**Fractal Geography of the Riemann Zeta and Related Functions**

Chris King Aug 2016 v1.10

Emeritus, Mathematics Department, University of Auckland

**PDF** v1.10 (16 Mb)

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**Abstract:***
The quadratic Mandelbrot set has been referred to as the most complex and
beautiful object in mathematics and the Riemann Zeta function takes the prize
for the most complicated and enigmatic function. Here we elucidate the spectrum of Mandelbrot and Julia sets
of Zeta, to unearth the geography of its chaotic and fractal diversities,
combining these two extremes into one intrepid journey into the deepest abyss
of complex function space.*

Generated with the author's Mac application **Dark Heart Viewer Package**. **Download** (Tiger to El Capitan) with application, manual and source code

**Latest article: The Physics and Numerical Exploration of Zeta and L-functions 2016**

See also: A Dynamical Key to the Riemann Hypothesis

and Experimental Observations on the Riemann Hypothesis

**Introduction:**

This paper completes a discovery process I
began in 2009, using computational applications I had developed, looking at the
Ôdark heartsÕ [1] - the
Mandelbrot parameter planes - of a wide variety of complex functions, including
the zeta function, to explore the world of complex functions as widely as
possible and elucidate universal properties.

This year, as I began to re-explore the
parameter planes, using a more versatile second generation version of the
application, I became literally sucked into the zeta abyss by an unending
stream of intriguing new and surprising features, which rapidly grew to the
point where I realized I was dealing with an entire geography of complex function
space, spread before me, vast and diverse, like the continents of Europe and
Asia combined. These are, as far as I know, hitherto unexplored, apart from
WoonÕs 1998 paper [2] setting out
a basic description of the Julia set of zero and the outlines of the Mandelbrot
set as in fig 1.

The current paper provides a full
investigation of the dynamics emerging from all types of critical point, from
those on the real line to the ones adjacent to the critical line x = ½.
The software to perform these investigations consists of an open source XCode
application for Mac downloadable from: http://dhushara.com/DarkHeart/.
For those unfamiliar with complex numbers, discrete dynamics, or the zeta
function, there is an extended introduction at the end of the paper.

Fig 1: (Left) The zeta function parametrized by
additive colors angle (green/yellow) and amplitude (blue waves and red) so that
0 is black/green, 1 is blue and large values are red and yellow with waves of
blue. (Centre) Parameter plane of from the
quasi-critical point 1000 on the asymptotic plateau in the right half-lplane
with singularity ÔislandÕ (inset) connected by a fractal thread. The bands of
blue and yellow distinguish points iterating to . Strictly speaking the green areas should also be black, as they iterate to the positive half-plane and become fixed, although far outside the iteration limit of the method. (Right) the Julia set of bounds basins of
attraction to the fixed point , containing the non-trivial zeros. Smaller island replicates
of the main connected component surround successive trivial zeros (inset). The
frond spacing as we ascend to larger imaginary values is spaced irregularly
with the zeta zeroes.

We are going to use graphic imagery to
explore and confirm the dynamics.
The approach is unashamedly numerical, depending on finite
approximations of arbitrary accuracy, using computational algorithms. It is
also intentionally Zen in its mathematical approach - Ôsymbolically silentÕ in
its primary use of graphical representations, with minimal symbolic
abstraction, to elucidate the geography as fully as possible before describing
it. This, supported by the software design is qualitative mathematics in
action, using the ÔartÕ to establish the ÔmathÕ.

Fig 1b: When attractor period coding is
used, it becomes clear that both the additive and multiplicative zeta parameter
planes are forming periodicities on their boundary in just the same manner as
the quadratic Mandelbrot set (inset), and that their Julia sets (inset) although
not having the same periodic geometry as polynomial Julia sets also have the
correct periodicities and display connectivity related to their site on the
parameter plane. This provides a signature example of a critical point on an
exponential plateau.

.

Since we know from Galois that we canÕt
solve fifth degree polynomials symbolically, let alone equations involving
transcendental functions, and the Riemann hypothesis that zetaÕs non-real zeros
all lie on the critical line *x* = 1/2 remains unsolved, despite having been proved for more abstract systems
[3]
,
even though all such zeros of zeta are palpably on the critical line, it is
clear that the symbolic approach, despite its capacity for abstract
generalization, has limitations when dealing with irregular systems of infinite
complexity. Hence the research approach taken in this paper.

**A Bridge over Turbulent Waters**

** **

The Zeta function is defined for , either as a sum over powers of the integers, or as a
product over primes . The sum formula is extended to by expressing it in terms of the eta functionÕs alternating
series** **. It is then extended again by
analytic continuation to , where , is the gamma function, generalizing the integer factorial *n*! The result is the
most complicated enigmatic complex function known to the human mind.

Fig 2: (a)
as a real
and complex function. (b) The additive Mandelbrot set of
with complex
exponential fronds. (c) The central frond for critical point *z*=1 has local quadratic bulbs. The corresponding view for asymptotic plateau quasi-critical
*z*=1000 is in inset (b). The bulbs have dendrites (d) supporting quadratic Mandelbrot satellites (e) whose left period 3 bulb has a quadratic period 3 Julia kernel (f,g) and orange plateau matching location * in inset (b).

To make a transition to the perplexing
situation posed by the extreme complexity of the zeta function, let us look at
a function that displays pivotal features of the situation in a simpler form.

Consider .
This is an exponential function with an extra *z*
term, which gives it a critical point at *z* =1,
since for *z* = 1. All transcendental functions can be represented as power
series equivalent to an infinite polynomial. .
Every fully differentiable ÔanalyticÕ complex function can be so represented.
This is similar to zeta but different in an important way. A power series
consists of polynomial terms, fixed integer powers of *z*, but a Dirichlet series like zeta consists of a spectrum of
exponential functions of integers. The situation is reversed, with weird and
wonderful consequences.

However our exponential does form a Rosetta stone for zetaÕs
dynamics. In fig 2 is shown the function, the Mandelbrot set of from the
critical point *z* = 1 (see the end section if
this is unfamiliar to you) and a period 3 Julia set. The function tends to zero
in the right half plane and to infinity in the left. However, it is sinusoidal
in the imaginary direction because an exponential of imaginary *y* is sin(*y*) + *i *cos(*y*), a sinusoidal function whose angle varies with *y*, neatly making complex exponential and trigonometric functions
imaginary versions of one another. Notice also the dimple at zero, indicating - the one zero of the function.

Looking at the additive Mandelbrot set of from 1, we see it is similar to our unfamiliar zeta case, with exponential fronds representing the waves of the imaginary exponential, zoomed laterally by the real exponential. Now the central frond looks a little different. When we zoom in on it (c), we find it has bulbs just like the quadratic Mandelbrot of fig 35, and these bulbs lead to dendrites containing satellite Mandelbrots identical to the quadratic case, as expressed in Douady and HubbardÕs seminal article on polynomial-like mappings
[9]
. Moreover, when we look at the Julia set
of the left-hand 3 bulb on the above satellite, the Julia set (f, g) has a tiny
period 3 quadratic Julia ÔkernelÕ, set in a fractal web connected to other like
kernels.

Now there is another Ôquasi-criticalÕ point
of this function, where it tends to 0 at (+)infinity. If we had instead used
1000 as our starting point, we would have found a subtly different Mandelbrot
set, with no
quadratic bulbs, which at the same point as our little Mandelbrot satellite was
in the middle of an orange tongue. The Mandelbrot set correctly classifies the dynamics in the positive half plane
while describes the
local polynomial dynamics in the Julia set, as can be seen in fig 2. Julia set
dynamics is thus regionally defined in terms of two distinct critical points.

The individual functions in the zeta sum
are integer exponentials looking like *f* except for the absence of the forking at zero caused by the *z* term, having imaginary wavelengths varying logarithmically with *n*, since . It is the overlapping of these wave functions which gives
rise to the irregular pattern of the zeros on *x*
= ½.

**Chasing the Critical Points and their
Parameter Planes**

To understand the complex dynamics of zeta
we need to examine its critical points. These are precisely the zeros of the
derivative of zeta, whose *z* values are the slope
of zeta at *z*, as shown in the right of fig 3.
Just as zeta has so-called ÔtrivialÕ zeros along *y* < 0 and Ônon-trivialÕ ones on the critical line *x* = ½, the critical points of zeta are of the same two
divergent types, which I will term ÔrealÕ and ÔunrealÕ, one series along the negative
real* *axis and the other close to, but not on,
the critical line.

Fig 3 shows the first few critical points
on *y* = 0 lying between the trivial zeros, and
those in the complex plane lying between the non-trivial zeros. While the
ÔrealÕ criticals have oscillating values forming an exponentially varying series
, the ÔunrealÕ ones have similar critical values to one another, irregularly wandering between 0.4 and 1. We will name the criticals by rounding down, so the reals we consider are *z*-2, *z*-4, *z*-7,* z*-9, *z*-11, *z*-13, *z*-15, *z*-17 etc. Notice that the Ôminiscule criticalsÕ up to *z*-13 lie in the central valley of dzeta where the absolute derivative
is less than 1, with *z*-15 forming a transition
point and the ÔvastÕ ones, from *z*-17 on, are
lost in tiny pockets in the exponentiating highlands. We name the ÔunrealsÕ
looking at their positive imaginary values e.g. *z*23, *z*31, *z*38, *z*42, É *z*65. They also tend to
be located in regions where absolute dzeta is less than 1.

