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\documentclass{article}
\usepackage{color}
\usepackage{graphicx}
\usepackage[normalem]{ulem}
\usepackage{morefloats}
\usepackage{hyperref}
\hypersetup{
colorlinks=true,
linkcolor=blue,
filecolor=magenta,
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}
\newtheorem{conjecture}{Conjecture}
\author{Barry Brent}
\date{15h 29 March 2017, preliminary draft}
\begin{document}
\title{Experiments with the dynamics of the Riemann zeta function}
\maketitle
\begin{abstract}
We collect experimental evidence
for several propositions,
including the following:
(1) For each Riemann zero $\rho$
(trivial or nontrivial)
and each zeta fixed point
$\psi$
there is a
nearly logarithmic
spiral $s_{\rho, \psi}$
with center $\psi$
containing $\rho$.
(2) $s_{\rho, \psi}$ interpolates a
subset $B_{\rho, \psi}$ of the
backward zeta orbit of $\rho$
comprising a set
of zeros of all iterates of zeta.
(3) If zeta is viewed as a function on sets,
$\zeta(B_{\rho, \psi}) = B_{\rho, \psi} \cup \{ 0 \}$.
(4) $B_{\rho, \psi}$ has nearly
uniform angular
distribution around
the center of $s_{\rho, \psi}$.
We will make these statements precise.
\end{abstract}
\bibliographystyle{plain}
\section{\sc introduction}
\subsection{Overview.}
For complex $w$, let $\zeta^{\circ -}(w)$
denote the backward zeta orbit
of $w$ so that
$\zeta^{\circ -}(w) = \{s \in \bf{C}$ s.t.
some iterate of zeta takes $s$ to $w \}$.
If the sequence
$B = (a_0, a_1, a_2, ... )$
satisfies $a_0 = w$
and $\zeta(a_n) = a_{n-1}$
for all $n \geq 1$, we
say that $B$
is a branch of $\zeta^{\circ -}(w)$.
If $B$ converges with $\lim B = \lambda$, say,
we conjecture that $B$ is unique and we write
$B$ as $B_{w, \lambda}$.
We collect numerical evidence
supporting the following claims, which
will be made precise below:
(1) that for each of a countable set
of non-real
zeta fixed points $\psi$ and each
nontrivial Riemann zero
$\rho$,
$B_{\rho, \psi}$ exists, is unique, and
is the center of a nearly logarithmic
spiral,
say $s_{\rho, \psi}$,
interpolating $B_{\rho, \psi}$;
(2) that the members of
$B_{\rho, \psi}$
are distributed nearly uniformly on
$s_{\rho, \psi}$;
(3) and that there is another set of
real zeta fixed points $\psi = \psi_{-2n}$
near the trivial zeros $-2n$, $-2n \leq -20$,
such that consecutive members of
$B_{\rho, \psi_{-2n}}$
rotate around $\psi_{-2n}$
through an angle of $\approx \pi$ or $2 \pi$,
depending upon the parity of $n$,
so that the members of
$B_{\rho, \psi_{-2n}}$ lie on
a curve that is very nearly a straight line
passing through both $\psi_{-2n}$
and $\rho$.
\newline \newline
We treat in detail
relationships between
zeta basins of attraction and
the branches of $\zeta^{\circ}(\rho)$
that we have observed experimentally.
The resulting plots are included here because
they suggested the presence of the spirals
which are the main subject of the article, and
so--we speculate--they may eventually also suggest
the ideas needed to analyze these spirals;
for we have not proved any theorems.
\newline \newline
Our experiments were done with \it Mathematica \rm and
spot-checked with
\it Sage\rm. Data files and \it Mathematica \rm notebooks are
posted on the ResearchGate site \cite{Br}.
\subsection{Possible bearing on the Riemann hypothesis.}
Here are several scenarios.
(1) A clear understanding of the spirals $s_{\rho, \psi}$
with
zeta fixed point centers $\psi$ and
passing through Riemann zeros
$\rho$ might lead to a sort of dictionary, so that
the Riemann hypothesis might be
put in a form that speaks of zeta fixed points instead of
zeta zeros. (2) If (as we conjecture)
the spirals $s_{\rho, \psi}$
are approximated by logarithmic
spirals, it might be possible to
confine the $s_{\rho, \psi}$ to
lie within
spiral-shaped ``error bands'' about
logarithmic spirals.
Because each zero $\rho$ lies on each
$s_{\rho, \psi}$,
each zero would lie in each one of
a countable collection of these
error bands in the complex plane
(one for each zeta fixed point $\psi$);
then the zeros would
be confined to
the intersection of these error
bands, and this region
might be small. Our very incomplete
knowledge
of the $s_{\rho, \psi}$
is founded on prior knowledge of
the locations of the zeros, so that
this scheme is tainted with circularity;
but perhaps this taint might in
some way
be removed. (3)
Under the conjectures stated in this
article, measuring from the fixed points $\psi$,
a zero $\rho$ lies at the ``first'' intersection
(in terms of arc length, say)
of $s_{\rho, \psi}$ with the critical line;
and so the Riemann hypothesis
might be restated in terms of
these intersections.
Each $\rho$ appears to lie on
all of the $s_{\rho, \psi}$,
and so we might eventually
obtain a countable family of
conditions on these intersections,
which could, possibly, be
in some way usefully
combined. Section 6.1.3 discusses
some data that support
Conjecture 4, which codifies a part of
this scenario.
\subsection{More definitions.}
Let $\zeta$ denote the Riemann zeta function
and let us write the iterates of
a function $f$ as
$f^{\circ 0}(z) = z$ and
$f^{\circ (n+1)}(z)= f(f^{\circ n}(z))$
for $n =0, 1, ....$
An $n$-cycle for $f$ is an
$n$-tuple $(c_0, ... , c_{n-1})$
such that $f(c_{n-1}) = c_0$ and
$f(c_k) = c_{k + 1}$
when $k \neq n - 1.$
The forward orbit of $w$ under $f$
is the sequence
$(w, f(w), f^{\circ 2}(w), ... )$.
The backward orbit of $w$ under $f$
is the set
of complex numbers $s$ such that
$f^{\circ n}(s) = w$ for some
integer $n \geq 0$.
Let the symbol $f^{\circ -}(w)$
denote this
backward orbit; if
$w$ does not belong to a cycle,
$f^{\circ -}(w)$ carries the
structure of a rooted tree in
which the root is $w$ and the children of
$s \in f^{\circ -}(w)$
are the solutions $t$ of $f(t) = s$.
We will call any path in
$f^{\circ -}(w)$
that begins at $w$ a branch of
$f^{\circ -}(w)$
(also: ``a branch of the inverse of $f$''.)
Such a branch, then, is a sequence
$(a_0, a_1, a_2, ....)$ with
$a_0 = w$ and $a_n = f(a_{n+1})$ for
each non-negative integer $n$.
Since the Riemann hypothesis has been verified
within the range of our observations,
we write without
ambiguity $\rho_k$ for
the $k^{th}$ nontrivial Riemann zero
ordered by height
above the real axis and
$\rho_{-k}$ for its complex conjugate.
If $\zeta^{\circ n}(z) = 0$
and $\zeta^{\circ n-1}(z)$ is a nontrivial
Riemann zero, we call $z$ a
nontrivial zero of $\zeta^{\circ n}$.
For typographical reasons,
we will occasionally write
$\zeta_n$ for $\zeta^{\circ n}$;
within our article, there should no confusion
with other common uses
of this symbol.
\newline \newline
For $z \in \bf{C} \rm \cup \{\infty\}$,
$A_z :=\{u \in \bf{C}$ s.t.
$\lim_{n \to\infty}
\zeta^{\circ n}(u) = z\}$
(the ``basin of attraction'' of $z$ under
zeta iteration.)
Let $\phi \approx -.295905$
be the largest negative
zeta fixed point.
Then $A_{\phi}$ is a fractal \cite{W};
each nontrivial Riemann zero
appears to lie in an irregularly
shaped bulb of $A_{\phi}$
(Figures 1.1, 3.2, and section 3 more generally.)
\newline \newline
For a spiral
$s$ with center $\gamma$,
let $\alpha_s$ be the point on
the intersection of the critical line
and $s$ closest to $\gamma$.
We define a real-valued function
$\theta(z)$
on complex numbers $z \in s,
|z-\gamma| \leq |\alpha_s - \gamma|$
by requiring that
$\theta(z)
\equiv \arg(z - \gamma) \pmod{2\pi}$,
and that $\theta(z)$
increases continuously and monotonically as $z$
moves around $s$ from $\alpha$ in the direction
of decreasing $|z-\gamma|$.
In other words,
$\theta(z)$ behaves up to a multiplicative
constant like a winding number.
If a sequence $(a_1, a_2, ...)$
lies on a spiral
$s$ with center $\gamma$
and for some pair of real numbers $A > 0, B >0$
and all
$k = 1, 2, ..., |\theta(a_k) -
\theta(a_{k+1})| < A e^{-Bk}$,
we will say that the $a_k$
are distributed nearly uniformly around
$s$.
\newline \newline
For complex $z$,
let $r(z) := |z-\gamma|$. Let $m, b$ be real numbers,
so that $r(z) = \exp (m\theta(z) + b)$
describes a logarithmic spiral
with center
$\gamma$ and typical element
$\exp (m\theta(z) + b) \exp (i \theta(z))$.
Suppressing the dependence on $m$ and $b$, let
$$d_{rel}(\gamma, z) :=
\left | \frac{z-\gamma-\exp (m\theta(z) + b) \exp (i \theta(z))}{z-\gamma}
\right|.
$$
We say that $s$ is $c$-nearly logarithmic for
real positive $c$ if
$$\max_{z \in s, 0 < |z - \gamma| \leq |\alpha_s - \gamma|} d(\gamma, z)
< c$$
for some $m$ and $b$.
\newline \newline
We require a notion of ``very nearly a straight line.''
Suppose (1) a complex curve $C$ of finite arc length
has an initial point $z_I$ and terminates at a point $z_T$,
(2) that there are real numbers $m$ and $b$
such that
$$
\lim_{z \in C, z \to z_T}
\left|\frac{ \Im(z) - (m \Re(z) + b)}{\Im(z)}\right| = 0,
$$
and (3) that the convergence has exponential decay as
$|z - z_T|$ decreases from $|z_I - z_T|$ to zero.
Then we say that $C$ is very nearly a straight line.
\newline \newline
We need measures of the absolute
and relative deviations
of points $a_k$ in a branch $B_{\rho,\psi}$
of the backward orbit of a
nontrivial Riemann zero $\rho$ from
a logarithmic spiral fitted to that branch
using \it Mathematica\rm's FindFit command.
Suppose the $a_k$ are interpolated by a spiral
$s_{\rho, \psi}$ centered at $\psi$
such that for
$z \in s_{\rho, \psi},
r(z) = |z - \psi|$
and $\theta(z) = \arg (z - \psi)$.
Further, suppose that $s_{\rho, \psi}$
has a log-linear model
$\tilde{r}(z) = \exp (m\theta(z) + b)$
for real numbers $m$ and $b$,
in which we have
fitted the points $(\theta(a_k), \log r(a_k))$
to a straight line.
Then we will write
$$d_{abs}(\rho,\psi,k):= |a_k - \psi-\tilde{r}(a_k) e^{i \theta(a_k)}|$$
and
$$d_{rel}(\rho,\psi,k):=
\left |\frac {a_k - \psi -
\tilde{r}(a_k) e^{i \theta(a_k)}}{a_k - \psi}\right |.$$
(This is an abuse of our earlier notation for
$d_{rel}(\gamma, z)$ which should not be confusing.)
