RZViewer showing a third degree transcendental L-function using the functional equation method in RZViewer
Riemann Zeta Function Viewer 2.0 for Mac Flight Manual
Current Issue 2.0.0 Mac OS Tiger to el Capitan
Dark Heart Package 2.0 for Mac Latest full overview of DH and RZ Viewer 2.0
Download all viewers with Riemann Zeta and Dark Heart viewer source codes
Uses Mellin transform integrals in Dirichlet, Hecke, elliptic curve and modular form L-functions to give a smooth representation of a wide variety of L-functions permitting investigation such as confirming the Birch Swinnerton-Dyer conjecture. Includes an experimental general Dokchitser type inverse Mellin transform algorithm for motivic L-functions, operating successfully on proof-of-concept test examples.
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In addition to its scintillating Mandelbrot and Julia exploration facilities, version 1.7 supports a full spectrum of abstract zeta and L-functions, including those of elliptic curves and modular forms. It has direct onboard parametrization of Riemann and Hurwitz zeta functions, Dirichlet L-functions, Dedekind zeta, and Hecke L-functions as well as Davenport-Heilbronn counterexamples. It also contains a full suite of examples of L-functions of elliptic curves, modular forms and Maass forms, including the most recent third degree transcendental L-functions so far discovered.
Includes template upload C-scripts running directly off Mac Terminal generating parameter files for RZViewer using coefficients generated from open source cross-platform Sage and PARI-GP scripts for the L-functions of elliptic curves and modular forms. The package also includes Tim Dokchitser's Computel algorithm and PARI-GP to generate further coefficient templates and bitmaps to run under RZViewer.
See http://dhushara.com/DarkHeart/RZV/ for research, updates and source code.
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Research investigations using RZViewer its associated Matlab toolbox and its companion Dark Heart Viewer:
- A Dynamical Key to the Riemann Hypothesis (with movies)
- Fractal Geography of the Riemann Zeta Function (with movies)
- Experimental Observations on the Riemann Hypothesis
- Exploding the Dark Heart of Chaos
By clicking, you can also investigate the
dynamical parameter plane (Mandelbrot set) of the critical value 1, and the Julia sets of z=f(z)+c (or z=cf(z)) for (x,y)=c.
Zeta, its additive Mandelbrot set and Julia set of c=0.
You can also investigate related functions eta, xi,
gamma, and psi (digamma), the derivatives of zeta, xi and gamma and Newton's method
on these as well (see below).
Basics:
Drag a rectangle to enlarge a portion of the current image.
Click a point to cycle:
Function > Mandelbrot > Julia(c)
at the clicked point c on the parameter plane (Mandelbrot set).
The controls all have floating help panes to guide you.
The window can be resized to suit in real time,
resulting in a refresh and recalculation.
Standard settings:
Threads enables a multi-core machine to work faster using as
many threads as coprocessors. Max
Iterations gives how many
iteration steps of a point on the Julia or Mandelbrot set before maximum
iteration cutoff. Recalc refreshes and recalculates the existing
screen. Reset
returns to the RZ function and default scale but doesn't alter any other
settings. Keep scope on click keeps the window centre and scale when clicking
between Mandelbrot, Julia and function, so the other parameters can be changed
without changing the point of view.
To facilitate locating critical points for calculating Mandelbrot sets, tick dZ and explore the derivative function for a critical point (a zero of the derivative looking like a dimple). Use dragging to blow up to high resolution. Tick Keep scope and slowly click twice right on the zero without moving the mouse. The c-values will now appear in the Julia portrait. Then tick Hold, untick dZ and click twice through the original function to the Mandelbrot set. The c-values are retained in the text field by Hold. Now untick Hold and click the Set R,I button and you will get the Mandelbrot sets of the given c-value.
To facilitate fast fractal iteration, the default is
100 function series terms and a cutoff of 64 iterations for each point of the
Mandelbrot and Julia sets, but these can be adjusted for greater accuracy.
Video of Julia set Explosions as we move c from ~ -19 to ~ -15. Notice the independent variations of the dynamics in the central bay and the right half-plane.