Fig 3: The ÔrealÕ critical points of zeta lying along
the *x*-axis (left) and
the ÔunrealÕ ones close to the critical line (centre). The derivative of zeta
(right) shows the two series of critical points as its zeros at the nipples and
dimples along the negative real axis and vertically along the outer edge of the
blue curve where dzeta has absolute value 1.

However, we are not just looking for
critical points, but the places where critical points might iterate to,
Mandelbrot satellites that classify interesting Julia set dynamics, which might
be somewhere else all together. Looking for tiny regions in a complex
exponential fractal can be worse than trying to find a needle in a haystack, so
we need at least minimal GPS navigation.

In the quadratic case (see end
section), the critical point at zero iterates . For *c *~ 0 we are in the main
cardioid, where all points head to the fixed point 0. We can solve for this
fixed point. The simplest case is the critical point itself being fixed . Here the *c* value turns out
to be the same as the critical point, but in general, this *c* value, which we call the Ôprincipal pointÕ, could be different from
that of the critical point.

More generally, we can try to solve for *c* values that become eventually fixed or eventually periodic with
period *n* in *m*
steps. These points are the repelling Misiurewicz points, forming the tips and *n*-connection points of the period *n*
dendrites, as well as the attracting main and satellite Mandelbrot sets, in the
quadratic case. We will call these collectively ÔM-pointsÕ. Solving for fixed
critical values, fixed at the second step, gives giving or *c* = 0, -2. These are our original point and the tip of the dendrite
on the negative real axis. If the absolute derivative is less than 1 the point
is attracting. We thus need to check what these do, by checking whether . The first is (super)-attracting since its derivative is 0.
The second however is repelling, since its derivative is -4. Hence it doesnÕt
lead to a Mandelbrot, but the tip of a dendrite.

For our purposes, we seek the simplest of
these solutions for the most horrendous function. We wonÕt be able to solve all
the equations but we might be able to get a numerical solution and even one
that we can display graphically in a useful form.

The very simplest - the critical point
being fixed is , or in the multiplicative case . Since is critical, its derivative is zero, so
it is super-attracting and must have a critical value in the Mandelbrot set or
its satellites. These Ôprincipal pointsÕ can be far from the critical point, even in regions of dzeta
where the values are exponentiating towards the infinite.

If we go one step further and look for *c* values for which the critical value is a fixed point, call them
Ôfixed valuesÕ, for the additive zeta Mandelbrot as in fig 1, we
seek . Solving we get , or , where is the critical
value. For the multiplicative case , we get .

In both cases these Ôtransfer functionsÕ of
*c* are just transformed copies of zeta,
translated, or scaled, in the domain and raised, or sunken, in the range. We
can identify the principal point among the fixed values, because it has a
Ôdouble twistÕ in its angle, leading to two yellow angle rays. To see how to use this to advantage with RZViewer click here.

Fig 4: Transformed zeta transfer functions for the
critical points *z*23 and
*z*-13 compared with that
of dzeta with the angle colouring omitted to emphasize the transition across
abs(*z*) =1. As noted
(right) the loci in the bay for *z*23 have absolute derivative greater than 1 so should be
Misiurewicz points, while those of
*z*-13 are attracting and
should lie in the Mandelbrot set, or its satellites. Principal points are
identifiable by their Ôdouble raysÕ top in *z*23 and left in *z*-13.

We can now make a graphical portrait of the
transfer functions, locate their zeros and explore the neighbourhood for its
local fractal geography. However we also need to know if the fixed points are
attracting and thus lie in satellite Mandelbrots, or repelling and thus
Misiurewicz points, by testing the derivative of zeta. Fortunately we have a serendipitous
graphical way to do this, because the colouring scheme for zeta was chosen to
highlight absolute value 1, so applied to dzeta, it gives a colour test of the
derivative for attracting or repelling. This can be scaled to examine it more
closely, or the derivative can be calculated numerically. The method only gives
one basic set of candidates, further of which could be found by solving for
later fixed or periodic points.

** **

**A: The Additive World**

We are first going to explore the geography
of the additive Mandelbrot sets for the various
critical points of zeta and how they interact with one another and with the
Julia sets they define. Subsequently we will explore the more bizarre dynamics
of the multiplicative parameter planes of .

** **

**1: Far East - the
Asymptotically-Critical Plateau**

We begin with the Mandelbrot set in fig 1,
originating from the nominal quasi-critical point 1000 on the plateau of zeta
converging to 1 in the right half plane.

This does not display polynomial bulbs or
satellite quadratic Mandelbrot sets but consists of fractal enclosed
representations of bounded Mandelbrot regions, interpenetrated by the chaotic
escaping set, fractally replicating the exponential ferns, whose global form is
illustrated by the anti-Mandelbrot island around the singularity in the inset
of fig 1.

Fig 5: Locations on the Mandelbrot set classify the
asymptotic dynamics in the right half-plane. The quantitative step-colouring of
each location on *M*
coincides closely with the step colouring of dynamical escape on the plateau.
Points in and its fractal islands* *remain bound (black).

What the plateauÕs parameter plane is measuring is
the dynamics of differing *c* values, iterated
from 1000 far in the right half plane. This is illustrated in fig 5, showing
the way atlas addresses on define the asymptotic step
dynamics in the right half plane. As we pass through fractal regions of , this results in a fractal sequence of dynamic ÔexplosionsÕ
of the right half-plane whenever a path in crosses a
boundary between a black and a coloured region, dramatic in movie format.

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.

Points *c* in
the right half-plane iterate to the fixed point *c* + 1 because = 1, so iterating from the quasi-critical point, . This is true for all *z* with large positive real
part, so the iteration is fixed*.**
Numerically, the principal point is **, which again places it in the asymptotic limit, which
coincides with the picture of the Mandelbrot set engulfing the positive
half-plane.*

Although the dynamics consists of fractal exponential fronds, these do
display the same mediant-based fractional winding adding fractional rotation
periods that the quadratic Mandelbrot bulbs have. In
fig 6 we show that the same mediant winding sequences, we see in fig 35 for the
quadratic Mandelbrot appear on the bays in the fronds bounding *M*.

Intriguingly the Farey tree of mediants appears in one variant of the Riemann hypothesis. Farey sequences consist of all fractions with denominators up to *n* in order of magnitude - viz
. Each fraction is the mediant of its neighbours
. Two versions of RH state
[8]
:

We can colour *M* according to how many steps it takes to reach within of a fixed point
or periodicity and colour by the number of steps in blue and add redness for
the period. This immediately shows up the periodicities of the bays
neighbouring the boundary, which can be confirmed to correspond to Julia sets
with rotational periodicity the same number, confirming the sequences of
periods of the bulbs and the mediant relationship in the fractal progression.

Fig 6: Shading the bulbs for the critical plateau *z*=1000 demonstrates their periodicity, as confirmed by the Julia set portraits, which display the corresponding rotational periodicities in their spirals, confirming an upward set of odd periodicities 3, 5, 7, 9 ... and a downward set of integer periodicities 3, 4, 5, 6 ..., each with mediant fractality, viz (3,4)=7, (4,5)=9, (3,5)=8, (5,7)=12. Period 3 is shown in the top right of (c).

** **

**2: Real Critical Points, from Miniscule
to Vast**

While the zeta zeros on the real line are
regarded as the trivial solutions of , the critical points on the *x*-axis
are anything but trivial, and each displays qualitative features of zeta that
give each critical point a distinct role in the dynamics. When we have a function
with more than one critical point, to understand the dynamics, we have to
investigate the Mandelbrot set of each critical point. The critical points
contribute to different dynamical features of the whole Julia set, as
illustrated in figs 5 and 10. The dynamics also involves interactive effects
between the critical points which causes their Mandelbrot sets to appear merged
or amorphous and the dynamics in different parts of the Julia set to be
influenced by each of the critical points.

In this respect the situation is very
different from the quadratic case, where the Mandelbrot set is an infinite
atlas of the dynamics of the Julia sets, each of which has a single type of
dynamics determined by the *c* value of the single
critical point. In the case of
zeta, with an infinite collection of critical points, the relationship between
the Mandelbrot and Julia dynamics is structurally analogous to a Fourier
transform. As before, the Mandelbrot set for a given critical value is a
spatial ÔintegralÕ of Julia dynamics over continuously varying *c* values. However the Julia set dynamics is now determined by a
countably-infinite collection of critical points, each of which can fractally
dominate the dynamics around its M-points. Examples of Julia set dynamics
responding to many critical points are illustrated in several of the figures.