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1]{leaf1point1.png}
\vskip .07in
\sc{fig. 1.1: $A_{\phi}$}
\end{figure}
\subsection{Summary of observations.}
We express
some of our observations as
explicit conjectures.
\begin{conjecture}
For each pair of positive integers $L$ and $n$,
there is
a Riemann zeta $L$-cycle $\Lambda$
such that
the $\lambda \in \Lambda$ with
maximum imaginary part is close to the $n^{th}$ nontrivial zero
$\rho_n$ (but we will not define this use of
the term ``close''
more precisely in this draft)
and with the following properties.
\newline \newline
(1) Each $\lambda \in \Lambda$ is
a repelling fixed point of
$\zeta^{\circ L}$ lying
in the intersection of boundaries
$\partial A_{\phi} \cap \partial A_{\infty}$
in the usual topology on $\bf{C}$.
\newline \newline
(2) For each $\lambda \in \Lambda$
there is a complex number $z_{\lambda}$
and a natural number $0 \leq j_{\lambda} \leq L-1$
such that
\newline
(a) $\zeta^{\circ j_{\lambda}}(z_{\lambda}) = \rho_n$.
\newline
(b) For some small positive $c$,
$\lambda$ is the
center of a
$c$-nearly logarithmic spiral
$s_{z_{\lambda}, \lambda}$
interpolating a branch $B_{z_{\lambda},\lambda}$
of $(\zeta^{\circ L})^{\circ -}(z_{\lambda})$
(but we will not be more precise about this use of the term ``small'' in
this draft.)
\newline
(c) $\lim B_{z_{\lambda},\lambda} = \lambda$.
\newline
(d) $\bigcup_{\lambda \in \Lambda}
B_{z_{\lambda},\lambda}$ is a branch $B_{\rho_n,\Lambda}$ of
$\zeta^{\circ -}(\rho_n)$.
\newline
(e) The members of $B_{z_{\lambda},\lambda}$
are distributed nearly uniformly on
$s_{z_{\lambda}, \lambda}$.
\newline \newline
(3) If $z \in \zeta^{\circ -}(\rho_n)$
then for
some positive integer $j$ and some positive integer $L$
and some $L$-cycle $\Lambda, \zeta^{\circ j}(z) \in B_{\rho_n,\Lambda}$.
\end{conjecture}
Conjecture 1 is essentially a
description of patterns we observed reliably
in numerous experiments; clause 2e, in particular,
is plausible on its face in view of the spiral plots
exhibited in Figures 5.2, 5.3, 5.4, 5.7, 5.9, and 6.1. Figure 5.5 supports,
in particular, our use in this clause of the term ``nearly uniform'' as
we have
defined it above.
\begin{conjecture}
(a) If $L = 1$, so that the unique element of
$\Lambda$ is close to $\rho_n$,
then we
write $\Lambda =
\{\psi_n \}$, and we have that
$j_{\psi_n} = 0$ and
$z_{\psi_n}= \rho_n$.
In this situation, in clause (b)
of Conjecture 1 we can take $c< e^{-4}$;
furthermore, if $L = 1$,
then the infimum of the valid values of $c$
goes to zero as $n \to \infty$.
\end{conjecture}
Some evidence for this conjecture appears in section 6.1.1.
\newline \newline
When $L = 1$ we have restricted our claims in
this conjecture to
spirals $s_{\rho_n, \psi_n}$ because that is the case
we have checked most thoroughly, but we have also checked less thoroughly
the case in which the zero is fixed (usually, $\rho_1$)
and $\psi_n$ varies over positive $n$. It appears to us
that the conjecture generalizes to all pairs
$\rho =\rho_{n_1}, \psi = \psi_{n_2}, (n_1, n_2) \in \bf{Z}^{\geq 1} \times
\bf{Z}^{\geq 1}$.
\newline \newline
Because $\lambda \in \Lambda$ would be repelling, it
would also be an attracting
fixed point of a local
branch of the functional inverse
of $\zeta^{\circ L}$,
and then the convergence
would follow
from standard results,
for example, Theorem 2.6 of \cite{HY}.
\begin{conjecture}
(1) There are
repelling zeta fixed points
$\psi_{-2n}$ near the trivial zeros
$-2n \leq -20$.
\newline
(2) For any nontrivial Riemann zero
$\rho$, members of
$B_{\rho, \psi_{-2n}}$
lie on a curve which is
very nearly a straight line
segment.
If $n$ is even,
the endpoints are $\rho$ and
$\psi_{-2n}$; otherwise,
the curve passes through $\psi_{-2n}$
and terminates at $\rho$.
\newline
(3) If $2n \equiv 0$ (mod $4)$,
then $$\arg \frac {d\zeta(\psi_{2n})}{dz}
\approx 2 \pi;$$
If $2n \equiv 2$ (mod $4$),
then $$\arg \frac {d\zeta(\psi_{2n})}{dz}
\approx \pi.$$
\end{conjecture}
% realfxpts26feb16.nb
(See Figure 5.7. Some evidence for
this conjecture appears in section 6.1.2.)
These observations are
consistent, of course,
with the proposition that
$B_{\rho, \psi_{-2n}}$
is interpolated by a spiral.
Our computations of $\zeta(x)-x$,
$x$ real, indicate
that the $\psi_{-2n}$ are real
(the graph crosses the $x$-axis near each
trivial zero we examined.)
It was conceivable that there might
be a broader relationship of the same kind
between the derivative of zeta at a given
fixed
point and the structure of the associated spirals
centered at those fixed points elsewhere
in the complex plane, but our experiments
have not verified any such relationship.
It is suggestive, of course,
that these relationships are
exact when zeta is considered
as a function of real numbers
and $\psi_{-2n}$ is replaced
by $-2n$.
\newline \newline
The following conjecture codifies
in part the scenario of section 1.2
\begin{conjecture}
(1) The relative deviation $d_{rel}(\rho_n,\psi_n,0)$
of the nontrivial Riemann zero
$\rho_n$ from a logarithmic spiral
fitted to the elements of
$B_{\rho_n,\psi_n} = (a_0, a_1, ...)$
satisfies
$$\lim_{n \to \infty} d_{rel}(\rho_n,\psi_n,0) = 0.$$
In particular, if we write
$$D_{rel}(N) = \log \frac 1N \sum_{n=1}^N d_{rel}(\rho_n,\psi_n,0),$$
then there exist two exponents
$0 < e_1 < e_2 < 1$ such that
$-(\log N)^{e_1} < D_{rel} (N)< -(\log N)^{e_2}$
for $N = 1, 2, ....$
\newline
(2) The absolute deviation $d_{abs}(\rho_n,\psi_n,0)$
satisfies
$$\lim_{n \to \infty} d_{abs}(\rho_n,\psi_n,0) = 0.$$
In particular, if we write
$$D_{abs}(N) = \log \frac 1N \sum_{n=1}^N d_{abs}(\rho_n,\psi_n,0),$$
then $\frac 1N < D_{abs}(N) < \sqrt{\frac 1N}$.
\end{conjecture}
We have provided some support for
clause (1) in Figure 6.5 of section 6.1.3, using $e_1 = .8$
and $e_2 = .85$; clause (2) is supported by the
plot in the right panel of Figure 6.7 in the same section.
\newline \newline
We examined the possibility that
branches of
$\zeta^{\circ -}(z), \zeta(z) \neq 0$
for arbitrary $z$ on the critical line converge to
the same fixed points as
branches of $\zeta^{\circ -}(\rho), \rho$
a non-trivial zero.
We tested various such $z$
and found spiral
branches converging to the fixed points of zeta.
So it ought be
possible to explain the spirals
with a theory that avoids
any appeal to special properties of the
Riemann zeros. We are not
going to describe these experiments
in any further detail in this article.
\newline \newline
An analogy from fluid mechanics led us to
check for invariance of branches of
$\zeta^{\circ -}(z)$
for these $z$
under rotation about
the fixed points at their centers.
We found that the deviation from
this sort of invariance is
systematic and can itself
be described by referring
to (other) logarithmic
spirals.
\newline \newline
We made a brief survey of functions
other than zeta to gauge the
extent of the spiral phenomenon,
which we will not describe in
any further detail
than the following.
Functions as simple as cosine appear
to exhibit this behavior.
We also observed it in, for example, the
Ramanujan $L$-function.
We hope to carry out another survey
with a different software package.
\subsection{Prior work.}
Many authors
have examined
the Riemann zeta
function with computers.
Notable citations from the perspective of
this article are
Arias-de-Reyna \cite{A},
Broughan \cite{B},
Cloitre, \cite{C},
Kawahira \cite{Ka}, King \cite{Ki, Ki2}
and Woon \cite{W}.
\section{\sc methods}
\subsection{Quadrant plots.}
We will be displaying
colored plots (say, ``quadrant plots'')
depicting, for a
point $w$ of $\bf{C}$ and a
meromorphic function $f$,
the quadrant of $f(w)$. We use
quadrant plots in three ways:
(1) to determine small squares
containing exactly one solution of
an equation of interest, so that
this information can be used by standard
equation-solving routines to find a solution
to several hundred digits of precision (which we find
is necessary, for example, to locate zeta cycles)
lying in a particular region;
(2) to superimpose quadrant plots upon plots of
the basin of attraction $A_{\phi}$. These
two kinds of plot typically interlock in a way that
helps us to understand the meaning of
many small irregular features of $A_{\phi}$;
and (3) to show how the quadrant plots
spiral as we reduce the size of the plot
window about a fixed point of zeta or one of
its iterates. Observation of
these spiral motions was our
first indication that forward orbits near
fixed points do lie on spirals.
\newline \newline
In
quadrant plots, the
boundaries of
single-colored regions are
$f$ pre-images of the axes--curves
corresponding to
zero sets of $\Re(f(s))$ and $\Im(f(s))$;
the apparent intersections signal the
presence of zeros or poles of $f$.
By adjusting the color scheme to
distinguish between
regions where $|f|$
is large or small,
we can try to distinguish zeros
from poles. Some apparent
intersections
are revealed to be illusions by
a change of scale.
Similar but colorless methods
for plotting zeros were
put to use in \cite{A}.
\newline \newline
The visualized region is partitioned
into small squares,
each of which is represented by
a pixel.
We choose a test point $s$
in each square.
The pixel representing the square
is colored according to the rules in
Table 1.
In the table,
the region $D$ is a disk with center $s = 0$ and
large radius $r$ (chosen as may be convenient.)
We denote the complement of $D$ as $-D$.
\newline \newline
\hskip 1in
\begin{tabular}{|c|c|} \hline
Location of $f(s)$& Color of pixel depicting region containing
$s$\\ \hline \hline
real and imaginary axes & black \\ \hline
$D \hskip .05in\cap $ Quadrant I & rich blue\\ \hline
$ - D \hskip .05in\cap$ Quadrant I & pale blue\\ \hline
$D \hskip .05in\cap $ Quadrant II & rich red \\ \hline
$ - D \hskip .05in\cap$ Quadrant II & pale red\\ \hline
$D \hskip .05in\cap $ Quadrant III & rich yellow \\ \hline
$ - D \hskip .05in\cap$ Quadrant III & pale yellow\\ \hline
$D \hskip .05in\cap $ Quadrant IV & rich green\\ \hline
$ - D \hskip .05in\cap$ Quadrant IV & pale green\\ \hline
\end{tabular}
\vskip .1in
\sc{table 1: coloring scheme for quadrant plots}
\rm
\vskip .1in
\hskip -.19in
The junction of four rich colors
represents a zero,
the junction of four pale colors represents a pole,
and
the boundary of two appropriately-colored regions is an
$f$ pre-image of an
axis. An example is shown in Figure 2.1:
$s \mapsto (s - 1)^2 (s - i) (s + 1)^5/(s+i)^3$.
(We have superimposed a pair of axes on this
quadrant plot.)
