images show the zeros superimposed on the corresponding iteration fractals.

Rather than showing the zeros as single points, I used the “delta” function of Hurwitz_zeta_machine to selectively colour those points in the image that are within 0.5 units of the actual zeros. The qualifying points are coloured using a shade of grey that varies with distance; the qualifying points farthest from the zeros are black, and those closest to the zero are white.

Each individual image can be viewed as a horizontal slice through the above image.

The resultant animation is too large to be uploaded to Tumblr, so I have uploaded it to an alternative image hosting site: link to animation.

Here are two frames from the animation. The first shows 12 zeros of the Riemann zeta function lined up on the critical line, the second shows 16 zeros of the Hurwitz zeta function ζ(s, a), with a = 0.5. Note how four of the zeros are lined-up on the line Re(s) = 0:

The following points are worth noting:

When a = 1, ζ(s, a) is the Riemann zeta function and the zeros are lined-up on the critical line.

When a = 0.5, four of the zeros are lined up on the line Re(s).

The zeros are always located on yellow / orange patches of the fractal surface, which correspond to regions that fall within the basins of attraction of the relevant function’s attractor fixed point.

Specifically, the zeros are associated with “segments” that span one or two “segment endpoints”. See my earlier posts on features of iteration fractals based on the Riemann zeta function for an explanation of these phrases.

The yellow / orange regions are often divided by large blue / cyan / green regions that grow and shrink, but always avoid the zeros.