**(a)
Continental Divide: The Critical Point ***z***-15**

The
critical point *z* ~ -15.339 commands a pivotal
role in the dynamics of the central basin. When we examine its principal point
and fixed values in the central valley, fig 7(b), we find they are in the main
body of its Mandelbrot set , close to the shores of the three bays we can also
see in the Mandelbrot set of fig
1, originating from the three bounding fronds. Pivotally its principal point is
right off the shore of the innermost bay, and it is here we find a sequence of
quadratic bulbs and the cusp familiar in the quadratic Mandelbrot set (fig 35).

Fig 7: (a) Base of the central valley for *z*-15 showing quadratic bulbs perturbed by
cubic and higher dimensional interference. (b) The critical value fixed points,
including the principal point, all lie in the central valley close to the
boundary, thus dominating the bulb dynamics. (c) The period 3 side bulb gives
rise to a Julia set (d,e) with obvious period 3 dynamics. The dynamics are
perturbed by adjacent critical points both of which are in a cubic relationship
to z-15 and possibly other ÔunrealÕ criticals. The features of (c-e) share
dynamical morphology with regions on the Mandelbrot set (g) and the
corresponding Julia set (h) of the cubic function . (f, h) Satellite Mandelbrot sets of the two functions also
share cubic morphology.

As we move into the cusp, fig 8 lower, we
find high periodicity dynamics characteristic of classic quadratic regions such
as Ôseahorse valleyÕ (6 in fig 35), but as we move further away from the cusp the dynamics becomes more complicated, with the largest bulb having an appendage from the base of a kind also seen in cubic functions where the critical maxima and minima are close enough that their dynamics interferes. Although zeta has no degenerate critical points which are multiple zeros of the derivative (compare fig a6), in fig 7 comparison is made between regions of the cubic function
and this region,
in terms of both its Mandelbrot and Julia dynamics, confirming the similarities
in a Julia set from the period 3 sub-bulb (*) and in satellite Mandelbrots from
each. This dynamic interference possibly
originates from *z*-13 as it shares features here
with *z*-15, however many other critical points
could be involved. For example, the unreal critical *z*95 has a deformed version of the *z*-15
structure, which also has the same cubic ÔwingsÕ.

Many of the ÔunrealÕ criticals also have
critical values close to the critical value of *z*-15
of 0.52 and fixed values in similar locations (see figs 21, 22), so that the
dynamics surrounding the central bay consists of the superposition of a
countable infinity of perturbations – a little like the humming on
telegraph wires in the desert consists in principle of summed vibrations along
the transmission line.

Fig 8: (Above) The structures in fig 7 are not
fractally repeated in the fractal valley at y~13 (left) which has distinct
dynamics from the central valley, although z-15 does have a Mandelbrot island
at the starred point (right). Fractal repeats do occur for z-17 on (figs 17-19)
and the dynamics is more similar for *z*-2 (fig 16). (Below) An exploded view in the cleft of
the main basin of *z*-15
showing high periodicity dendrites.

Each frond is a fractal replicate of the
entire dynamical parameter plane, so has a fractal replicate of the central
valley, increasingly to one side, as we move up successive fronds. For many
critical points such as *z*-17, fig 17, the
fractal valleys replicate the central valley dynamics, but not in *z*-15, as shown in fig 8 above, where two adjacent generations of
fractal valley each have their own distinctive dynamics. Nevertheless the
fractal valley in fig 8 does support a Mandelbrot satellite in a corresponding
location to the satellite we find in fig 9 at the base of the central valley.

There are also multiple fractal replicates
of the central valley interspersed down the base of the valley (see figs 9 and
15) and into the crests and troughs running along the negative real axis, which
will be useful in elucidating the dynamics.

** **

Fig 9: Fractal basin on the real line at *x* ~ -17.95 (top, enlargement right) has a
satellite of the critical point at

*x*
~ -15. The Julia set of its upper period 3 bulb (below) generates a Julia set
with a period 3 kernel in its base.

** **

**(b) Gently Undulating Lowlands : The
Miniscule Criticals z**

** **

We deal with the miniscule criticals as a
group, because, in many ways, they behave like a higher degree polynomial of
degree between 4 and 6 depending on the situation.

We
start by looking at *z*-2, and *z*-13 at the large bulb we
investigated in fig 7. When we examine the miniscules, we find this has become
a towering amorphous structure, I will call the ÔantÕ, indicating interference
between several critical points. On *z*-13 this has bulbs, indicating these regions are
quadratically sensitive to it, which also have satellite Mandelbrots on their
dendrites (*), as does *z*-15. *z*-2 also has satellites, indicating the
ÔantÕ region is sensitive to most of the miniscules. Notably there is no such
structure on *z*1000.
In *z*-2
this region is a fractal replicate of the central bay, with three beaches
separated by fronds. The ÔheadÕ region even has ÔhornsÕ which effectively
replicate the ant structure in the central bay within itself. When we look at the
whole central bay of *z*-2
or *z*-7 we
find the ÔantÕ is a fractal replicate of the bay repeated for each of the three
fronds and fractally on all scales and is also present in the bays of the
fronds in fig 16. Each of these also has the complex quadratic structure
involving several critical points we find in the ant. Period 3 bulbs on each of
the satellite Mandelbrots generate period 3 Julia kernels, confirming the
satellite of each critical is determining the Julia dynamics in the period 3
web of each set, despite the fact that the Julia set is sensitive only to the
location of the *c* value and not the critical
point that generated the satellite. This shows each of the critical points are
collectively determining the Julia dynamics.

** **

Fig 10:
Critical points from z-2 to z-15 all show fractal polynomial structures
on the boundary of the central valley (a–c), including satellite
Mandelbrot sets (e-g). These are not only perturbed by the Ôminiscule criticals
z-2 – z-13 but by many of the unreal critical points, many of which have
critical values surrounding that of z-15.
The critical points z-2 to z-9 have critical values very close to 0 and
thus form an atlas of the dynamics in the central valley and the zeros. (d) z-2 classifies the differing
central valley dynamics between points 1 and 2 in (a). Period 3 bulb dynamics
of the satellite Mandelbrots of the three critical points each show distinct
regions of period 3 dynamics in their Julia dynamics (h-k), confirming all
three critical points leave their mark on the Julia set. Only that of z-2
continues into the central basin.*z*-2/*z*-7 showing the ÔantÕ is one of three fractal bay
structures, which are repeated fractally on all scales. (m) A satellite for *z*-7 at the head of the ant, confirming the
head and head cusp are sensitive to the ÔminisculesÕ, particularly *z*-2 as shown in fig 11. Its period 3 Julia
kernel web (n) is covering the central valley.

** **

The
effects of each however differ. *z*-2, and with it, the lesser miniscules, form an atlas of the
dynamics passing close 0, as their critical values are very close to 0. Hence they determine dynamics in the
central bay. Sampling the point 1 on the Ôear of the *z*-2 ÔantÕ which lies outside the *z*-13 ÔantÕ gives a connected
black centre while the point 2 lying outside all three has a chaotic centre.
Notice that the ÔantÕ is absent in and indeed the asymptotic plateau in all the Julia sets is
brown indicating escape there. But only in *z*-7 and *z*-2 is the web of the period 3 kernel
connected across the central basin. We thus can see in the Julia sets the
regional actions of three distinct critical points simultaneously, central
basin, asymptotic plateau and local polynomial.

The
collective evolution of the miniscules and their undulation in the Mandelbrot
sets with their critical values is clearly portrayed in the dynamics of the apex of the innermost bay, where the fixed value in the
neighbourhood of *z* = -16 points to the blunt
frond apex for all of the miniscules, which undulate in position with their
critical values, but becomes a quadratic cusp for *z* = -15, which also dominates the local dynamics of unreal critical
points, the asymptotic plateau, and *z*-19.
However *z*-2 does have a quadratic cusp with
bulbs at the head of the ant and its sibling bays.

Fig 11: z-15 as critical transition All the critical points from *z*-2 to *z*-13 have their fixed value corresponding to the zero at -16 converging to the tip of the basal frond, however when we reach *z*-15, the entire basin boundary turns into a quadratic cleft. This is conserved by critical points with critical values in the range 0.4-33.8 as illustrated below for *z*1000 and the non-trivial critical *z*95 and is also true for *z*-19, with a critical value of 33.8,
indicating that *z*-15 is influencing the dynamics of all these in this region. At *z*-17, the central valley becomes flooded (see fig 17). However * z*-2 has a quadratic cusp on the head of the ÔantÕ (top
right). All these structures differ from the naked cusps at the tip of
exponential fronds in the black ocean of the Mandelbrot set (lower right).
Images all to scale of 0.02, except for the centre left and right.

This video shows an overview and two blow-ups of the additive zeta Mandelbrot set, as the critical value is varied from *z*-17 past the miniscules and *z*-15 to just beyond *z*1000. It has two dynamic layers interacting, one evolving and the other a 'static' representation of the Mandelbrot of the asymptotic half-plane. The blow-ups then look at the formation of the cusp, moving again through the miniscules and just past *z*1000. The program is designed to correctly handle using critical values, rather than critical points, so the process can be viewed in one transition rather than oscillating with the maxima and minima. The interaction between the two layers remains an intriguing and not fully explained phenomenon.