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.8]{leaf2point1.png}
\vskip .07in
\sc{fig. 2.1: quadrant plot of $s \mapsto
(s - 1)^2 (s - i) (s + 1)^5/(s+i)^3$}
\end{figure}
\newpage
\subsection{High precision equation solving under geometric constraints.}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.3]{leaf2point2left.png}
\hskip .07in
\includegraphics[scale=.3]{leaf2point2center.png}
\hskip .07in
\includegraphics[scale=.3]{leaf2point2right.png}
\hskip .07in
\vskip .1in
\sc{fig. 2.2: zooming in on $\lambda_1$ via quadrant plots of $\zeta^{\circ
3}(s) - s$}
\end{figure}
We illustrate the application of quadrant
plots to solve equations under geometric constraints
by showing how we found the three-cycle
$\Lambda$ described in section 5.4. In Figure
2.2, we have superimposed plots of $A_{\phi}$
and quadrant plots of
$s \mapsto \zeta^{\circ 3}(s) - s$
on small squares near the first non-trivial Riemann zero.
The resolution has been kept low to speed up the computations;
high resolution is not particularly helpful in this
situation. The left panel is a square with side length $20$
and center $\rho_1$. The central panel depicts a square
with side length $2$ and center $\rho_1 + 3.4 +.1i$; we have
adjusted the center to keep in view a particular four-color
junction visible in the left panel. It represents a solution
of $\zeta^{\circ 3}(s) - s = 0$, namely the three-cycle element
$\lambda_1$ we are trying to compute. At this stage, if we used,
for example, the \it Mathematica \rm command
FindRoot constrained to search within this square,
it might land on any one of the several four-color
junctions we see in the central panel. So we
change the center of the plot again, this time to
$\rho_1 + 3.46 +.103 i$, and we make a
square plot centered there with side length $.02$.
This is shown in the right panel of Figure 2.2.
Next we use a slow, ``handmade'' routine
to approximate $\lambda_1$ by searching within this square.
Then, using this approximation as the beginning value for a
search with FindRoot, we obtain a solution with $500$ digits
of precision:
\newline
$3.9589623348847434673516458439896123461477039951866801455882506555054$
\newline
$331235719797619129160432526832126428515417856326242408422124490775895$
\newline
$37219976674458409141742662175701089081252727395073714398968532356378$
\newline
$12255138302084634149524670809965144703541657360428502230820135428609$
\newline
$38536894453944241116438492746243199878001238993540770158034816978947$
\newline
$866042863811536518002674033394246742728451523022955079328623947833520$
\newline
$567532298244004442294156837342370982002330874074322076777185746207730$
\newline
$323482406094614280046$
\newline
$+14.23622856322181332287122301085588169299871236208494399568695437825$
\newline
$6069322896396761526007136189745757467102551375667154010366364994538731$
\newline
$7916271113823253751110948972775762217941663830770714262041755664035323$
\newline
$9671078789529204404394764315531582588051352309327292004343654135172820$
\newline
$7780017861238006999109644383198471665302823015355865202971277187847669$
\newline
$1974168218415293165267046606327405458655765280027732495125802150527924$
\newline
$57282410834191507107658393848458313664113623935800293262678700791600125$
\newline
$465010766853i$.
\newline
Let us denote this approximation of $\lambda_1$ as $a$.
A numerical check indicates that
$|\zeta^{\circ 3}(a) - a|$ agrees with zero to $495$ decimal places.
\newline \newline
Our main reason for requiring so much precision is that we will be
repeatedly solving equations of the form $\zeta(u) = v$ for $u$, in each
case
replacing $v$ with the previous $u$, to construct lists of
(usually) $100$ elements of a branch of the backward orbit of a nontrivial
Riemann
zero, looking for the $u$'s near pre-selected
fixed points. As the procedure
repeats $100$ times, there is an accumulation of numerical error, and
in this situation very high precision is needed to maintain enough
accuracy to ``see'' the spirals formed by these branches in our plots.
\section{\sc a tour of A sub phi}
We are interested
in $A_{\phi}$ because
plots of this set
make visible the underlying
structure of the
network of $\zeta^{\circ n}$
pre-images of the critical line for
all $n$ at once:
(1) the nontrivial
zeros of the
$\zeta^{\circ n}$
lie in bulbs
of $A_{\phi}$ on filaments $F$
decorating
the border of $A_{\phi}$, and
(2) one
$\zeta^{\circ n_{_F}}$
pre-image of the critical line
transects each such $F$.
(Claims 1 and 2 are not, of course
logically equivalent;
we are summarizing computer
observations that we will describe
in more detail below.)
Thus the structure
of union of rooted trees
visible in plots of $A_{\phi}$
is apparently graph-isomorphic
to a corresponding
structure for the
point set
$$
\bigcup_{\Re(z) = \frac 12}
\zeta^{\circ -}(z) = \mathcal{U}\, \mbox{(say.)}
$$
This observation informs our
discussion of the trees $T$ in the
next section. We pretend
that we have stated
a rigorous
definition of the decoration notion
and definite conditions
for the membership of
a given complex number in a given filament.
In view of the relationship between
$\mathcal{U}$
and $A_{\phi}$, this
should not cause problems: each
filament $F$ may be identified with
one (of the many)
$\zeta^{\circ n_{_F}}$ pre-images of the critical line,
the definition of which
could be made precise.
But we should say explicitly that ``$A$ decorates $B$''
is a transitive relation and that
the filaments are subsets of $A_{\phi}$.
\newline \newline
In Figure 3.1, for example, the
points at the junctions of four colors
represent zeros of $\zeta^{\circ 2}$;
the zeros in the long filaments are
nontrivial.
\begin{figure}[!htbp]
\centering
\includegraphics[scale = .4]{leaf3point1left.png}
\hskip .05in
\includegraphics[scale = .4]{leaf3point1right.png}
\vskip .07in
\sc{fig. 3.1: left: $A_{\phi}$ at the edge of the main cardioid; right: superimposed
quadrant plot of $\zeta^{\circ 2}$}
\end{figure}
The right panel of
Figure 3.2 shows a quadrant
plot of $s \mapsto \zeta(s) - s$
superimposed on a plot of $A_{\phi}$;
the fixed points of zeta appear as the
junction of four colors.
The left panel
depicts the nontrivial Riemann zeros using
the same scheme (a quadrant plot of zeta.)
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.4]{leaf3point2left.png}
\hskip .1in
\includegraphics[scale=.4]{leaf3point2right.png}
\vskip .07in
\hskip 0in
\sc{fig. 3.2: left: quadrant plot of zeta; right: quadrant plot of $\zeta(s)
-s$; both superimposed on
$A_{\phi}$}
\end{figure}
\newline \newline
The filled Julia set of zeta
(the points in $\bf{C}$
with bounded orbit
under iteration by zeta)
is $\bf{C}$ $- A_{\infty}$.
The basin
$A_{\phi}$ appears to be
dense in $\bf{C}$ - $A_{\infty}$.
The sets $\bf{C}$ - $A_{\infty}$
(Figure 3.3) and $A_{\phi}$,
regarded as regions in
the complex plane,
are indistinguishable
in our plots
but they are not identical.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.85]{leaf3point3.png}
\vskip .4in
\sc{fig. 3.3: \bf{C} - $A_{\infty}$}
\end{figure}
For example,
there is an infinite number of real
zeta
fixed points (\cite{W}, Theorem 1)
that belong to
$(\bf{C}$ $-A_{\infty})-A_{\phi}$.
In addition, there
appear to be infinite families of
non-real zeta
$k$-cycles for each integer $k \geq 1$
in $(\bf{C}$ $-A_{\infty})-A_{\phi}$.
\newline \newline
Zero lies in $A_{\phi}$ (\cite{W}, Theorem 1.)
This set is a fractal decorated with
numerous long filaments
(Figure 3.1.)
Zeroes of the $\zeta^{\circ n}$
lie on the filaments.
Because zero is an element of $A_{\phi}$,
we know that the whole
backward orbit
$\zeta^{\circ -}(0)$
lies in $A_{\phi}$.
Because the
pre-images of
nontrivial Riemann zeros under
iterates of zeta lie on the
filaments,
the itinerary of a point in the backward orbit
of a nontrivial zero $\rho$ visits several
filaments at the edge
of $A_{\phi}$ before coming to
$\rho$.
(Some but not all pre-images of the trivial
zeros also lie on filaments.)
\newline \newline
The set $A_{\phi}$
seems to
comprise \newline
\newline
(1) a heart-shaped, seven-lobed central body,
which we will call the main cardioid.
\newline \newline
(2) two major filaments
of bulbs of various irregular shapes
that emanate from the main cardioid,
transected by the critical line and containing
one nontrivial Riemann zero in each bulb
(right panel of Figure 3.2.)
\newline \newline
(3) infinitely many
blunt processes and long filaments
decorating the main cardioid
and each of the irregular bulbs.
The filaments comprise smaller
copies of the bulbs,
which, in turn, are decorated with
similar filaments,
\it ad infinitum. \rm
(Figure 3.1.)
Thus, when we plot them, the set of
filaments decorating $A_{\phi}$ exhibit
a visible tree structure.
\newline \newline
The visible features
described in (1) - (3) were evident in
Woon's plots of $\bf{C}$ $- A_{\infty}$
\cite{W}.
The filaments appear to be
zeta-iterate pre-images (close copies) of
the two major filaments.
Pre-images of the real axis pass through
the blunt processes and contain
pre-images of the trivial
zeros. For example,
the right panel of Figure 3.1,
superimposes a quadrant plot
of $\zeta^{\circ 2}$ on the
left panel, so that
junctions
of four differently-colored regions
each represent a zero of $\zeta^{\circ 2}$. There are
three long filaments depicted in this image
containing zeros, the immediate zeta images
of which are
nontrivial Riemann zeros; but between the lower two
such filaments is a blunt
process transected by a zeta pre-image of the real axis,
and we can see another series of
$\zeta^{\circ 2}$-zeros lying along this curve.
These are zeta pre-images of the negative
even numbers.
\newline \newline
(4) at each trivial zero $< -18$,
a microscopic, more or less distorted
copy (zeta-iterate pre-image)
of the entire assemblage
described in (1) - (3).
(By ``microscopic'' features
we mean features so small
that they
can only be visualized by a
change of scale from that of Figure 3.3.)
In Figure 3.4, we show copies
of the main cardioid
near the trivial zeros $-28, -26, -24, -22$
superimposed on quadrant plots of
$\zeta^{\circ 2}$ in the same squares.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.45]{leaf3point4r1c1.png}
\hskip .05in
\includegraphics[scale=.45]{leaf3point4r1c2.png}
\vskip .05in
\includegraphics[scale=.45]{leaf3point4r2c1.png}
\hskip .05in
\includegraphics[scale=.45]{leaf3point4r2c2.png}
\vskip .1in
\sc{fig. 3.4: $A_{\phi}$ copies near
$s = -28, -26, -24, -22$ superimposed on $\zeta^{\circ 2}$
quadrant plots}
\end{figure}
The size of these features decays exponentially
with distance from zero. Their left-right orientation
alternates.
We speculate that the alternation
can be derived from the
alternating sign of the real
derivative $\frac {d \, \zeta(x)}{dx}|_{x = -2n}$.