Menus:
About gives a complete instruction summary. Save saves the current window's image as a tif file and all the settings for a given calculation in a text file (see format below), enabling you to save all the parameters for future investigation and redrawing. It will overwrite any files of the same name. Load Parameters will load a previous calculation and redraw it. Load Movie Seq loads a sequence text file to generate a series of images to make a movie (see format below). Save Movie saves the sequence as a series of LZW compressed tifs. Images in the main window can also be captured and saved directly to png format using shift apple 4. Print enables an image to be printed or exported to pdf format. Page Setup needs to be landscape in 80% to fit the standard window on one A4 page. Help directs you to the About menu.
Video of fireworks in the zeros from L(5,4), rotating each character by the factor implied by their position on the unit circle.
Advanced settings:
Pop-up Menus: Color schemes can be changed in real time using the color pop up
menu. All the other controls require window recalculation.
Functions related to zeta, including eta, xi, gamma, psi and
others discussed below can also be explored using the function pop-up menu. For
completeness, an alternative Gamma product formula, can be used instead of the default
Lancos approximation, by pulling down the pop-up menu to choose a power of 10
terms in the product, but is only slowly convergent, so is perilously slower
and less accurate unless 10,000 iterations are used.
Buttons and Text Fields: A function terms text field is provided to set how many terms are calculated in the chosen sum or product function, with a default of 100 for speed of initial exploration. This can be adjusted to exact large values which have anomalous behavior (see end). Current window parameters and c values can be read out from and written into the text fields and set by pressing the appropriate button. L(m,n) text fields enable examining a wide variety of Dirichlet L-functions as well as Zeta and Eta. Numbers in the real text fields are double precision and may appear in floating format e.g. -7.000000186963007e-05 manifestly too large to see in their entirety in the text box without dragging or selecting all and copying. The Julia c values will be overwritten by the Mandelbrot critical point origin and current Julia click point each cycle but can be manually re-entered as desired.
Tick boxes: dZ portrays the derivative in function mode so one can find critical points for portraying Mandelbrot sets. Full Terms is located on the main page as a control and as an attribute. It forces the current function to use all the specified series terms. Otherwise it breaks off if the terms get smaller than 0.001 the size of the sum. Occasionally not having Full causes strip dislocations in a few regions. The control has immediate effect during drawing and is useful if a slow blow up of a Julia set needs to have full iterations just for the complex boundaries of the kernel. Mellin uses a Mellin transform in the central region if the function admits an analytic continuation. Like Full Terms, it is active during drawing. The logFn. tick box examines the log of the selected function. Split sets the functional equation split (-1> -100, 0>0, 1>0.25, 2>0.5). There is an x,y~0 orbit trap option highlighting points on the orbit which iterate close to the axes. Mobius transforms the complex plane, mapping the line x=1/2 onto the unit circle, so you can see the whole extent of the Dirichlet functions. For the exact mapping see below. Escape only tests only for escaping points, to avoid spurious periodic solutions from overwriting escaping orbits, but is slower because attracting orbits have to run to the Max interations for each point. crVal uses critical value instead of critical point for Mandelbrots. This is useful for generating parametric movies running between critical values. For finding critical values see blue skiews method. Tick box +c/xc changes the parametrization from f(z)+c to cf(z) giving a second parameter plane and collection of Julia sets for each function. Angle turns on and off the green/yellow display of the angle (argument) of f(z) in function view for each mode. Rotate rotates the image by pi/2 to assist viewing the critical line.
Parameter planes of 
and 
showing extensive differences
Sliders: Attractor bound adjusts the size
of the epsilon neighbourhood testing for fixed point or periodic attractors. Escape
bound sets the real absolute bounds on points
escaping to infinity.