We now turn to decoding the collective
dynamics of the miniscules and their influence on the dynamics near the real
line. Mandelbrot satellites of the miniscules occur in a number of fractal
regions n the real line, several of which are fractal replicates of the central
valley (see fig 14), which have complex amorphous regions which originate from
the overlapping effects of the critical points on one anotherÕs dynamics. These
regions can be distinguished from a number of fractal regions that are simple
basins with a single periodicity, by colouring according to the incipient
period. This shows the compound sets have varying periodicities. We can then
look for satellite Mandelbrots to confirm they are a Mandelbrot compound
structure, as illustrated in fig 12.

Fig 12: Complex sets displaying overlapping effects of
several critical points show their nature through each of them possessing
well-formed Mandelbrot satellites, despite having an amorphous morphology. (Top
left) region connecting a frond to the central valley. (Top right and below)
fractal replicates of the central valley displaying differing degrees of
critical point interference (3 and 4 in overview fig 14). Highlighting
incipient periodicities (middle row) helps to distinguish complex sets from the
blue exponential islands (left) all of which have simple fixed point dynamics.

Figures 12, 13 and 14 show how the relative
dynamics of the miniscules can be revealed in stages, by examining each of the
regions in positions labeled 1-4 in the top of fig 14. These are each fractal
replicates of the central valley and expose the relative dynamics of the
miniscules, all of whose principal points are submerged in the central bay.

Fig 13: (Left) the local Mandelbrot islands of the
first four critical points on the real axis in the fractal replicate 1 in fig
14. (Right) Central valley region of the
corresponding Julia sets approximating the *c* values confirms all four parameter planes
influence the Julia dynamics.

Video of the oscillating Mandelbrot sets and their Julia progression

The largest at position 3 is the most
merged and shows a quadratic satellite only for *z*-13, the most far-flung of the set. The next at position 4 has a
greater degree of separation, as shown in fig 12, but still the dynamics is
merged, with only *z*-11 showing a clear
satellite, despite others having sub-satellites on their surrounding dendrites,
confirming this region is a compound Mandelbrot.

Things become much clearer in region 1, where
we can see from fig 13 that each of the first four criticals have satellites
which are oscillating in position in relation to their critical values, as we
saw in fig 11. We lay those of the
first four critical points in this region out in sequence, so we can see each
as a well-formed Ôblack heartÕ, each with sensitivities to the location of the
Ôblack heartsÕ of the other critical points. The evolution of these satellites
is confirmed by the varying dynamics of the corresponding Julia sets shown on the
right.

When we move to region 2 of fig 14 we find
a clear case of fractal separation of the satellites, which now follow a
sequence we shall also see extended for all the real criticals in relation to
the fronds. Each of the successive criticals forms a graded sequence, with one
max-min pair to each of the three frond-pairs, moving from the outermost in the
bay to the innermost, laying bare the dynamics of the miniscules, which was
submerged in the central bay.

Fig 14: Fractal replicate of the central valley (2) at
top shows the evolution of the ÔminisculeÕ critical points, which is otherwise
hidden because their fixed values fall into the central bayÕs blackness. A
consistent evolution of the Mandelbrot satellites down the fronds is shown,
with one hump and one trough for each successive frond.

The satellites have base periodicity 3 in
their central region, when compared against the period 3 satellite on the
negative real dendrite of the quadratic of fig 35, as shown in fig 15. Evidence of this can also be seen in
the Julia sets, by comparison with a base Julia set for the quadratic
satellite. This explains how these satellites can exist in a region where the
derivative is large, because for a period 3 cycle we calculate the derivative
by the chain rule as a product of the derivatives at the three points in the
cycle, one of which is close to zero and has a tiny derivative.

Fig 15: Period-sensitive colouring of Mandelbrot
satellites from replicates 1 (top) and 2 (lower left) for z-4 both coincide
with the period 3 satellite on the quadratic Mandelbrot set, confirming they
are period 3. This both coincides with the forms of the Julia sets in fig 14,
which show real period 3 dynamics and explains how they can exist in a region
where the derivative has absolute value greater than 1, because other steps in
the period 3 cycle include points close to 0 with tiny derivatives, ensuring
the period 3 derivative, calculated by multiplying the three derivatives, by
the chain rule, confirms the 3-period is attractive overall. For example in the
region 2 satellite approximation gives
-17.8120 > -19.8882 > -4.9145 É , with overall derivative -8.8565*101.3019*1.5049e-05
= -0.0135.

This evolution is replicated in the fractal
bays present in each frond, as illustrated in fig 16 where there is a
homologous evolution in the base of the valley y ~ 20. The dynamics in this
valley are very similar to those of y ~ 13, in fig 8, for both *z*-2 and *z*-15.

Fig 16: ÔA Garden Enclosed is my BelovedÕ – Base
of fractal valley around y ~ 20 (1 right). The evolution of the ÔminisculeÕ
critical points z-2 (*) - z-13
(last larger scale) is also presented in the fractal valleys of the
fronds. The basin area also
supports satellites, as shown in the inset right located at 2 and is a partial
homolog of the ÔantÕ.

** **

At least some of the miniscules also have Mandelbrot kernels in the fractal web of the pole singularity. These can be difficult to find, but fig 16a shows an example for *z*-13 just off the singular island. The multiplicative parameter planes however have a rich singular web with many Mandelbrot kernels as illustrated in fig 26.

.

Fig 16a: Mandelbrot kernel in the singular web of *z*-13. Positions shown with a (*).

.

**(c) Lofty Peaks of Altiplano – The
Vast Criticals**

** **

We now enter a sparse mountainous landscape
heading outside the central valley, where the critical values and derivatives
become exponentially huge and the transfer function begins to cause large
translations, far into the positive and negative reals.

**
**

Fig 17: Once we arrive at z-17, the landscape becomes
sparse, the central valley becomes submerged and the fronds truncated. The
innermost pair of fronds meet in a Mandelbrot set at the principal point (a,b),
whose period 3 bulbs generate a Julia set (c,d) with period 3 kernels. The
valleys in each frond also have fractal replicates of the Mandelbrot set in the
central valley in the same relative position (e,f), Julia set (g), however this
is not at the fixed value, which corresponds to the tip of the corresponding
frond (h), with Julia set (i) having a touching frond pair. Zeta Misiurewicz
points thus include the tips of fronds as well as dendrite *n*-hub points (see figs 20-22), as laso noted
in fig 11.

** **

The case of *z*-17 in fig 17 shows the entire central valley flooded back to the
fourth frond pair where the two fronds meet in a single Mandelbrot satellite. A
*c* value in the period 3 bulb of this gives an
equally sparse Julia set with a period 3 Julia kernel held between the same two
fronds. This process is fractally replicated in the valley in each frond, with
an isolated satellite at the same frond pair. This is however not the location of the fixed values , which
lie at the tips of successive fronds and generate Julia sets in which a frond
pair are just touching at their tips. This is consistent with the principal
point being the only fixed value guaranteed to be attracting.

Fig 18: *z*-19 shows a further alpine displacement. The central
valley is now displaced from a second valley – the Ôprincipal valleyÕ
containing the principal point (1 f) and fixed values far to the left, (a). Its
Mandelbrot set (b,c) is now at the horizontal fusion between successive
conjoined frond ridges (b). The Julia set of a period 3 bulb of the principal
Mandelbrot (g,h) shows homlogous structure. The fixed value at 2 in (a) points
to a fractal recursion of sub-valleys at 1 in (d) rather than to the locus 2 in
(d) where there is a Mandelbrot satellite in the same relative position as in
the principal valley (e) with period-3 Julia (i,j,k). This continues with
fractal replicates in successive ÔunrealÕ fronds (l).

When we move on to *z*-19, the displacements have become even more acute. The transfer
function now places the principal point and fixed values far into the negative,
forming a shadow valley the Ôprincipal valleyÕ separate from the central
valley. It is here we find the
principal Mandelbrot set now nestled horizontally between two successive fused
frond ridges, rather than vertically in a frond pair as previously. As before,
his pattern is fractally replicated in the valley in each frond.

** **

Fig 19: *z*-21 and *z*-23. The pattern of exponentiating maxima and minima
corresponding to the series of fronds now continues with the maxima and minima
following the structures of z-17 and z-19 displaced by ever huger positive and
negative real translations.

The alternating pattern between *z*-17 and *z*-19 becomes a continuing
sequence, evident in *z*-21 and *z*-23, where the central valley has now become entirely lost from
view, enabling us to predict the dynamics of all subsequent criticals.

Fig 20: The unreal critical at z23 has its principal
point well into the black ocean (+ top left), so we do not see distinctive features
in its neighbourhood. The fixed values in the central valley lie outside the
black ocean and correspond to two different types of M-points, the top two
appearing as dendrite hubs and the other two are recursive fractal centres. For
example the fixed value of the lower image (*) points to a valley at the base
of a fractal valley ad infinitum.
The top three all have fractal symmetries consistent with period 3. The
Julia set of the top centre one (right) shows that this point is also an
organizing centre of Julia dynamics in the neighbourhood of the fixed value
(top right).