\newline \newline
Because these copies exist
on the left half of the real axis,
its zeta pre-images also contain
complete copies of $A_{\phi}$. The upper left panel of
Figure 3.5 depicts the first bulb in the major filament
in the upper half plane. It is a $10$ by $10$
square centered at $\rho_1$. Along its
border we see
an apparently infinite set of filaments alternating with
an apparently infinite set of blunt processes.
\begin{figure}[!htbp]
\centering
\vskip 0.1in
\hskip .2in
\includegraphics[scale=.45]{leaf3point5r1c1.png}
\includegraphics[scale=.45]{leaf3point5r1c2.png}
\vskip .1in
\hskip .2in
\includegraphics[scale=.45]{leaf3point5r2c1.png}
\includegraphics[scale=.45]{leaf3point5r2c2.png}
\vskip .2in
\hskip 1.1in
\sc{fig. 3.5: a copy of $A_{\phi}$ near a blunt process of $A_{\phi}$; left:
with superimposed quadrant plot of $\zeta^{\circ 3}$; right: superimposed
quadrant plot of $\zeta^{\circ 4}$}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.45]{leaf3point6r1c1.png}
\includegraphics[scale=.45]{leaf3point6r1c2.png}
\vskip .2in
\includegraphics[scale=.45]{leaf3point6r2c1.png}
\includegraphics[scale=.45]{leaf3point6r2c2.png}
\newline
\hskip .2in
\sc{fig. 3.6: darker regions are $\zeta^{\circ n}$
pre-images of $\Re(s) > \frac 12$ near the main
cardioid; $n = 1$ in row 1 column 1; $n=2$ in row 1 column 2; $n=3$ in row
2 column 1; $n = 4$ in row 2 column 2}
\end{figure}
Our tests demonstrate that the filaments are
transected by $\zeta^{\circ n}$
pre-images of the critical line
for $(n = 1, 2, 3, ...)$,
and that the blunt processes are
transected similarly by
$\zeta^{\circ n}$ pre-images of the real axis.
The other three panels depict a small
copy of $A_{\phi}$ to the right of
the largest
blunt process on the right side of the bulb
shown in the upper left panel.
In the lower panels, a quadrant plot of $\zeta^{\circ 3}$
in the left panel and of $\zeta^{\circ 4}$
in the right panel have been
superimposed upon this copy.
Evidently, it is
a $\zeta^{\circ 3}$ pre-image of $A_{\phi}$.
\newline \newline
(5) Our observations indicate that for
each filament $F$ decorating $A_{\phi}$
there is a positive integer $k_F$
(say, the degree of $F$)
such that each bulb of $F$ contains
one
nontrivial $\zeta^{\circ k_F}$ zero,
and no nontrivial zeros of $\zeta^{\circ k}$
for any $k \neq k_F$.
Even under the Riemann hypothesis, it would not be
necessary from
first principles that degree $k$ filaments
are transected by $\zeta^{\circ k -1}$
pre-images of the critical line,
even though that is the simplest possibility.
But it seems to be
the case. In Figure 3.6, the $\zeta^{\circ k -1}$
pre-images of the critical line transecting
degree $k$ filaments decorating
the main cardioid are shown for
$k = 1, 2, 3$ and $4$.
\section{\sc branches interpolated by spirals}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.33]{leaf4point1r1c1.png}
\hskip .7in
\includegraphics[scale=.33]{leaf4point1r1c2.png}
\vskip .1in
\includegraphics[scale=.33]{leaf4point1r2c1.png}
\hskip .7in
\includegraphics[scale=.33]{leaf4point1r2c2.png}
\vskip .1in
\includegraphics[scale=.33]{leaf4point1r3c1.png}
\hskip .7in
\includegraphics[scale=.33]{leaf4point1r3c2.png}
\vskip .1in
\hskip .23in
\includegraphics[scale=.33]{leaf4point1r4c1.png}
\hskip .7in
\includegraphics[scale=.33]{leaf4point1r4c2.png}
\newline
\vskip .1in
\sc{fig. 4.1: zooming in on $\psi_{\rho_1}$; column 1: quadrant plots of
$\zeta(s) - s$; column 2 (top to bottom): quadrant plots of $\zeta^{\circ
n}, n = 3, 4, 5, 6$}
\end{figure}
For any integer $L > 0$
there appears
to be an infinite set of
zeta cycles $\Lambda = ( \lambda_0, ..., \lambda_{L-1})$
that pick out a linearly ordered subset
$B_{\rho, \Lambda} = (a_0, a_1, a_2, ...)$
of $\zeta^{\circ -}(\rho)$
such that (1) $a_0 = \rho$,
(2) $a_n = \zeta (a_{n+1}), n = 0, 1, 2, ....$
and (3) for each $j = 0, 1, 2, ..., L-1$,
the subsequences
$b_j = (a_j, a_{j + L}, a_{j + 2L}, ...)$
appear to converge to $\lambda_j$.
The sequence $b_j$ is a branch of $\zeta_L^{\circ -}(a_j)$.
In most of the cases
that we have examined,
each $b_j$ appears to be
interpolated by a
spiral $s_{\rho_N, \, \lambda_j}$
with center $\lambda_j = \lim b_j$.
The $\lambda_j$ are repelling
fixed points of $\zeta_L$.
\newline \newline
Now we offer (speaking loosely) a geometric description
of some of the $b_j$ in terms of the
basin of attraction $A_{\phi}$.
(It applies to most, but not
all, instances we have examined to date.)
A variety of filaments decorate
$A_{\phi}$, but here we restrict attention to those
that decorate the main
cardioid. We assign the set of filaments
a structure of union of
rooted trees $T$ as follows.
A filament $F \in T$
is the parent of a filament $G \in T$
if and only if $G$ decorates $F$ and there is
no intermediate filament $H \in T$ such that
$G$ decorates $H$ and $H$ decorates $F$.
The filaments containing
nontrivial Riemann zeros have no ancestors,
but they are not unique in this respect.
\newline \newline
Now fix integers
$L \geq 1, N \neq 0$. There is an infinite
set of zeta $L$-cycles
$\Lambda = (\lambda_0, \lambda_2, ..., \lambda_{L-1})$
such that for each integer
$\Delta = 0, 1, 2, ..., L-1$,
there is a tree
$T_{\Delta}$ of filaments decorating the main
cardioid of $A_{\phi}$, and a
path $P_{\Delta} = (F_0, F_1, ...)$ in $T_{\Delta}$
with
$k_{F_m} = \Delta + m L$
and such that
(if $m > 0) \, F_m$
decorates the $|N|^{\rm th}$ bulb of
its parent $F_{m-1}$.
As in the first column
of Figure 4.1,
the filaments in
$P_{\Delta}$ spiral around $\lambda_{\Delta}$.
In our graphic visualizations,
the apparent size of $F_m$
decays
exponentially with $m$.
Something like this would seem to be a
necessary condition of the relation
$\lambda_j = \lim b_j$
we mentioned above.
\newline \newline
Each bulb of $F_m$ contains
a nontrivial zero $w$
of $\zeta^{\circ \Delta + mL}$ and
$\zeta^{\circ \Delta + mL-1}(w)$ is
a nontrivial Riemann zero. Which one?
Let $w_{m, N}$ be
the nontrivial $\zeta^{\circ \Delta + mL}$ zero
belonging to the $|N|^{\rm th}$
bulb of
the filament $F_m$ in $P_{\Delta}$.
For $m > 0$,
$\zeta_L(w_{m, N}) = w_{m - 1, N}$,
so the sequence
$(w_{0, N}, w_{1, N}, ...)$
is a branch of $\zeta_L^{\circ -}(w_{0, N})$.
In our observations, the
zeta image of bulb $|N|$ of a filament
$F$ in $T_{\Delta}$
with $k_F > 1$
is bulb $|N|$ of its parent filament
in $T_{\Delta}$.
Therefore
$\zeta^{\circ \Delta + mL -1}(w_{m, N}) = \rho_{\pm N}$.
\newline \newline
The observation that
$\zeta_L(w_{m, N}) = w_{m - 1, N}$
suggests that
there should be graph isomorphisms between
subgraphs of the rooted tree graphs
associated to the $\zeta_L^{\circ -}$ on
one side and subgraphs of the
trees $T$ decorating the main cardioid of
$A_{\phi}$ on the other.
We should mention that the situation
for \it copies \rm of the main
cardioid such as the ones
illustrated in Figures 3.4 and 3.5
is different; except to say that
the zeta images of copies are also copies,
we will not discuss it
further in the present article.
\section{\sc spirals interpolating a branch of
the backward zeta orbit of a Riemann zero}
\subsection{Single spirals interpolating a branch.}
When $L = 1$, $\Lambda = \{\lambda_1 \}$ where
$\lambda_1$ is a repelling
zeta fixed point; there appear to be
at least three categories of such points:
$\psi_{-2n}$ (say) lying near the
trivial zeros $-2n = -20, -22, ...$;
zeta fixed points $\psi_{\rho^*}$ near each nontrivial
Riemann zero $\rho^*$,
and eight
fixed points lying at the boundary of
the main cardioid (right panel, Figure 3.2.)
How near? In the case of
the $\rho^*$, one can form an impression by
keeping in mind that this figure
depicts a $120$ by $120$ square (section 8.)
The distances $|-2n -\psi_{-2n}|$ are a great dealer smaller;
we omit the details.
\newline \newline
There are exactly two filaments $F$
with degree $k_F = 1$ decorating the main cardioid;
one of them contains
$\rho_N$ in its $|N|^{\rm th}$ bulb
$ = \beta_N$, say.
Our observations are consistent
with the following proposition.
The point $\psi_{\rho_{_N}}$
lies at the border of
the $|N|^{\rm th}$
bulb of
a filament $F^*$
with $k_{F^*} = 2$ decorating
$ \beta_N$.
This ramifies: if $\psi_{\rho_N}$
lies at the border of
the $|N|^{\rm th}$
bulb of
a filament $F^*$
then there is a child filament
$F'$ of $F^*$ such that
$\psi_{\rho_N}$
lies at the border of
the $|N|^{\rm th}$
bulb of $F'$
and $k_{F'} = k_{F^*}+1$.
\newline \newline
This is illustrated by the left column of
Figure 5.1 for $N = 1, 2, 3, 4$. It depicts
quadrant plots
of $s \mapsto \zeta(s) - s$, so that $\psi_{_{\rho_N}}$
shows up as four-color junctions on
the depicted filament (say, $F_2$.)
The right column shows
quadrant
plots of
$\zeta^{\circ 3}, \zeta^{\circ 4}, \zeta^{\circ 5}, \zeta^{\circ 6}$
in rows 1, 2, 3 and 4, respectively,
all superposed on plots of $A_{\phi}$.
The squares have side length
$.2, .02, .002,$ and $.0002$
in rows 1, 2, 3 and 4, respectively.
The center of the squares in row $N$ is
$\psi_N$, so the panels are
depicting the region around this
point at smaller and smaller scales.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.33]{leaf5point1r1c1.png}
\hskip .7in
\includegraphics[scale=.33]{leaf5point1r1c2.png}
\vskip .1in
\includegraphics[scale=.33]{leaf5point1r2c1.png}
\hskip .7in
\includegraphics[scale=.33]{leaf5point1r2c2.png}
\vskip .1in
\includegraphics[scale=.33]{leaf5point1r3c1.png}
\hskip .7in
\includegraphics[scale=.33]{leaf5point1r3c2.png}
\vskip .1in
\hskip .23in
\includegraphics[scale=.33]{leaf5point1r4c1.png}
\hskip .7in
\includegraphics[scale=.33]{leaf5point1r4c2.png}
\newline
\vskip .1in
\sc{fig. 5.1: top to bottom: $\psi_{\rho_n}, 1 \leq n \leq 4$; column 1:
quadrant plots of $\zeta(s) - s$; column 2: quadrant plots of $\zeta^{\circ
2}$}
\end{figure}
\newline \newline
Figure 4.1 displays the original indications
we had that some branches of
the inverse of zeta
lie on spirals.
It zooms in on the
illustration of $\psi_{\rho_1}$
in the left panel of
the top row of Figure 5.1.
The center of that panel is the fixed point,
lying on the border
of a filament $F_3$ (say)
decorating the lower border of
the largest full bulb.
In the right
panel of the top row of Figure 4.1,
a quadrant plot of
$\zeta^{\circ 3}$ has been superimposed
on a plot of the corresponding region of
$A_{\phi}$; we see that $F_3$ contains zeros
of $\zeta^{\circ 3}$. (Simple tests
% document the tests!
show that they are
nontrivial zeros in the sense of
the introduction.)