There are several colour schemes for function, Mandelbrot and Julia:
Function: (0) Absolute-Angle mode rgb = red logarithmic abs(z), blue cosine abs(z), green angle(z). This is the most informative although not the most appealing (1) Real-Imaginary mode rgb = real, imaginary and angle, (2) X-ray mode with red, cyan real~0 and imag~0, by exp(-x^2), adjustable by the attractor bound, overlaying bg = abs(z) and angle(z) Highlights gram points with real=0 on x=1/2 and zeros at the intersections of real~0 and imag~0. (3) Positive-Negative mode. Highlights the positive and negative real (r~c), imaginary (g~m) and angle by 1-cos (b~y). (4) Log-angle mode This method uses abs(log(1/(1-Z(z))) for the absolute value, adjustable using the escape bound slider, retaining the angle of Z(z). This still highlights the zeros since log(1/(1-0))=log(1)=0 but highlights variations in the function to the right of the critical line. It is good at highlighting the function in the right half-plane and in the critical strip in a way which evenly compares asymptotical large and near-unitary values. (5) Soft mode rgb = cos(angle) sin(angle), abs(z). Good for retrieving function data from tif file.
Function modes 0-3
Mandelbrot: (0) Sine wave colours, (1) attactor coded colours,
(2) RGB ranked colours (3) Potential function rainbow. Attractor coding gives escaping points tending to real infinity green through orange and points remaining finite coloured by decreasing blue by iteration, combined with redness corresponding to the attractor period. (4) attractor coded inverse quadratic. (Highlights geometrical details lost in high iteration numbers).
The 'critical' point for the Mandelbrot set is chosen
to represent a prominent critical landmark. For zeta, eta, mu, and sigma it is
chosen to be 1000 corresponding to the limit 
as real(z)
+∞. For gamma it is 1.465. For Xi it is 0. Other critical points can be chosen using the advanced settings and inputing CX CY values under Mandelbrot mode, just as c values can be inserted for any Julia set. A list of critical points for zeta eta and xi is provided below, or can be approximated using the values of observed zeros of dzeta etc. You can also input CX and CY in function mode to generate a transform of the function, whose zeros are at the fixed points of the critical values [f(cv)-v=0 or f(c+v)-v=0] and hence candidates for being in Mandelbrot kernels of the critical points.
You can flip easily between Mandelbrot mode and transformed function mode using 'keep scope on click'. To exit reset. This can potentially find a finite number, out of the infinite collection of kernels, but sometimes the one located is vanishingly small, leading to blowing the mantissa of the floationg point doubles before we reach it.
Not all the functions have meaningful Mandelbrot sets. The zeta attracting fixed point is at -0.2959050055752.
Colour scheme 1 uses attractor coded colours. At left the incipient and actual attracting periods of bays in the multiplicative Mandelbrot set are shaded with increasing redness. The periods can then be verified by inspecting local Julia sets. Right an additive Julia set of 
distinguishes escaping points (blue) from
attracting periods in the negative reals (red) and positive (green) shaded by
iterations to reach 
-periodicity, aiding in investigating their bifurcations.
Julia: (0) Sine wave colours, (1) attactor coded colours,
(2) RGB ranked colours, (3) rainbow with potential function on escaping points.
Colour coded Julia attractors have
blue shaded to escaping real, red shaded to periodic attractor, green
non-negative periodic attractor, with grey indeterminate. The red and green are
tinged with blue to indicate the period. (4) attractor coded inverse quadratic.
For exploring the higher zeros of zeta, a high number
of series terms can be set using the slider. Values of the RZ function outside
the critical strip [0,1] are set to 0 after y=±450i because the sin/cos component of the analytic continuation overflows to Inf. This leaves the zeros in the critical strip still well rendered up to values in the millions with the highest number of zeta iterations set in the slider.
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Blue skies exploration of an additive Mandelbrot kernel near z95=0.78+95.29i. Top left to right. (1) The derivative function wih the chosen critical point starred. (2) The modified zeta function showing fixed values. (3) A further 'north' section of the same function showing the principal point starred close to the location of the original critical point. (4) The Mandelbrot kernel with bulbs. Bottom left to right: The hashed and starred locations point to a satellite Mandelbrot island whose period 3 bulb generates a Julia set (lower right) with period 3 kernels (mid right).