**(3) Shang-ri-La – The Unreal
Criticals**

** **

We now turn our attention to the unreal
criticals interspersed between the notorious non-trival zeros on the critical
line x = ½, a little to the right of the zeros, with values from *x* ~ 0.78 - 2.4.

The locations of the critical points are
generally to the right of the critical line and since their critical values are
small their principal values lie close to the critical points in the Mandelbrot
ocean. However some of them that are close enough to the chaotic landscape
create local bays with quadratic Mandelbrot shorelines similar to the dynamics
of *z*-15 in the central bay.

The first unreal critical *z*23 has a real part of *x* = 2.4 and shows little evidence of
polynomial dynamics in the bay. All its fixed values in the central valley lie
in chaotic territory, either at apparent dendrite hubs or loci of an endlessly
recursive fractal process. The derivative function confirms these should all be
repelling and thus constitute Misiurewicz points. All of those off the real
axis appear to have period 3 symmetry.
The Julia sets generated by these fixed values display a centre at the
same fixed value with homologous dynamics.

The dynamics in the Julia set of fig 20
demonstrates that critical points far away from the central valley, can
influence the dynamics there around their fixed values. The Julia parameter is
simply the M-point corresponding to the fixed value on the boundary of the
central valley, not the unreal critical point *z*23,
yet the distant unreal critical is leaving its mark on the Julia set defined by
a *c* value in the central valley. This shows the
ÔArizona effectÕ – the humming you hear on the telegraph wires out in the
silent desert is a superposition of vibrations potentially coming all the way
from California. In a similar way, the complex boundary of the central valley
for each critical point is a combined ÔwhisperingÕ of all the critical points, both real and unreal, which is why it is complex and sometimes highly amorphous. We shall see later that there are often Mandelbrot satellites in the neighbourhood of repelling fixed values, but for additive unreal criticals, these may all be submerged in the Mandelbrot ocean.

** **

Fig 21: Dynamics of z31. (Upper sequence) shows the local basin of the critical point
with quadratic bulbs, and a series of exploded views from the starred points to
a Mandelbrot satellite whose Julia set has a (low resolution) period 3 kernel
web. (Lower sequence) the central valley has two fixed values lying within the
black ocean and only one (centre) on the boundary, pointing at a triple vertex
of three clefts at the branching structure inset. The same structure on *z*-7 is a fractal version of the central bay
of the same kind as the ÔantÕ of fig 10 and the branched pattern is also
visible as distortions of the *z*-15 quadratic bulbs in the top of *z*95 in fig 11.

With *z*31,
the second unreal critical, with a real value of 1.29, we begin to see richer
polynomial dynamics. The bay bounding the region of the principal point now has
a series of quadratic bulbs and these have dendrites supporting well-defined
Mandelbrot satellites, which also give rise to period 3 Julia kernels from
their period 3 bulbs as shown in fig 21.
In this case two of the fixed values in the central valley lie in the
ocean and only the centre one tends to a boundary M-point, this time at the
triple vertex of three frond tips, (Mandelbrot cusps), with a Julia set having
homologous dynamical centres.

As a third example, we have *z*95, which has a low real value of 0.78 and lies in a small focused
bay with prominent quadratic bulbs, having dendrites supporting chains
Mandelbrot satellites, whose period 3 bulbs generate confirmatory period 3
Julia kernels, establishing classic polynomial dynamics associated with an
isolated critical point.

Again, this has only one of its fixed
values in the central basin on shore, where it forms a fractal centre, again of
period 3 nature. This suggests that much of the complex amorphous structure in
the shoreline of the central bay is a product of the interaction of a large
number of the unreal criticals with similar fixed value locations to those of *z*-15 acting together in superposition.

** **

Fig 22: Dynamics of z95 (* top left). In the upper
sequence is shown the location of z95=0.78+95.29i with critical value
0.43+0.078i with real part lower than that of *z*-15. The low real part of the critical pointÕs coordinates causes it to be nestled closely towards the shore of the ocean, giving rise to a well-formed quadratic basin highlighted to show the iterations around the unreal critical (+). A series of exploded views from the bulb (*) leads to a well-formed satellite Mandelbrot, whose Julia set has well-defined web (lower right) of period 3 kernels. The lower sequence shows only the most left-hand fixed value lies on shore and gives rise to a fractal centre, again of period 3 symmetry, whose Julia set again has a homologous dynamic around the location of the fixed value

** **

**B: The Multiplicative Universe**

** **

We now enter the universe of the
multiplicative Mandelbrot parameter planes a very different
cosmos, where many of the patterns we have discovered to date will be manifest
in new and different ways.

** **

Fig 23: The global outlines of the multiplicative
Mandelbrot set for the asymptotic plateau show fundamental similarities to the
additive case with some notable differences. The large fronds are now phased
with the critical points, rather than the zeta zeros which can now be found in
the region of the blue bays. The
central valley now has a thick chaotic exponential crown. As with the additive
case (fig 5) points on this parameter plane classify asymptotic dynamics in the
right half plane, leading to explosions during motion on curves in parameter
space.

** **

**(1) The Far Horizon**

** **

The first case we examine, is that of the
asymptotic plateau, where, as with the additive case, the parameter plane forms
an atlas of the dynamics on the asymptotic plateau as shown in fig 23. As
previously, paths in the parameter plane
can result in explosions in motion videos of Julia sets.

Video of explosions in the multiplicative Julia sets in the crown along the line from -10+3*i* to -1+3*i*.

Fig
24: The large fronds of *z*1000 are phased with the
critical points of zeta (large dimples) and the regular angular variations of
dzeta (right) while the zeros (small dimples) are located in fractal sub-bays
on the shanks of the fronds. *z*1000 is ideally placed to model zeta itself because its critical value is 1 so there is no rescaling. Lower the first 600 non-trivial zeros showing intermittent zeros have periodic dynamics, iterating to themselves (blue) with period 4 (black) determined by the
sign of the real part of the third iterate (red), while others diverge to

The additive and multiplicative parameter planes have superficial similarity, but these hide significant differences. The fronds are much larger and neatly phased around the critical points with the regular angular variations of dzeta, while the zeta zeros (and indeed all the real zeros from -18 on) are located in the zero point of the first order fractal valleys, since
and the fractal valleys are immediate pre-images of the central valley surrounding zero, although the zeros, and their neighbourhoods, subsequently diverge, because of their large *c* values. Hence each fractal valley is a dynamic map of the neighbourhood of each zeta zero. The zeros iterate
, which is in the exponentiating left half-plane. Around half (268/600) numerically iterate far into the right half-plane and enter strongly attracting period 4 cycles, since they are drawn back close to
, as with 1000. E.g.
. Since the iteration passes through 0, the pseudo-Mandelbrot set
from the non-critical 0 forms an atlas of the eventual dynamics of all the zeros for every value of *c*,
as in fig 24b, giving the zero dynamics for each multiplicative Julia set.

**
** **
**

**(2) Tierra Firma – The Rift Valley**

** **

If we examine the changes in the region around the central valley as we go through the real criticals, the results are surprising and unexpected. The entire picture of the right half plane being asymptotically fixed in the black ocean is lost and we find the exponentiating regions are now approaching from both positive and negative reals, with a rift valley in the centre with two opposing kinds of dynamics - on the left the central bay with submerged dynamics amid fractal replicates of the singularity island and on the right fractal structures of islands interspersed with Mandelbrot sets, arising from interaction with the pole at *z*=1.

In fig 25 the rift valleys are shown for *z*-2 through to *z*-17. From there the
rescaling takes the dynamic onto such small scales, the central valley ceases
to exist. In fig 26 we examine the fractal structures in the opposing side of
the rift valley for *z*-2 through to *z*-11, where the satellites form part of a lattice of islands whose
symmetries are uniquely determined for each critical point.

** **

Fig 25: The Rift Valley: As we move through the
smaller real critical points, the central arena flexes wildly in tectonic
tsunamis, with ÔevolvingÕ exponentiating regions (lighter yellow and ochre)
overlaying and interacting with a ÔstaticÕ resonance with the asymptotic
limitÕs dynamics (darker ochre shades), folding around the complex plane, so
that they also appear in the positive real half plane, giving rise to fractal
structures for positive real values. Once again *z*-15 forms a boundary between the miniscule criticals and the vast criticals beyond. These features arise as a result of the singularity. They are shared by the Dedekind zeta function but not the weight 12 modular form delta over *SL*(2,*Z*), whose parameter
planes are all of the type in the following section.

Video of the enfolding of the multiplicative Mandelbrot set, as the 'critical' value runs from -1 to 1. There are two distinct layers in the dynamics, the 'evolving' one responding to the critical point, overlaying and interacting with the 'static' one resonant wth the asymptotic limit, which merge again in the last frame at critcal value 1 of points in the limit.

Fig 26: The positive arena of the rift valley displays
a series of generic fractal lattices, supporting a network of Mandelbrot
satellites (*) from *z*-2
to *z*-11 each with their
own unique structure. Centre bottom the Julia set for *z*-9 which can be compared with that of *z*-2 in fig 30. Fractal versions of these structures also occur in fractal valleys above and below the rift valley in *z*-7 (4th row) and *z*-4. These features appear to arise as a result of the singularity as they are shared by the Dedekind zeta function but not the weight 12 modular form delta over *SL*(2,*Z*), whose parameter planes are all of the type in the following section.