In the lower rows,
the zoom is repeated and $\psi_{\rho_1}$
is seen to lie near a
still-smaller filament decorating the
bulb near the center of the
figure just above it. The right column
depicts quadrant plots of $\zeta^{\circ 4}$,
$\zeta^{\circ 5}$ and $\zeta^{\circ 6}$
for the squares opposite them
in the left column. So in Figure 4.1 we are
seeing zeros of these functions (again
nontrivial.)
They also appear
(in virtue of the shapes of the
underlying $A_{\phi}$-bulbs)
to be
$\zeta^{\circ 3},
\zeta^{\circ 4}, \zeta^{\circ 5}$
pre-images of nontrivial Riemann zeros.
The rapid reduction of scale from one row
to the next attests to a similar
reduction of the distances of these
pre-images from $\psi_{\rho_1}$
(which, as we have remarked, is not surprising.)
The possibility that they may be traveling
on spirals emerges from a look at the angles
that the filaments $F_3$ and (say) $F_4, F_5,
F_6$ make with the horizontal.
These observations led us to do the numerical
tests described in the last section.
\newline \newline
%****************see notebook figures-2to0.nb**************
We made a survey of the spirals $s_{\rho,\psi_{\rho^*}}$
for various choices of $\rho$ and $\rho^*$.
We made a table of $\psi_{\rho_n}, 1 \leq n \leq 100$
with $500$ digits of precision
and we used a table of nontrivial Riemann zeros
with $300$ digits of precision
made by Andrew Odlyzko \cite{O}. We made tables of
the $z_k$ in various
$B_{\rho, \psi_{\rho^*}}$ to high precision,
proceeding inductively.
We set $z_1 = \rho$ and, given
a value of $z_k$, after using
\it Mathematica's \rm FindRoot command
to solve $\zeta(s) = z_k$ in the vicinity of $\psi_{\rho^*}$,
we set $z_{k+1}$ equal to the solution. We began with $300$
$z_k$ for each $B$ and used tests of reliability of
each $z_k$ to truncate the list; typically,
we ended up with a least $100$
consecutive $z_k$.
\newline \newline
We use polar coordinates $(r(z), \theta(z))$
to denote a typical point $z$ on $s_{\rho,\psi_{\rho^*}}$
such that (1) $r(z) = |z - \psi_{\rho^*} |$,
(2) $\theta(z)$ is chosen so that
$\theta(z)
\equiv \arg(z - \psi_{\rho^*}) \pmod {2\pi}$,
and (3) $\theta(z)$ varies continuously and monotonically
as $z$ moves around the spiral in a fixed direction.
In other words, $\theta(z)$
behaves up to a mutiplicative
constant like a winding number.
Then $r(z)$ appears to decay exponentially with $\theta(z)$.
(Of course we are only able to check this
for $z \in B_{\rho,\psi_{\rho^*}}$, that is,
for the $z_k$ we propose are interpolated by
$s_{\rho,\psi_{\rho^*}}$, because no other test for
membership in $s_{\rho,\psi_{\rho^*}}$ is available
to us.)
Therefore it was not practical to plot the spirals
$s_{\rho,\psi_{\rho^*}}$ without re-scaling,
so we plotted the points
$(\log r(z), \theta(z))$ instead.
This procedure
everts the apparent spirals:
if $k, j$ are such that
$r(z_k) < 1$ and $r(z_j) < 1$, then
$r(z_j) < r(z_k)$ implies
that the plotted point
$(\log r(z_j), \theta(z_j))$
is further from
the center of the re-scaled interpolating spiral
than the point
$(\log r(z_k), \theta(z_k))$: the
reverse of the situation before re-scaling.
(The direction of winding
of the spiral is also reversed because the
logarithms take negative values.) The
points near the center of the re-scaled spiral
depict $z_k$ for smaller values of $k$
for which $z_k$ is closer to
$\rho$ and further from $\rho^*$.
They are crowded so closely, in spite of our
re-scaling, that
the interpolating curve near $\rho$ is
obscured.\newline \newline
\begin{figure}[!htbp]
\vskip 0.1in
\hskip .7in
\includegraphics[scale=.3]{leaf5point2r1c1.png}
\hskip .7in
\includegraphics[scale=.3]{leaf5point2r1c2.png}
\vskip .1in
\hskip .7in
\includegraphics[scale=.3]{leaf5point2r2c1.png}
\hskip .7in
\includegraphics[scale=.3]{leaf5point2r2c2.png}
\vskip .1in
\hskip .7in
\includegraphics[scale=.3]{leaf5point2r3c1.png}
\hskip .7in
\includegraphics[scale=.3]{leaf5point2r3c2.png}
\vskip .1in
\hskip .7in
\includegraphics[scale=.3]{leaf5point2r4c1.png}
\hskip .7in
\includegraphics[scale=.3]{leaf5point2r4c2.png}
\vskip .1in
\hskip .7in
\includegraphics[scale=.3]{leaf5point2r5c1.png}
\hskip .7in
\includegraphics[scale=.3]{leaf5point2r5c2.png}
\newline
\vskip .1in
\hskip 0in
\sc{fig. 5.2: backward orbit branches for
$\rho_n \, (1 \leq n \leq 5)$
centered on two cardioid zeta fixed points}
\end{figure}
Figure 5.2 depicts branches of backward orbits of
$\rho_n (1 \leq n \leq 5)$
spiraling around two fixed point
$\approx -14.613 + 3.101i$
(left column) and
$\approx - 5.28 + 8.803i$
(right column)
on the
border of the main cardioid;
we omit the 500 digit decimal
expansions, which are easy to compute
using \it Mathematica\rm's
FindRoot command.
Figure 5.3 depicts branches of
backward orbits of $\rho_n$
spiraling around $\psi_{\rho_{_n}} (1 \leq n \leq 10)$.
(We omit their precise expansions for the same reason.)
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.3]{leaf5point3r1c1.png}
\hskip .7in
\includegraphics[scale=.3]{leaf5point3r1c2.png}
\vskip .1in
\includegraphics[scale=.3]{leaf5point3r2c1.png}
\hskip .7in
\includegraphics[scale=.3]{leaf5point3r2c2.png}
\vskip .1in
\includegraphics[scale=.3]{leaf5point3r3c1.png}
\hskip .7in
\includegraphics[scale=.3]{leaf5point3r3c2.png}
\vskip .1in
\includegraphics[scale=.3]{leaf5point3r4c1.png}
\hskip .7in
\includegraphics[scale=.3]{leaf5point3r4c2.png}
\vskip .1in
\hskip .4in
\includegraphics[scale=.3]{leaf5point3r5c1.png}
\hskip .7in
\includegraphics[scale=.3]{leaf5point3r5c2.png}
\newline
\vskip .1in
\sc{fig. 5.3: backward orbit branches
for $\rho_n$ near $\psi_{\rho_n}, 1
\leq n \leq 10$}
\end{figure}
\subsection{An example.}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.4]{leaf5point4r1c1.png}
\hskip .1in
\includegraphics[scale=.4]{leaf5point4r1c2.png}
\vskip .07in
\includegraphics[scale=.4]{leaf5point4r2c1.png}
\hskip .1in
\includegraphics[scale=.4]{leaf5point4r2c2.png}
\vskip .07in
\sc{fig. 5.4: logarithmic scaling of
$s_{\rho_1,\hskip .03in \rho_1}$}
\end{figure}
%I have to think more about the following remark.
%Because typically in our plots
%$\log r(z) < 0$, rescaling everts the
%spiral (points closer to
%the center appear farther away) and
%reverses its orientation.
The first panel of Figure 5.4 plots the point set
$B_{\rho_1,\psi_{\rho_1}}$
(re-scaled as described,
and shifted to place the apparent spiral's
center at the origin.)
It offers the
appearance that the $z_k$ (in red) form arms something
like those of
a spiral galaxy;
this seems to be a result of
nearly regular
growth of $\theta(z_k)$ with $k$.
But $z_k$ for consecutive $k$ do
not lie in adjacent positions on these arms;
in the second panel,
the $z_k$ are connected by chords in the same
order as they appear in the sequence
$B_{\rho_1, \psi_{\rho_1}}$:
vertices $v, w$ representing $z_v, w = \zeta(z_v)$
are connected by a chord.
The lower left panel of Figure 5.4 is a plot
of $\log |z_k - \psi_{\rho_1}|$ vs. $k$;
it is clear that the distances of the $z_k$
(colored red)
from the fixed point $\psi_{\rho_1}$ at the center
of the spiral are decaying exponentially.
The lower right panel of Figure 5.4
depicts a spiral curve (colored blue)
that approximately interpolates the
$z_k$; we found this curve using
the NonLinearModelFit command in
\it Mathematica\rm.
The equation of the curve is
%figures -2to0.nb
$$ \log r(z) = a + b \theta(z) + c \exp(d \theta(z)) ,
$$
with
$$a \approx 0.05575203301551956560399459579161529353,$$
$$b \approx -2.39481894384498740085074310912697832305,$$
$$c \approx -2.8680355917721941635331399485184884 \times 10^{-120},$$
and
$$d \approx 0.97375124237020440301256901292961731822.$$
% a, b, c and d here
%correspond to b, d, a, c,
% respectively, in the Mathematica notebook.
The absolute value of $c$ is so small that
this is quite close to being
the equation of a logarithmic spiral.
In the section on error terms below
we will compare directly the loci of the $z_k$ with
logarithmic spirals.
\newline \newline
In Figure 5.5, we study the
variation in $\theta(z_k)$. The
left panel plots $\delta_k = \theta(z_{k+1}) - \theta(z_k)$
against $k$ and shows that the $\theta(z_k)$ are
very nearly periodic in $k$.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.4]{leaf5point5left.png}
\hskip .01in
\includegraphics[scale=.4]{leaf5point5right.png}
\vskip .07in
\sc{fig. 5.5: left: $\delta_k$ vs. $k$
right: $\log | \delta_{k+1} - \delta_k|$
vs. $k$.}
\end{figure}
The right panel,
which plots
$\log |\delta_{k+1} - \delta_k|$ against $k$,
shows that the departure from periodicity
in $\theta(z_k)$ actually appears to
decay exponentially
with $k$.
However for other choices of
$\rho$ and $\rho^*$
this no longer holds, and so it is an
open question whether or
not it would hold even in this example
for very large $k$, that is, very close
to the center of the spiral.
We remark that the $z_k$ could, of course,
be distributed along a nearly-logarithmic spiral
while also being distributed in a completely
irregular or at least non-periodic way in the
theta aspect, so the two questions are at least
superficially independent.
\newline \newline
Now suppose $(r_1, \theta_1)$ and $(r_2, \theta_2)$
lie on a true logarithmic spiral
$\log r = a + b\theta$. The constants $a$, $b$ are
determined by any two points of the spiral, hence,
if two pairs of points determine
different values of $a$ and $b$,
then the curve that the three (or four)
points comprising the pairs lie on
is not a logarithmic spiral.
We used this idea to test
$B_{\rho_1,\psi_{\rho_1}} = \{z_1, z_2, z_3, ... \}$
for the property of being
interpolated by a logarithmic spiral.
We performed the
test by solving for $a$ and $b$
using the pairs $(z_1, z_k), k = 2, 3, ....$
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.3]{leaf5point6left.png}
\hskip .01in
\includegraphics[scale=.3]{leaf5point6center.png}
\includegraphics[scale=.3]{leaf5point6right.png}
\vskip .07in
\sc{fig. 5.6: parameters $a, b$ for logarithmic
curve $\log r = a + b \theta$ induced
by successive
points of $B_{\rho_1,\rho_1}$.
left: $a$ vs. $b$;
center: $b$ vs. $k$; right: $a$ vs. $k$}
\end{figure}
In Figure 5.6 we have plotted the resulting
values of $a$ against $b$,
$b$ against $k$, and $a$ against $k$.