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Blue-Skies Method for Finding Mandelbrot Kernels for Zeta Functions
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This method enables exploration of a wide range of Mandelbrot sets of zeta functions intuitively taking advantage of the software algorithms without having to do complex arithmetic to look for the needle in the haystack.
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Step 1: Explore the critical points of the function using the dz derivative option. Choose a critical point and scale the image by dragging rectangles to focus on a small region around the critical point. If the critical point has strip-like artefacts due to function term round-off, tick the full terms option while the process is drawing the immediate neighbourhood. If the imaginary value is large, you may need to increase the function terms to get an accurate critical point.
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Step2: Tick the keep scope option and click the function twice right on the critical point to get Julia representation. This will insert the c value of the critical point into the Julia / Mand c text fields. Now tick Hold to lock these values in.
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Step 3: You now need to decide whether you are exploring an additive or multiplicative Mandelbrot set. Untick +c/xc if you are using multiplicative.
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Step 4: Untick dz so we are dealing with the original function (this didn't affect the Julia step because dz is active only in function mode) and do a Reset and then press Set R/I. This causes the critical point values to modify the zeta function causing the fixed values and principal point of the critical point to be zeros of the modified zeta transfer function.
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Step 5: Have a look initially at the principal point, which due to the maths is a double zero so has two yellow angular rays meeting in it. This is where one is most likely to easily find a principal Mandelbrot kernel. In the additive case the principal point will be quite close to the critical point because it only has an additive shift, but in the multiplicative case it may be rescaled and you will have to search for the double zero. You can insert a new Y value and click Set X,Y,S to move further up or down the imaginary axis.
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Step 6: With keep scope ticked, click the point once to go into Mandelbrot mode. Click Set R/I again to make sure the Mandelbrot is using the critical point you found. Drag a rectangle to zoom in on the neighbourhood of the point. This should give you a Mandelbrot kernel with reasonably well-formed bulbs, dendrites, and satellite black hearts if the number of function terms and iterations is high enough to be accurate at the values used.
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Step 7: You can now explore the region of the Mandelbrot kernel or its islands and can check the corresponding Julia 
kernels by clicking once on a spot with Hold unticked to pick up the actual c value you are now investigating. The Julia set may have only very tiny connected periodic kernels representing the local dynamics near the Mandelbrot kernel and in the multiplicative case these may appear only in a region of the Julia set close to the principal point.
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Non-trivial critical values of the Dedekind zeta function
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Step 8: You can also find a critical value to use with crVal by locating the critical point as a zero of dz scaling down to a microscopic domain around it and unchecking dz and refreshing to give the value of the function, which is stationary at the critical point. This can then be found by saving a bitmap and viewing it as a text file where the stationary value will be at the midpoint (see bitmaps below). Use the smallest size screen for a smaller bitmap. Selected critical values for zeta are included below.
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The Functions:
The package is designed to facilitate research into the widest spectrum of zeta-related and L-functions possible. Each of zeta, eta and all the L-function variants L(m,n) for m up to 63 for all n and for m up to 1000 for n=1, are encoded as a generalized zeta function which can then be differentiated, or have any of the functions from xi, through those like lambda and mu whose coefficients are important arithmetic functions, to the Newtons method's and dfferentiation applied to them. This creates a widest a diversity of exploratory functions as possible.
Zeta
Zeta is represented here in terms of Eta as:
for real(z)≥0
and by analytic continuation using:
for real(z)<0.
http://en.wikipedia.org/wiki/Riemann_zeta_function
Eta
Used to extend Zeta from real(z)>=1 down to real(z)>=0.
http://en.wikipedia.org/wiki/Dirichlet_eta_function
r example, for 
has residues 1, -1 at 0, 1
hence
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to avoid computational errors around the imaginary axis. This may result in some double zeros and double singularities along the critical line at the default values of series terms. The third zeros above and below have been coloured red to highlight the fractal interaction of the zeta zero basins.
where mu is the 
so that the zeta zeros are aligned with the critical line x = 1/2 (right).