**(3) Fractal Cosmology in the Galactic
Abyss**

** **

We now turn to the multiplicative transfer
functions to look for the principal points and fixed values of the miniscule
criticals. The transfer functions take the form and the
principal point is defined by . In both cases the process is being rescaled by being
divided by the critical value, which for the miniscules varies down to 4x10^{-3}.

The principal valley of the transfer
function is now huge – up to 20000 units across. Looking at *z*-2 to *z*-13 we find a consistent
picture in which the principal valley alternates between large positive and
large negative real values, ad the critical value changes sign, with a large
principal Mandelbrot set and repelling fractal structures at the other fixed
values, which support Mandelbrot satellites with higher periods within the
local fractal structure, as illustrated in fig 27, where the rift valley has now
become vanishingly small down the centre line of the figure.

** **

Fig 27: For the Ôminiscule criticalsÕ, the principal valley defined by the transfer function is absolutely huge, reaching sizes of 20000, because the transfer function scales zeta by dividing it by the critical value. The central arenas can be seen tiny in the distance down the centre of the figure. Alternating critical maxima and minima flip principal valleys between positive and negative half-planes as their critical values change sign from positive to negative. The
principal points each support a Mandelbrot set. As with the additive world, the
locations follow a graded evolution (*) and like the vast criticals of the
additive world, fixed values correspond to repelling M-points, which support
fractal regions which contain Mandelbrot satellites. Inset in *z*-7 is a satellite from the first unreal valley above the principal
valley.

** **

Again, there is a progression across the
fronds, with a maximum and a minimum in a progression down each frond accompanied
by flipping between positive and negative half-planes, because the local maxima
are positive and the minima are negative, causing negative scaling. In fig 28,
we confirm by incipient period colouring that the satellites have higher
periods, consistent with an attracting period, although the fixed value is
repelling.

** **

Fig 28: The principal Mandelbrot and two satellites of
*z*-4 with colour-coded
periods of 1, 2 and 3. Investigation of the periodicity of the Mandelbrot
satellites confirms the satellites have higher periods, which enables them to
be attracting even though dzeta may have absolute value greater than unity at
their location.

** **

**(4) NanoworldÕs Myriad Valleys**

** **

We can also explore the progression of the
critical points in miniature, by examining fractal replicates of the principal
valley in the regions to the left of the central valley, as illustrated in fig
29. The progression occurs in a similar pattern to the additive world as there
is no flipping of signs.

Fig 29: We can also examine the evolution of the
Ôminiscule criticalsÕ in microscopic fractal replicates of the principal valley
(*) lower right. The Mandelbrot satellite of the principal point (*) again
evolves through the frond locations.

Fig 30:
Three typical Julia sets of period 3 bulbs of Mandelbrot sets from the
three regions - the microscopic cleft
(left), the positive arena (centre) and the principal valley (right).

** **

**(5) Jewels in the Crown**

** **

The critical point *z*-15 again forms a transition between the dynamics of the miniscules
and the vast criticals. The positive half-plane has again become part of the
main Mandelbrot set, but because the critical value is 0.52, the transfer
function is scaled so the central valley region is scaled by ~2.

** **

Fig 31: Again *z*-15 forms a transition between the miniscule and vast
criticals. Because its critical value is 0.52, the principal valley size is
doubled, so the principal point and its Mandelbrot lie at the extreme rear of
the central valley (1). Again the fixed values are repelling and one can find
satellite Mandelbrots located in fractal replicates of the central valley (2)
and in successive fractal replicates of valleys in the fronds (3,4,5). There is
also evidence for other repelling M-points of *z*-15 in the crown (far right). The case of *z*-13 remains somewhat enigmatic with no
obvious Mandelbrot satellites in the positive arena and only in the two clefts
surrounding the central valley (* in fig 24).

This results in the principal point and its
Mandelbrot set being far back in the base of the bay and the fixed values being
high up in the fronds of the crown. As with the miniscules, the fixed values
are repelling, but satellite Mandelbrot sets are fractally repeated in the
valley of each frond and in each generation of sub-valley as shown in fig 31.

*z*-13 also
forms a transitional case, enigmatic for its lack of satellites in the positive
arena of the rift valley and also lacking any fractal structure in the base of
the central valley, due to the obstructing exponential node.

**(6) The Heavenly Heights**

** **

When we move on to the vast criticals, we find a situation similar to that of the additive case although geometrically inverted in scale. Because the critical values are now exponentially large, the principal valley is successively downscaled into a minute region near zero. At *z*-17 we find the central valley opposed by a copy of itself very
similar to the additive case of *z*-17, again
enclosing the Mandelbrot set of the principal point vertically in a pair of
fronds as in fig 32, fractally replicated in sub-bays as shown right.

** **

Fig 32: The ÔvastÕ criticals again follow an
evolution, but slightly different from the additive scenario. The central
valleys have now become microscopic because the critical values are huge and
alternate between positive and negative half-planes according to the sign of
the critical value, as with the ÔminisculeÕ criticals. In all cases the
Mandelbrot sets are pinched vertically between fronds, and as before the
process is repeated in the fractal valleys of each frond.

As with the miniscules, successive vast criticals have principal points flipping between positive and negative half-planes, because their critical values change sign. Fig 32 shows the evolution for the first four vast criticals, including Julia verification for *z*-19 and *z*-23. Rather than alternating
between vertical and horizontal frond placement, all principal Mandelbrot sets
are held vertically, but their locations flip between positive and negative
half-planes, but on miniscule scales, so that their Julia sets are generated
covertly from *c* values near zero.

** **

**(7) PandoraÕs Dystopia – The
Dissonance of the Unreal Criticals**

** **

To cap the bag, we have the principal
Mandelbrot sets of a series of unreal criticals up to *z*95. Rather than being in the Mandelbrot ocean, the principal sets can end up in a variety of locations, oceanic, far into the chaotic exponential region, or nestled in the tips of fronds. Some have irregular amorphous forms due to dynamic interference. Their locations can be far into the positive or negative due to the twisting of the transfer function by the complex number critical values, as shown for *z*65 in fig 33. Significantly, as with the real criticals, in figs 27, 31 and 32, one can find satellite Mandelbrot sets in the neighbourhood of (repelling) fixed values in neighbouring fronds to the principal point.

** **

Fig 33: We finally come to the
multiplicative unreal criticals where chaos bites back. Their transfer
functions can be tilted, (see *z*65) through division by the critical value so the principal points can
occur with large positive and negative real values. The first (*z*23) lies in the Mandelbrot ocean and has
quadratic bulbs at the shoreline. Some have perfectly formed Mandelbrot sets
around their principal points, which may lie deep within the exponential
chaotic region or be delicately supported by the tips of fronds. However in
several cases the principal sets are highly irregular. Well-formed period-3
Julia sets can be gained both from the bulbs of quadratic basins (*z*23) and Mandelbrot sets (*z*65) but the period-3 Julia kernel is
readily detectable only in the vicinity of the critical point where it is
largest, rather than in a disseminated web, so can easily be missed. With *z*95= 0.78+95.29i, the primary Mandelbrot is
at the principal point 40.7+241.71i and the Julia set has this as *c* value, but we need to examine it at the critical point to find the period 3 Julia kernel. It takes more terms of the zeta sum as we explore greater imaginary values, at last 500 for z95 to present without serious distortions of the Mandelbrot set, which becomes computationally time-consuming. Lower strip, a graph of the real and imaginary positions of the zeta criticals superimposed on the zeros of dzeta in the range up to gamma underflow to 0, ~450i. To take matters to the limit, we include principal sets for z223 = 2.500042+223.408567i and z266 = 0.791525+266.014624i, to similar scale, whose principal points lie at 19.696+ 232.45i and -217.38+475.88i respectively, the latter reaching to imaginary values on the border of underflow of the gamma function, and requiring up to 1500 zeta terms on 'Full'-terms iteration. Finally, if we were right when we saw with real criticals, that fixed values, although often repelling, had Mandelbrot satellites in their neighbourhoods, we should be able to find them for unreal criticals as well. The inset in *z*31 with principal point 1.8190+44.8408i shows one
instance, in a similar position in the lower frond at ~4.044+34.484i,
neighbouring the adjacent fixed value, implying the same pattern pertains. In *z*42 insets is also shown a satellite at ~8.794+46.720i, near the adjacent fixed value, while the principal point of the main set is at 10.0451 +51.7145i.

The Julia sets generated by *c* values the period 3 bulbs of the principal sets and satellites all
correctly display period 3 kernels, however they have an enigmatic quality,
unlike previous examples, where the period 3 kernel is disseminated through the
Julia set. In this case the kernels are strongly centered on the critical point
location (see *z*23 in fig 33) rather than the
central valley or principal point, so they can easily be missed. Accurate
calculation for these critical values and principal points requires much higher
numbers of zeta terms.