Evidently $a$ is roughly linear in $b$,
and both $a$ and $b$
appear to converge as $k$ grows
without bound. Thus the interpolating
curve is not a
logarithmic spiral, for then $a$ and $b$ would
be constants.
But the convergence of $a$ and $b$
suggests that the
interpolating spiral $s_{\rho_1,\psi_{\rho_1}}$
resembles a logarithmic spiral
more and more closely
as it winds inward towards $\psi_{\rho_1}$.
\subsection{Backward orbits near the trivial zeros.}
There appear to be
real zeta fixed points $\psi_{-2n}$
near each trivial zero $-2n \leq -20$.
Whether they lie slightly to the right or
to the left
of $-2n$ along the real axis appears to depend
upon the parity of $n$. This reflects
the alternating left-right orientations
of copies (zeta pre-images) of the
basin of attraction $A_{\phi}$ we see
in Figure 3.4.
\newline \newline
A branch $B_{\rho,\psi_{_{-2n}}}$
of the backward orbit of each nontrivial
Riemann zero $\rho$ lies on a
curve appearing to pass through or terminate at
$\psi_{-2n}$: if $-2n \equiv 0$ (mod $4$),
then the curve appears to terminate at
$\psi_{-2n}$. These curves closely resemble
straight line segments. Error terms
are discussed in section 6. If $-2n \equiv 2$ (mod $4$),
then (supposing, for the moment, that the curve
really is a line segment)
$\psi_{-2n}$ lies near its midpoint.
\begin{figure}[!htbp]
\centering
\vskip 0.1in
\includegraphics[scale=.4]{leaf5dot7r1c1.png}
\hskip .1in
\includegraphics[scale=.4]{leaf5dot7r1c2.png}
\vskip .1in
\includegraphics[scale=.4]{leaf5dot7r2c1.png}
\hskip .1in
\includegraphics[scale=.4]{leaf5dot7r2c2.png}
\vskip .1in
\includegraphics[scale=.4]{leaf5dot7r3c1.png}
\hskip .1in
\includegraphics[scale=.4]{leaf5dot7r3c2.png}
\vskip .1in
\includegraphics[scale=.4]{leaf5dot7r4c1.png}
\hskip .1in
\includegraphics[scale=.4]{leaf5dot7r4c2.png}
\vskip .1in
\hskip 0in
\sc{fig. 5.7: column 1: backward orbits of $\rho_1$ near
$\psi_{-2n}, 10 \leq n \leq 13$}
\sc{column 2: backward orbits of $\rho_{n-9}$ near
$\psi_{-2n}, 10 \leq n \leq 13$}
\end{figure}
In Figure 5.7,
several of the backward orbits are depicted,
re-scaled logarithmically as above.
\newline \newline
This observation is consistent with the
hypothesis that $B_{\rho,\psi_{-2n}}$ is
interpolated by a spiral such that the
$a_k \in B_{\rho,\psi_{-2n}}$
satisfy $|\arg(a_k-\psi_{-2n}) - \arg(a_{k + 1}-\psi_{-2n})| \approx 2\pi$
for all $k$
if $-2n \equiv 0$ (mod $4$), or
$|\arg(a_k-\psi_{-2n}) - \arg(a_{k + 1}-\psi_{-2n})| \approx \pi$
if $-2n \equiv 2$ (mod $4$). We discuss this
further in sections 5.5 and 6.2.
\subsection{Several spirals that
together interpolate a branch.}
\begin{figure}[!htbp]
\centering
\vskip 0.1in
\hskip 0in
\includegraphics[scale=.4]{leaf5point8r1c1.png}
\includegraphics[scale=.4]{leaf5point8r1c2.png}
\vskip .05in
\includegraphics[scale=.4]{leaf5point8r2c1.png}
\includegraphics[scale=.4]{leaf5point8r2c2.png}
\hskip 0in
\vskip.1in
\sc{fig. 5.8: quadrant plots of
$s \mapsto \zeta^{\circ n}(s) - s, 2 \leq n \leq 5$, near the bulb around
$\rho_1$}
\end{figure}
Figure 5.8 depicts members of zeta $n$-cycles
(zeros of $s \mapsto \zeta^{\circ n}(s) - s$) near the bulb
of $A_{\phi}$ containing $\rho_1$ for $2 \leq n \leq 5$.
As $n$ increases the pattern of
distribution of these zeros becomes
more and more obscure.
The situation near $\rho_1$ appears to be typical of
that near all nontrivial zeros of zeta iterates.
\newline \newline
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.4]{leaf5point9r1c1.png}
\hskip .1in
\includegraphics[scale=.4]{leaf5point9r1c2.png}
\vskip .07in
\includegraphics[scale=.4]{leaf5point9r2c1.png}
\hskip .1in
\includegraphics[scale=.4]{leaf5point9r2c2.png}
\vskip .1in
\hskip 0in
\sc{fig. 5.9: the branch of a $\rho_1$
backward orbit induced by a zeta
$3$-cycle}
\end{figure}
Figure 5.9 illustrates the branch
$B_{\rho_{_1},\Lambda}$ of
the backward orbit of
$\zeta^{\circ -}(\rho_1)$ induced by a $3$-cycle
$\Lambda = ( \lambda_1, \lambda_2, \lambda_3 ) $
with $\lambda_1 \approx 3.95896 + 24.2362i$ .
The red vertices of a
given chord of the graph
represent points in
the branch.
Geometrically, Figure 5.9 is
doubly abstract:
(1) The spirals have been positioned so that their centers are
placed at
$1000 + 0i, 1000(\cos(\frac{2\pi}{3})+ i \sin(\frac{2\pi}{3}))$
and $1000(\cos(\frac{4\pi}{3})+ i \sin(\frac{4\pi}{3}))$
for the sake of legibility; and
(2) the spirals are everted: they have been
re-scaled logarithmically, so that points closer to
the center appear, in the figure, to be
farther from the center.
The row 2, column 2 panel shows vertices representing
elements of $B_{\rho_{_1},\Lambda}$
and edges between vertex pairs
$(v, \zeta(v))$, while the other
panels depict the spirals separately;
these are portraits of
$b_j (j = 1, 2, 3)$.
In these three figures, each vertex pair
$(v, \zeta^{\circ 3}(v))$ is connected by an edge.
\subsection{Angular distribution of branches along the spirals.}
A structural invariant, which seems to
determine the number of arms visible in
our plots of branches of
$f^{\circ -}(z)$, is the function
$\delta_{f,z, \psi}: k \mapsto \arg(a_k - \psi) - \arg(a_{k+1} - \psi)$,
where the $a_k$'s are members of a particular
branch of $f^{\circ -}(z)$
converging to $\psi$.
\newline \newline
The values of $\delta_k$ in the case
discussed in
section 5.2 correspond in the
present notation to those of
$\delta_{\zeta,\,\rho_1,\, \psi_1}(k)$.
They appear to change very slowly
with $k$, and this behavior seems to be what gives rise to
the appearance of discrete arms in
plots of branches of $\zeta^{\circ -}(\rho_n)$.
\newline \newline
Among
the zeta fixed points very near
trivial zeros, the only values
of
\newline
$\delta_{\zeta,\rho_n, \psi_{_{-2n - 18}}}(k)$ that we see
(Figure 5.7) are
$\approx \pi$ and $\approx 2\pi$, distributed,
as we have noted above, according to the mod $4$
residue classes of the zeros.
\newline \newline
Because $\delta_{\zeta,\rho_n,\psi_{\rho_n}}(k)$
apparently converges
rapidly as $k \rightarrow \infty$
(as in Figure 5.5 where $n = 1$),
we take the value of
$\delta_{\zeta, \rho_n, \psi_{\rho_n}}(100)$ as a proxy for
\newline
$\lim_{k\rightarrow \infty} \delta_{\zeta,\rho_n,\psi_{\rho_n}}(k)$.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.85]{leaf5point10.png}
\vskip .07in
\sc{fig. 5.10: $$\frac{\delta_{\zeta,\rho_n,\psi_{\rho_n}}(100)-\pi/2}{\pi/2},
1 \leq n \leq 600$$}
\end{figure}
Then our calculations
are consistent with the
proposition that
$\lim_{k\rightarrow \infty} \delta_{\zeta,\rho_n,\psi_{\rho_n}} \approx \pi/2$
(Figure 5.10.) Very small differences in this limit as $n$ varies appear
to determine very different shapes for the discrete arms visible
in our plots.
\newline \newline
We have observed in all of our
experiments that the visible structure of
a branch of $f^{\circ -}(z)$ depends upon the fixed
point at its center and not on $z$, so
$\delta_{f,z, \psi}$ should depend only upon $f$ and
$\psi$. Contrary to the impression suggested
by our notation, it should be independent of $z$,
but we cannot exclude the possibility
that there are counterexamples to this idea.
\newline \newline
\subsection{Logarithmic models of spirals interpolating branches of
the backward orbit of zeta.}
The branches
$B_{\rho, \psi} = (a_0 = \rho, a_1, a_2, ...)$ of
$\zeta^{\circ -}(\rho)$
for nontrivial Riemann zeros $\rho$
converging to zeta fixed points $\psi$
are interpolated by curves that
resemble logarithmic spirals.
We carried out experiments in which we
looked for approximations
of these interpolating curves
by such spirals. We chose
branches of the argument
function, varying with $k$ and evaluated at
$a_k - \psi$, such that
an angle $\theta_k$ was assigned to
$a_k - \psi$ which was the least such angle
$ > c + \max_{j <k} \theta_j$
for $c = 0$ or $1$.
The angle
$\theta_0$ was the value
of $\arg (a_0 - \psi)$
from the branch of argument
chosen automatically by \it Mathematica. \rm
The choice of
$c$ was dictated by requiring
that $\theta_k$ act like
a winding number about $\psi$
evaluated at the points $a_k$. For $\psi$
near a trivial zero, $c = 1$ was chosen;
for $\psi$ near a nontrivial zero, $c = 0$.
\newline \newline
For $r_k = |a_k -\psi|$
plots of the sets of pairs
$(\theta_k,\log r_k), k \geq 0$,
appear to lie on curves resembling
straight lines. For spirals centered at
zeta fixed points $\psi_{\rho_n}$ near the $\rho_n$,
we approximated
these lines
using \it Mathematica\rm's FindFit
command.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.4]{leaf5point11left.png}
\hskip .1in
\includegraphics[scale=.4]{leaf5point11right.png}
\vskip .07in
\hskip 0in
\sc{fig. 5.11: slopes $m_n$ (left panel) and intercepts $b_n$
(right panel) in log-linear models $\widetilde{r} = \exp (m_n\theta + b_n)$
of spirals interpolating
branches
of $\rho_n$ centered at $\psi_{\rho_n}$ with $r = |z - \psi_{\rho_n}|$
and $\theta = \arg (z - \psi_{\rho_n})$}
\end{figure}
Figure 5.11 is a pair of plots of $m_n$ and $b_n$ against
$n$ for models $|z- \psi_{\rho_n}| = \exp (m_n \theta + b_n)$
fitted to branches of $\zeta^{\circ -}(\rho_n),
1 \leq n \leq 600$. In particular,
we write $\widetilde{r_k} = \exp (m_n \theta_k + b_n)$
for our estimate of $r_k$. These seem to be
first-order approximations to
genuine interpolating curves; the error-term
will be
discussed in the next section.
\newline \newline
Investigating spirals centered at
the zeta fixed points $\psi_{-2n}$ that
lie near the trivial zeros was carried out in
a different way. We were able to
collect a substantial amount of
data (meaning for the first $200$ members of the
branches) for the
$\psi_{\rho_n}, n \leq 600$ using
$500$-digit precision.