**
**

**Appendix: Fractal Geography of Eta, Xi and L-functions**

As noted in the introduction, zeta is defined through the Dirichlet Eta function:
which shares the same trivial and non-trivial zeros but also has regular zeros where
. Riemann also introduced a symmetric function xi, which also has the non-trivial zeroes
. Because of their close relationship with zeta a summary of their fractal geographies is included for comparison.

Fig a1: Additive Mandelbrot set of xi from *z*=1/2 (a) has a small centre-left frond (a1, b) with multiple quadratic bulbs (c) supporting satellites (d).

Their period 3 bulbs generate period 3 Julia sets (e, f). Other fronds (a3, g) also have satellites (h) correctly generating Julia dynamics (i).

Xi has critical points running up the critical line, but the function is very close to 0 throughout, so only the central critical at * z*=1/2 leaves any mark on the Mandelbrot dynamics. This results in a very simple profile for xi similar to that of our initial example
, with local quadratic dynamics and global exponential dynamics in the additive case, and only exponential fronds in the multiplicative case, which is omitted.

Eta has a more intriguing geography, because it shares the zeta zeros, and also has an additional regular set of unreal zeros, resulting in a more widely scattered set of unreal criticals. Its real criticals form a shorter sequence, with much clearer manifestations of quadratic dynamics throughout. Its unreal criticals are even more varied, although similar qualitatively to zeta. The details of the fractal geography of eta's dynamics is summarized in the figures and captions.

Fig a2: Additive Mandelbrot and Julia dynamics for the first four real criticals of eta. These follow a similar trend to those of zeta, with *e*-3 having bulbs in the outer basin, *e*-5 bulbs on the inner, *e*-7 having a Mandelbrot held between the inner two fronds and *e*-9 having a principal Mandelbrot far into the negative (1). However all four also have Mandelbrot satellites supporting Julia dynamics along the real axis and from the bulbs in *e*-3 and *e*-5.

Fig a3: While the unreal criticals that have larger positive real parts are too far into the Mandelbrot ocean to display quadratic dynamics on the shoreline, eta's unreal criticals whose real parts are less than 1 form additive quadratic basins between their local fronds, with satellites generating corresponding Julia dynamics. Upper sequence *e*72. Lower sequence *e*33 and *e*99.

Fig a4: Eta's multiplicative Mandelbrot sets show a similar evolution to zeta with important differences. *e*-3 has both a principal Mandelbrot set in the positive half-plane and unlike the chaotic crown of fractal islands in zeta, a central bay with quadratic bulbs and satellites. *e*-5 has a principal set in the negative half-plane and *e*-7 and *e*-9 have principal sets on alternating sides in the manner of zeta's vast criticals. There is no sign of the series of lattice structures zeta has in the positive arena of the rift valley. The dynamics of *e*-7 in the central valley shows a strong correspondence with *e*1000, as we noted in zeta's dynamics as well.

Fig a5: Eta's unreal criticals form a varied sequence of cases, from the first one (*e*14), with a large real part lying in the Mandelbrot ocean with quadratic bulbs supporting satellites, through the next (*e*20) touching tips of frond dendrites to the next (*e*25) interior to the chaotic domain. Some critical points (*e*60, *e*60.8) are scattered laterally (1.09 and 2.42), one of which has a much larger real value but their principal sets are still in the chaotic escaping region. In the most extreme case examined (*e*76), with a real part of 2.52, we are just back to a location enfolded in the frond tips.

** **

The Dirichlet *L*-functions
where c are a cyclic set of Dirichlet characters generated by a finite residue group, display several new properties of the fractal geography of zeta functions. Some have a complex twist, resulting in twin conjugate functions, which are asymmetric about the *x*-axis, as is *L*(5,2) in fig a6. While there are no degenerate zeros with multiple roots in zeta, as it can be expressed in the form
, showing all the zeros are of first order, including those derived from gamma's singularities, the principal characters of numbers with *k-*distinct prime factors have a multiple zero at the origin with a neighbouring critical point having degenerate higher degree Mandelbrot sets, as noted in fig a6.

**
**

Fig a6: Function profiles, Mandelbrot and sample Julia sets from *L*(5,2), *L*(777,1) and *L*(210,1) show several manifestations of *L*-function fractal structure. The central *L*(5,2) Mandelbrot set is twisted asymmetrically because the coefficients exist in two conjugate forms. *L*(777,1) and *L*(210,1) show examples of degenerate cubic and quartic Mandelbrot sets due to the multiple degenerate critical points showing as zeros in the derivative (inset). These derive from the fact that each is a product of distinct primes - 777=3.7.39 and 210=2.3.5.7, resulting in factoring zeta by 3 and 4 product terms (1-*p*^{-z})^{-1} respectively.

**
**

We can look at *L*(5,2) a different way, by expanding it as a power series with the same *a _{n} *rather than a Dirichlet series. This results in a complex function formally defined on the interior of the unit disc. We can the investigate he function by finding its critical points using the derivative function and profiling the Mandelbrot plane of each critical to analyze the Julia sets.

**
**

Fig a7: (a), (b) the power series equivalent of *L*(5,2), and its derivative showing the three critical points investigated. (c) The three corresponding Mandelbrot sets. (d) When these are overlapped using RGB coloring we see the three overlap in the small white region. Points in this region such as that in (e) have three connected components simultaneously of period 1, with additional islands corresponding to the infinite sequence of other critical points close to the boundary. (f) Shows a series of Julia sets, the first two on the lower main body of 2 to either side of the cusp. The left hand one is also in 1 while the right hand one is not and thus has a cantor Julia component. The third is period 7 in 2 and period 3 in 1. The fourth is in the yellow period 3 bulb common to 1 and 3.

**
**

Fig a8: Some *L*-functions also have coefficients generated from a Fourier series and so can be examined as such. Modular forms, such as delta shown here, have a modular relationship on the positive imaginary half-plane
. (a) Modular form delta, (b) its derivative with a critical point (*), (c) the transfer function showing the principal point (*) (d) the Parameter plane with a Mandelbrot kernel (*), (e,f) enlargements and (g,h) a period 3 Julia kernel (*).

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Fig a9: Dedekind zeta function (a) based on the Gaussian integers (integer coefficients in C) displays similar fractal features to the Riemann zeta function in a more simplified form. (b1-4) shows the additive basin for critical point -7.67 with satellite Mandelbrot kernels. (c) The basin for -6.65 showing shifted quadratic sensitivity to the centre and the loss of the small basins at the end of the valley. (d1-4) The principal Mandelbrot kernel for -8.69 showing a period 3 Julia kernel. (e1) The multiplicative parameter plane of -1.37 showing two Mandelbrot kernels and on the right a kernel in the fractal web resulting from interaction with the pole singularity. The complementation of vast parameter kernels and those in the singular web occurs for the other miniscule criticals -2.48, -3.54, -4.59 etc.

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Fig a10: The weight 12 modular form delta (a) over *SL*(2,*Z*) shares many features with the previous example, but lacks any of the multiplicative features arising from the singular web. (b1,2, c, d1,2) Additive basins of trivial criticals -9.65, -8.64, and -7.53, with satellite kernels (*). (e1,2) Vast parameter plane of miniscule critical -1.4 has no corresponding singular web, as in the Riemann and Dedekind zeta functions. (f-h) Non trivial principal satellites of 6+14.21*i*, 6+ 18.6*i*, and 6+25.9*i*.

Fig a11: *L*-function of the elliptic curve
has a triple degenerate zero which gives rise to a cubic Mandelbrot kernel, in turn giving rise to cubic Julia kernels.

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Continuing into the realm of abstract *L*-functions of elliptic and other curves, we find new fractal properties emerging from the genus of the embedded curves, in which curves of higher genus give rise to degenerate higher degree Mandelbrot and Julia sets throughout their central valleys.

** **

Fig a8: *L*-function of the genus-3 curve
has an entire set of degenerate cubic zeros. Its critical points correspondingly give rise to cubic Mandelbrot satellites with cubic Julia sets.

** **

**Introduction for Non-mathematicians:**

The aim of this paper is to present a tour of the zeta function which anyone with some passing understanding of maths can enjoy and appreciate, as an exciting intrepid journey on complex math space. The paper is based on an easy to use Mac application developed by the author using XCode, downloadable from: http://www.dhushara.com/DarkHeart/. You can thus all experience this journey yourselves and even make video records of your adventures, using the RZViewer application.

** **

Fig 34: Critical points of real functions, complex
addition and multiplication and the simplest Julia set.

** **

**Complex Numbers, Fractals and Chaos**

Just about everyone has a good idea of the
real number line and understands how they add and multiply, and how functions
such as *y*(*x*) = *x*^{2} can be depicted in graphs with humps and troughs.
Functions with several terms in powers of *x*,
such as *y*(*x*) = *x*^{3} –*x*, are the
polynomials. Virtually every other function we commonly use, including the
so-called ÔtranscendentalÕ trigonomeyric and exponential functions can be
written out as an infinite polynomial in a power series: e.g. , commonly
written as . Sin(x) behaves
precisely as a polynomial with infinitely many humps and troughs.

Complex numbers are more abstruse for many
people, who find their use of imaginary numbers perplexing and irritating. We
make numbers complex by trying to solve *x*^{2}
+ 1 = 0, which has no zeros on the real line, by adding to the number
system, forming the unit along a new y-axis in a 2-dimensional number plane.
These numbers turn out not just to solve one equation, but provide existence of
the zeros for all polynomial equations. Moreover they become essential in the
study of wave processes in physics. Complex numbers add real and imaginary
parts, just like real numbers, but multiplication has a new twist to it. The sizes *r* of the numbers multiply, but their angles add: . This causes complex numbers and complex functions to twist
as well as expand and contract, so *z*^{2}
is *r*^{2} the size but at double the
angle, as shown in fig 34.

Enter chaos and fractals. One of the most
basic ways of representing dynamic change is to map a function into itself in a
feedback loop that causes a kind of stroboscopic ÔdiscreteÕ flow in a series of
jumps of iteration of the feedback loop . When we do this, the discrete flow falls into two kinds of
dynamics - order and chaos - each of which tends to have opposing regions in
the space in which the process is taking place. In a regime of order, the
discrete iteration steps down a basin of attraction converging towards a
defined fixed equilibrium, or periodic oscillation, rather like a stroboscopic
picture of the water flowing out of a bathtub. In the chaotic regime, the
discrete flow is divergent, so that arbitrarily close trajectories diverge
exponentially –commonly known as the butterfly effect – arbitrarily
small perturbations leading to global instabilities.

Discrete chaos happens in real and complex
numbers alike, but in complex numbers the process becomes fully fledged, due to
the twisting action of complex numbers and in the 2D plane of complex numbers,
the dynamics becomes apparent in a way we can all experience.

Fig 35: (Left) The Mandelbrot set of surrounded by
its Julia sets. (Right) How the critical point of the function determines
whether the Julia set is connected or not. The critical point *x* = 0 (see upper left) is the last point to
escape. When the Julia set is connected (upper right) the Ôfilled-inÕ Julia set
(black) contains the critical point and critical value . The inverse images of a large circle are all closed curves
and the complement of is a disc. In
the disconnected case the inverse images bifurcate to form a figure 8 and the
circles repeatedly double. The centre of the first figure 8 is the critical
point and the critical value c
escapes to infinity. The Mandelbrot set is formed by iterating the critical
point for each c value thus forming a Ôconnectivity atlasÓ of the Julia sets.

**Complex Discrete Dynamics**

In 1918 Gaston Julia devised a method for
iterating rational functions of complex numbers to produce a discrete dynamical
system by feeding back a function again and again into itself. A rational
function is a fraction of two polynomials so itÕs a little more general than
polynomials like . The two regions of order and chaos
became the complementary Fatou and Julia sets, and over time Julia sets proved
to be fractals – geometric sets of points in which a process is endlessly
replicated recursively in the manner of a snowflake.

The simplest Julia set is that of simply the unit
circle illustrated top right in fig 34. For numbers with absolute value greater
than 1, repeated squaring makes them bigger and bigger and they race off to
infinity. For smaller numbers squaring makes them smaller and they converge to
zero. But for numbers on the unit
circle squaring keeps doubling the angle without changing the size, so each
point races round the circle chaotically at a different exponentiating rate. We
can even see the butterfly effect of chaos because any two numbers have the
difference of their angles doubled for each squaring step, so no matter how
close they start out, after a finite number of steps they will get to be half a
revolution apart and be on opposite sides of the circle.

Then Benoit Mandelbrot, who had invented
the term fractal, began to study the set of points *c* in the plane, for which the Julia sets were topologically
connected. This is a supremely subtle and difficult question, but it turned out
to have an elegantly simple computational solution. For a quadratic, such as , the Julia sets consist either of a totally disconnected
Cantor set [1], with all
points outside escaping to infinity, or a connected fractal, with points inside
drawn into their own basins. The critical points, lying horizontally on the
humps or troughs of the function, are always the last points to escape, so if
the Julia set is connected, the critical point must be trapped inside. In this
simplest case, the quadratic has just one trough at the origin, where the
derivative, or slope, is zero. So if we iterate for each *c* from the critical point and test whether it does escape in a finite
number of steps, we can colour the *c* value of
the Mandelbrot set (traditionally black) if the critical point didnÕt manage to
escape and thus portray it, colouring the complement by how many steps it did
take to escape.

The Mandelbrot set and its complement is
thus a fractal atlas of the dynamics of each and every Julia set - a fractal
whose fractal regions are infinitely varied, leading to the claim of it being
the most complex and beautiful mathematical object in existence. The heart
shape also shows the effect of multiplicative twisting of complex numbers, with
the cusp, and each of the bulbs, displaying different forms of fractional
rotation highlighted by the number of dendrites emerging from each bulb. The
ÔM-pointsÕ, Misiurewicz points - tips and intersection hubs of dendrites if
they are repelling, and the roots of the Mandelbrot set and its satellites if they
are attracting, form key points in our investigation, which are accessible
because they are eventually fixed or eventually periodic in a finite number of
steps, as opposed to the asymptotic dynamics of points in a basin of
attraction, or chaotically wandering.
The fractional rotation periodicities of the bulbs and dendrites also
follow mediants so that, between
period *b* and period *d* bulbs, is a smaller period *b*+*d* bulb.

The Mandelbrot set, as complex dynamic
atlas generalizes to polynomials, rational functions, and even transcendental
functions, such as trigonometric and exponential functions, all of which have
discrete dynamics, which can be described in terms of parameter planes forming
an atlas of their Julia sets, with generic correspondence to the quadratic
case, representing a simple hump, or trough, corresponding to an isolated zero,
and occasional degenerate higher dimensional cases. In rational and
transcendental functions, the Julia set kernels form an infinite web, and in
the transcendental case, rather than chaos and the Julia set existing only on
the boundary of the infinite and finite basins, the closure of the entire
infinite basin is chaotic, but otherwise the classification of the forms of the
kernels of the Julia sets appears to be universal.

**The Zeta Function**

As noted at the beginning of the paper, the Zeta function[4] is defined in two radically different ways: [1]

The relationship between the sum formula
over integers *n* and the product over primes *p* was discovered by Leonhard Euler and is equivalent to prime
sieving. This has led to a deep connection between zeta and the theory of prime
numbers, and in particular to the, as yet unproven, and possibly unprovable,
yet ostensibly true, Riemann hypothesis - that all the Ônon-trivialÕ zeros of
zeta (i.e. those not on the negative real axis) lie on the critical line *x* = ½.

While both sides of [1] are convergent only
for real(*s*) > 1, we can see that we can
easily extend convergence to real(*s*) > 0
using DirichletÕs Eta [5],
by defining zeta as an alternating series:

This takes us all the way across the
critical strip, including the non-trivial zeros.

Riemann[6]
analytically extended the zeta function to the entire complex plane, except the
simple infinity at *z* = 1, by considering the
integral definition of the gamma function , leading to the formula , where , the generalization of the factorial function, is also
extended to the negative half-plane by the analytic continuation (although this
is not needed to define zeta) *.* A full
derivation of zeta is included in the companion paper [7].
The trivial zeros arise from those of the sine on the negative real axis, and
the non-trivial, from itself in the
critical strip 0 ² *x* ² 1. We now have the full picture in fig 1,
with exponential growth outside the central valley in the centre.

The zeta function is famous, not just for
its supreme complexity, but its enigmatic non-trivial zeros, which all appear
to lie symmetrically on the line *x* = ½.
Proving this has remained elusive, because their locations arise from an
infinite number of ÔholographicÕ superpositions of the powers of *n*, which vary
exponentially in size with *x* but have sinusoidal wave functions of *y* varying
logarithmically with *n*. Despite the apparent symmetry in the *x* direction, the
dynamics is determined by both the real and imaginary parts acting together,
and the imaginary parts form an irregular series of values, consisting of wave
transformations of the irregular distribution of the primes.

**Analytic Continuation and Mellin Integral Formulas for Zeta and L-Functions**

The following gives a summary of the analytic continuations and Mellin integral transforms for Riemann zeta, Dirichlet *L*-functions, Dedekind zeta, Hecke *L*-functions on Gaussian integers, and for the *L*-functions of elliptic curves and modular forms.

.

.

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**References:**

[1] Georg Cantor described the simplest case of a fractal: a unit
interval in which the open middle third is removed recursively from each
remaining interval *ad infinitum*.

[1] King C. (2009) Exploding the Dark Heart of
Chaos http://www.dhushara.com/DarkHeart/DarkHeart.htm

[2] Woon S (1998) Fractals of the Julia and Mandelbrot sets of the
Riemann Zeta Function arXiv:chao-dyn/9812031v1

[7] King C. (2009) Observations on the
Uncomputability of the Riemann Hypothesis

http://www.dhushara.com/DarkHeart/RH2/RH.htm

[8] http://en.wikipedia.org/wiki/Farey_sequence

[9] Douady A, Hubbard J. (1985) On the dynamics of polynomial-like mappings Ann Sci Ecole Norm Sup 18, 287-343.