On the other hand, even with
$1000$-digit precision,
we were able to collect data only
on the first twenty elements
of branches centered at the $\psi_{-2n}$
for $n \leq 30$ before
the use of the FindFit command
to get a linear model for the pairs
$(\theta_k, \log r_k)$
produced error messages
from \it Mathematica. \rm
\newline \newline
Fortunately, the branches spiraling about
the $\psi_{-2n}$ appear to
be better behaved than the
ones spiraling about the $\psi_{\rho_n}$.
For each pair $(n, n^*)$
the branch $B_{\rho_n, \psi_{-2n^*}}$
is apparently
interpolated both by a curve very nearly a
logarithmic spiral and
by another curve which is very nearly
a straight line passing through
the points $\rho_n$ and
$\psi_{-2n^*}$.
As we will see in the next
section, the fit of the branches to
the straight line passing through
these two points
is so good
that we use it as our second model,
together with the assumption
that the
$\theta_k = a\pi k \, +$ (a constant
depending only on $n$ and $n^*$),
with $a = 1$ or $2$ depending
as we have explained
only on the parity of $n^*$.
It was feasible to find
linear models for
the maps $k \mapsto \log r_k.$
%real rel error tests4.nb
Combining the assumptions
about the $\theta_k$
with the linear models we
construct for the $\log r_k$
gives logarithmic models for
the interpolating spirals.
We test these models in the next section.
\newline \newline
We chose
to examine the behavior of
branches
$B_{\rho_n,\psi_{-2n-18}}$
of $\zeta^{\circ -}(\rho_n)$
because, among zeta fixed points close to the
trivial zeros $-2n$, the greatest one
(the one that lies rightmost
along the real axis) is very close to
$-20$.
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[scale=.4]{leaf5point12left.png}
\hskip .1in
\includegraphics[scale=.4]{leaf5point12right.png}
\vskip .07in
\hskip 0in
\end{centering}
\sc{fig. 5.12: slopes $m_n$ (left panel) and intercepts $b_n$
(right panel) in models $\widehat{r_k} = \exp (m_n k + b_n)$ of spirals
interpolating
branches of $\rho_n$ centered at $\psi_{-2n-18}$}
\end{figure}
Figure 5.12 is a plot corresponding
to Figure 5.11
for the zeta fixed points $\psi_{-2n-18}$.
Here we have
$B_{\rho_n,\psi_{-2n-18}} = (a_0 = \rho_n, a_1, a_2, ...)$.
Writing $r_k = |\psi_{-2n-18} - a_k|$, we
took $m_n$ and $b_n$ to be, respectively,
the means of the slopes and intercepts of the chords connecting
consecutive pairs $p_k = (k, \log |a_k - \psi_{-2n-18}|)$.
Thus our model for $r_k = |a_k - \psi_{-2n-18}|$ is
$\widehat{r_k} = \exp (m_n k + b_n)$.
We discuss it further
this in
the following section.
\section{\sc error terms}
We will take the phrase
``error term'' to encompass complex-valued
deviations from a given estimate as well as
their absolute values.
Like the original
estimates, the curves followed by
complex-valued deviations appear to have
the form of logarithmic spirals.
This raises the prospect of an infinite regress,
which might perhaps lead to an exact
expression for the best interpolating curves,
but we have postponed any investigation of this
idea.
\subsection{Deviation of backward
orbit branches from logarithmic spirals.}
\subsubsection{Branches converging to fixed points near non-trivial zeros.}
(This subsection provides some of our evidence
for Conjecture 1.) For nontrivial
Riemann zeros $\rho_n$ and
the corresponding zeta fixed points $\psi_n$, we plotted the relative
complex-valued deviations $d_{rel(\rho_n,\psi_n,a_k)}$.
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[scale=.3]{leaf6point1r1c1.png}
\hskip .0in
\includegraphics[scale=.3]{leaf6point1r1c2.png}
\hskip .0in
\includegraphics[scale=.3]{leaf6point1r1c3.png}
\vskip .7in
\includegraphics[scale=.3]{leaf6point1r2c1.png}
\hskip .0in
\includegraphics[scale=.3]{leaf6point1r2c2.png}
\hskip .0in
\includegraphics[scale=.3]{leaf6point1r2c3.png}
\vskip .7in
\includegraphics[scale=.3]{leaf6point1r3c1.png}
\hskip .0in
\includegraphics[scale=.3]{leaf6point1r3c2.png}
\hskip .0in
\includegraphics[scale=.3]{leaf6point1r3c3.png}
\vskip .1in
\hskip .7in
\newline
\vskip .1in
\hskip 0in
\end{centering}
\sc{fig. 6.1: $B_{\rho_n, \psi_n}, n = 1, 28, 48$: column 1: original branches;}
\newline
\sc{column 2: deviations $d_{rel(\rho_n,\psi_n,a_k)}$ of $B_{\rho_n, \psi_n}$
from logarithmic spirals;}
\newline
\sc{column 3: $\log d_{rel(\rho_n,\psi_n,a_k)}$ vs. $k$}
\end{figure}
The runs depicted in Figure 6.1
portray both kinds of plots for
$\rho = \rho_n, \psi = \psi_{\rho_n}$
for $n = 1, 28$, and $48$.
It seems noteworthy that
the two kinds of plots
resemble each other so closely, but
inspection demonstrates that they
are not identical.
\newline \newline
The values of $\log d_{rel(\rho_n,\psi_n,a_k)}$ for $n = 1, 28$,
and $48$ are also plotted in
Figure 6.1 (column 3.)
The magnitude of the $d_{rel(\rho_n,\psi_n,a_k)}$
appears to decay exponentially
for $k <$ roughly $130$; for larger $k$,
however, the magnitude of the deviations
appears to grow exponentially without exceeding
$e^{-6}$ for $k \leq 200$. We think
that the shape of this curve,
which is typical, is an artificial
effect of the FindFit command on
a file of $200$ pieces of data: the fit is
best near the center of the data file.
% cplxfitcurve12may16no1.nb
% % and
% rlfixptloglinear22april16no5.nb
\newline \newline
For a fixed point $\psi = \psi_{\rho_n}$ near a nontrivial
zero $\rho = \rho_n$,
we used the initial $200$ elements of
each branch $B_{\rho, \psi}$ as a proxy
for $B_{\rho, \psi}$
to study the
relative deviations
of the curve interpolating it
from a logarithmic spiral.
Taking $\beta = 200$ for the moment, let us set
\newline \newline
$
max_{n, \beta} := \max_{1 \leq k \leq \beta} d_{rel(\rho_n,\psi_n,a_k)},
$
\newline \newline
$max_{n, \beta}^* := \sqrt{n/\log n} \times max_{n, \beta},$
\newline \newline
let $mean_{n, \beta}$ denote the mean of the
$d_{rel(\rho_n,\psi_n,a_k)}, k = 1 ,2, ..., \beta$
and let
\newline \newline
$mean_{n, \beta}^* := \sqrt{n/\log n} \times mean_{n, \beta}.$
\newline \newline
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.4]{leaf6dot2r1c1.png}
\hskip .2in
\includegraphics[scale=.4]{leaf6dot2r1c2.png}
\vskip .1in
\includegraphics[scale=.4]{leaf6dot2r2c1.png}
\hskip .2in
\includegraphics[scale=.4]{leaf6dot2r2c2.png}
\vskip .1in
\sc{fig. 6.2: row 1: $mean_{n, \beta}^*$ (left)
and $max_{n, \beta}^*$ (right), $\beta = 200, 1 \leq n \leq 600$;}
\sc{\newline row 2: smoothings of the corresponding plots in row 1}
\end{figure}
(We exclude $k=0$ in these definitions;
thus the values of these four numbers
tell us nothing about the fit of $a_0 = \rho$
to the spiral in question.)
\newline \newline
The panel in row 1 column 1 of Figure 6.2
is a plot of $600$ values of
$mean_{n, \beta}^*$.
The panel in row 1 column 2 is a plot of
$max_{n, \beta}^*$.
The plots in row 2 are smoothings of the plots in
row 1: for each $n$, they depict means of
$mean_{j, \beta}^*$ and $max_{j, \beta}^*$ over the
range $1 \leq j \leq n$.
\newline \newline
These plots are consistent with the proposition
that, for
$\beta = 200$, $mean_{n, \beta}$ and $max_{n, \beta}$
both
$= O \left ( \sqrt{\frac {\log n}{n}}\right )$
with both implied constants $<1$. More
optimistically, perhaps, the plots are consistent
with the hypothesis that
$mean_{n, \beta}$ and $max_{n, \beta}$
both
$= o \left ( \sqrt{\frac {\log n}{n}}\right )$.
\newline \newline
We tested the same idea after replacing $\sqrt{\frac {\log n}{n}}$
with powers $\left(\frac {\log n}{n}\right)^{\epsilon}$
for $\frac 12 < \epsilon <1$. It seems possible that the
supremum of $\epsilon$ for which these statements
might be true lies in the half-open interval $[.8, .9)$.
It also seems possible that this supremum is a
decreasing function of $n$. We omit the
relevant plots.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Branches converging to fixed points
near the trivial zeros.}
(This subsection provides
some of our evidence for Conjecture 2.)
Let the branch $B_{\rho_n, \psi_{-2n-18}}$ of
$\zeta^{\circ -}(\rho_n) = (a_0 = \rho_n, a_1, a_2, ....)$.
Before we test a logarithmic model for the decay of
$r_k = |a_k - \psi_{-2n - 18}|$,
we want to assess how well
$B_{\rho_n, \psi_{-2n-18}}$
fits the straight line passing through
$\rho_n$ and $\psi_{-2n-18}$.
We measured the vertical
deviation of the $a_k \in B_{\rho_n, \psi_{-2n-18}}$
from the straight line passing through
both $\rho_n$ and $\psi_{-2n-18}$
as a fraction of the heights of the $a_k$.
We make the following definitions:
\newline \newline
$M_n$ and $B_n$ are the slope and intercept respectively
of the straight line passing through $\rho_n$ and
$\psi_{-2n - 18}$
\newline \newline
and
\newline \newline
$d^{trivial}(n, k): =$
$$ \left|\frac{ \Im(a_k) - (M_n \Re(a_k) + B_n)}{\Im(a_k)}\right|,$$
\newline \newline
$mean^{trivial}_{n, \beta} = $ the mean of the
$d^{trivial}(n, k), 1 \leq k \leq \beta,$
\newline \newline
and
\newline \newline
$max^{trivial}_{n, \beta} = \max_{1 \leq k \leq \beta}d^{trivial}(n, k)$.
\newline \newline
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.4]{leaf6point3left.png}
\hskip .2in
\includegraphics[scale=.4]{leaf6point3right.png}
\vskip .1in
\sc{fig. 6.3: $\log mean^{trivial}_{n, \beta}$ (left)
and $\log max^{trivial}_{n, \beta}$ (right); $\beta = 20, 1 \leq n \leq
32$}
\end{figure}
The left panel of Figure 6.3 is a plot of
$\log mean^{trivial}_{n, \beta}, 1 \leq n \leq 32$ and $\beta = 20$.
The right panel is a corresponding of $\log max^{trivial}_{n, \beta}$.
Evidently, the $a_k$
lie near the specified lines, and agreement with the lines
improves rapidly as $n$ increases.
\newline \newline
Next we define for $a_k \in B_{\rho_n, \psi_{-2n-18}}$
\newline \newline
$r_{n,k} = |a_k - \psi_{-2n-18}|$,
\newline \newline
for $k > 1, m_{n, k}$ = the slope of the chord connecting
the ordered pairs $(k, \log r_{n, k})$ and $(k, \log r_{n, k-1})$ in
$\bf{R}\rm^2$,
\newline \newline
$m_{n, \beta} = $ mean of the $m_{n, k}, 1 \leq k \leq \beta$,
\newline \newline
and we define a $y$-intercept function $b_{n, \beta}$ analogously.
The error functions are defined as follows:
$$d^{model}(n, k, \beta):=\left|\frac {\log r_{n,k} - (m_{n, \beta} k +
b_{n, \beta}) }{\log r_{n,k}}\right|,$$
\newline \newline
$mean^{model}_{n, \beta} = $ the mean of the
$d^{model}(n, k, \beta), 1 \leq k \leq \beta,$
\newline \newline
and
\newline \newline
$max^{model}_{n, \beta} = \max_{1 \leq k \leq \beta} d^{model}(n, k, \beta)$.
\newline \newline
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.4]{leaf6point4left.png}
\hskip .2in
\includegraphics[scale=.4]{leaf6point4right.png}
\vskip .1in
\sc{fig. 6.4: $\log mean^{model}_{n, \beta}$ (left)
and
$\log max^{model}_{n, \beta}$ (right); $\beta = 20, 1 \leq n \leq 32$}
\end{figure}
The left panel of Figure 6.4 is a plot of
$\log mean^{model}_{n, \beta}, 1 \leq n \leq 32$ and $\beta = 20$.
The right panel is a corresponding of $\log max^{model}_{n, \beta}$.
Once more the fit is good and improves rapidly as $n$ increases.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Deviation of the Riemann zeros from
fitted logarithmic spirals.}
The plots in Figure 6.5 display
numerical information that support
Conjecture 4 and the scenario described in section 6.1.3.
They indicate the
possibility that as $n \to \infty$, the nontrivial Riemann zeros
$\rho_n$
become more well fitted to the logarithmic spirals
we have in turn
fitted to the branches $B_{\rho_n,\psi_n}$.
The left panel is a plot of $\log d_{rel}(\rho_n,\psi_n,0)$ against
$n, 1 \leq n \leq 600$;
The right panel plots $D_{rel}(N)$, the log of the running mean
of the $d_{rel}(\rho_n,\psi_n,0)$, as defined in
Conjecture 4,
$1 \leq N \leq 600$.
Figure 6.6 shows the corresponding plots
for $d_{rel}(\rho_n,\psi_{n+1},0)$.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.4]{leaf6point5left.png}
\hskip .2in
\includegraphics[scale=.4]{leaf6point5right.png}
\vskip .1in
\sc{fig. 6.5: left, $\log d_{rel}(\rho_n,\psi_n,0), 1 \leq n \leq 600$};
\newline
\sc{right, $\log \frac 1N \sum_{n=1}^N d_{rel}(\rho_n,\psi_n, 0), 1 \leq
N \leq 600$};
\newline
\sc{green curve: $-(\log N)^{.8}$, red curve :$-(\log N)^{.85}$}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.4]{leaf6point6left.png}
\hskip .2in
\includegraphics[scale=.4]{leaf6point6right.png}
\vskip .1in
\sc{fig. 6.6: left, $\log d_{rel}(\rho_n,\psi_{n+1},0), 1 \leq n \leq 600$};
\newline
\sc{right, $\log \frac 1N \sum_{n=1}^N d_{rel}(\rho_n,\psi_{n+1}, 0), 1 \leq
N \leq 600$};
\end{figure}
The next two plots treat absolute deviations $d_{abs}$, which
suggest the narrowing of the widths of the ``error bands''
mentioned in the speculations we ventured in the Introduction.
Figures 6.7 corresponds to the plots of relative deviations in Figure 6.5.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.4]{leaf6point7left.png}
\hskip .2in
\includegraphics[scale=.4]{leaf6point7right.png}
\vskip .1in
\sc{fig. 6.7: left, $\log d_{abs}(\rho_n,\psi_n,0), 1 \leq n \leq 600$};
\newline
\sc{right, $\log \frac 1N \sum_{n=1}^N d_{abs}(\rho_n,\psi_n, 0), 1 \leq
N \leq 600$};
\newline
\sc{green curve: $\frac 1N$, red curve :$\sqrt{\frac1N}$}
\end{figure}
Figure 6.8 displays information on absolute
deviations for fixed points $\psi_{-2n}$ nearest to the
trivial zeros $-2n$, with spirals terminating at $\rho_n$ (left panel)
and $\rho_{n+1}$ (right panel).
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.4]{leaf6point8left.png}
\hskip .2in
\includegraphics[scale=.4]{leaf6point8right.png}
\vskip .1in
\sc{fig. 6.8: left, $\log d_{abs}(\rho_n,\psi_{-2n-18},0), 1 \leq n \leq
600$;}
\newline
\sc{right, $\log d_{abs}(\rho_{n+1},\psi_{-2n-18},0), 1 \leq n \leq 600$;}
\newline
\sc{both green curves: $-n^{1.35}$, both red curves : $-n^{1.4}$}
\end{figure}
\subsection{Deviation from rotational invariance.}
It did not seem
plausible to us that
something special about Riemann zeros $\rho$
should force the
branches $\zeta^{\circ -}(z), z = \rho$
in particular, to be attracted
to repelling fixed points $\psi$
along logarithmic spirals.
As we remarked in the introduction, dynamical
systems theory leads one to expect
that the $\psi$ should attract
all of the nearby branches $\zeta^{\circ -}(z)$
whether or not $\zeta(z) = 0$,
and (one speculated) probably along
roughly similar curves.
Logarithmic spirals appear
in fluid mechanics
(see, \it e.g.\rm, \cite{Ma}, v.2, pp. 186-188
or \cite{SpAk}, p. 358.)
By analogy with the streamlines of
a vortex in a fluid,
we speculated that
the
existence of spiral curves connecting
zeros
of zeta
to repelling zeta fixed points might
be a consequence of
a scenario in which
there is
an infinite family of such
spirals related by rotations around
the fixed point.
By this we mean
a family of
spirals $s$ parameterized by real numbers $x$,
varying continuously with $x$
in a sense made explicit by condition (1) below,
such that if $s_x$ and $s_{x + \theta}$
are two such spirals with common center a
zeta fixed point
$\psi$, then
\newline \newline
(1)
$ s_{x + \theta} - \psi = e^{i \theta} ( s_x -\psi)$
\newline \newline
(condition (1) being an equation of homotheties) and
\newline \newline
(2)
$z \in s_x \Rightarrow$
(i) $\zeta(z) \in s_x$ and
(ii) there exists a
branch $B_z \subset s_x$ of $\zeta^{\circ -}(z)$
such that $\lim B_z = \psi$.
\newline \newline
In this scenario, the spiral curves would be
congruent in the sense of
Euclidean geometry and exactly one
spiral
would intersect the critical line
at each $\rho_n$
without appealing to special properties
of the zeros. We have verified the existence of
spiral branches $B_z$
of $\zeta^{\circ -}(z)$ for various $z$ on the
critical line other
than Riemann zeros without meeting a counterexample.
Like the spirals we have already described,
they are approximately logarithmic;
we omit the relevant plots.
\newline \newline
Condition (1) imposes rotational invariance
on the $s_x$. This suggests the
possibility that the branches of $\zeta^{\circ -}$
interpolated by them enjoy the
same property. Suppose
$u$ and $\zeta(u)$ lie on $s_x$ with center
$\psi$ and
let $R_{\theta,\psi}(z) := e^{i \theta}(z - \psi) + \psi$
be the function that takes $z$ to its image under
rotation by an angle $\theta$ around $\psi$.
Under perfect rotational invariance, not
only of the spirals $s_x$ but of the branches
of $\zeta^{\circ -}(z)$ for particular $z$
that they interpolate, the numbers
$R_{\theta,\psi}(\zeta(u)) - \zeta(R_{\theta,\psi}(u))$
must vanish. Therefore we studied the
$R_{\theta,\psi}(\zeta(u)) - \zeta(R_{\theta,\psi}(u))$.
We restricted ourselves to $u \in \zeta^{\circ -}(z)$
for various $z$,
not only because these are the main objects of
interest, but because our only
reliable information
about the spirals
comes from their interpolation
of the branches $B_{\psi} =
(a_0, a_1, a_2, ...)$
of $\zeta^{\circ -}(z)$,
and so our only useful candidates for points in $s$
are the members of such branches.
\newline \newline
Logarithmically scaled plots (which we omit) of the
discrepancies (say) \newline
$R_{\theta,\psi}(\zeta(a_n)) - \zeta(R_{\theta,\psi}(a_n))$
indicate that these numbers decay in modulus
exponentially and rotate
around the origin in a nearly linear fashion with $n$.
In other words, they themselves describe curves
that are approximated by logarithmic
spirals.
\section{\sc appendix: the figures}
\subsection{Figure 1.1}
Figure 1.1 a 120
by 120 square with center $1 + 0i$.
\subsection{Figure 2.1}
Figure 2.1 depicts a $6$ by $6$ square with center zero.
\subsection{Figure 3.1}
Figure 3.1 shows an 8 by 8 square
with center $-5 + 9.5i$.
\subsection{Figure 3.2}
Figure 3.2 shows a
120 by 120 square with center zero.
\subsection{Figure 3.3}
Figure 3.3 depicts
a 60 by 60 square
with center zero.
\subsection{Figure 3.4}
The upper left panel of Figure 3.4
shows a $2.4 \times 10^{-5}$
by $2.4 \times 10^{-4}$ square
centered at $-28$.
The upper right panel shows a $2.4 \times 10^{-4}$
by $2.4 \times 10^{-4}$ square
centered at $-26$. The lower left
panel shows a $.004$ by $.004$ square
centered at $-24$. The lower right depicts a
$.07$ by $.07$ square centered at $-22$.
\subsection{Figure 3.5}
The panel in row 1 column 1
of Figure 3.5 depicts a $10$ by $10$
square centered at $\rho_1$.
The other panels show a square
with side length $.006$
and center $\rho_1 + 4.1215 - .4015i \approx 4.6215 + 13.7332 i$.
\subsection{Figure 3.6}
Each panel of Figure 3.6 is a 30 by 30 square with center $ = -5$.
\subsection{Figure 4.1}
The squares depicted in Figure 4.1
have side length
$.2, .02, .002,$ and $.0002$
in rows 1, 2, 3 and 4, respectively.
The center of each square is
$\psi_{_{\rho_1}} \approx -2.3859 + 16.271i$.
\subsection{Figure 5.1}
All the panels of Figure 5.1
depict $A_{\phi}$ in
$2$ by $2$ squares. In rows 1 - 4,
the centers are $\psi_{_{\rho_1}}$ - $\psi_{_{\rho_4}}$,
respectively, where
$\psi_{_{\rho_2}} \approx -2.0369 + 21.9931i$,
$\psi_{_{\rho_3}} \approx -1.6935 + 26.5283i$,
and
$\psi_{_{\rho_4}} \approx -1.7496 + 30.8158i$.
\subsection{Figure 5.2}
In Figure 5.2, column 1 depicts branches of
$\zeta^{\circ -}(\rho_n), 1 \leq n \leq 5$,
centered at a zeta fixed point
$\approx -14.613 + 3.108 i$;
column 2 depicts branches of
$\zeta^{\circ -}(\rho_n), 1 \leq n \leq 5$,
centered at a zeta fixed point
$\approx -5.279 + 8.803i$.
\subsection{Figure 5.3}
In Figure 5.3 (referring to the caption),
the value of $n$ in row $a$, column
$b$ is $n = 2a + b - 2$.
\subsection{Figure 5.8}
Figure 5.8 depicts four views of
a $12$ by $12$ square
with center $\rho_1$.
\subsection{Figure 5.9}
Figure 5.9 depicts the branch
$B_{\rho_{_1},\Lambda}$ of
the backward orbit of
$\zeta^{\circ -}(\rho_1)$ induced by a $3$-cycle
$\Lambda = ( \lambda_1, \lambda_2, \lambda_3 ) $
with $\lambda_1 \approx 3.95896 + 24.2362i$.
\newpage
\bibliography{bibtexcite}
$\mbox{barrybrent@member.ams.org}$
\end{document}
```