--Paul Erdos

The Number Theory Room contains pictures of famous number-theoretic quantities. A theoretical explanation is in development and will someday be posted on the Theory Page. The unifying theme here is the underlying Modular group SL(2,Z) (proto-)symmetry (more precisely, a Fuchsian-group symmetry). The secondary goal is to provide a number-theoretic explanation for the structure of the interior of the Mandelbrot Set. A tertiary goal is to better understand the arithmetic series generated by iterated functions in terms of the associated Maclaurin series, and/or Bell series sums. I used to believe that I knew what an analytic function was. Not any more. I'm hopelessly confused now.

**Six Sigma**. Amazingly, Ramanujan was able to provide a number of
interesting identities for this function. I wonder if he knew that this
is what some of his functions looked like. Ref. Hardy & Wright.

**Imagine Six Sigma**.

**Phase Two**.

**Gee Two**. The modulus of the Weierstrass ellipic function
invariant g_{2} as a function of the square of the
nome q=exp (i pi tau). The zeros of this function are the blue dots.
These get faint as they approach the edge of the disk; a clearer
view of the zero can be gotten from the phase diagram, several images
below.

**Really Gee Two**. The real part of the Weierstrass ellipic function
invariant g_{2} as a function of the square of the nome q=exp (i pi tau).
The real part is negative in the black areas, and positive in the
colored areas. Blue denotes smaller values, passing through red for the
larger values.

**Imagine Gee Two**. The imaginary part of the Weierstrass ellipic function
invariant g_{2} as a function of the square of the nome q=exp (i pi tau).
The imaginary part is negative in the black areas, and positive in the
colored areas. Blue denotes smaller values, passing through red for the
larger values.

**Really Gee Three**. The real part of the Weierstrass ellipic function
invariant g_{3} as a function of the square of the nome q=exp (i pi tau).
The real part is negative in the black areas, and positive in the
colored areas. Blue denotes smaller values, passing through red for the
larger values.

**Imagine Gee Three**. The imaginary part of the Weierstrass ellipic function
invariant g_{3} as a function of the square of the nome q=exp (i pi tau).
The imaginary part is negative in the black areas, and positive in the
colored areas. Blue denotes smaller values, passing through red for the
larger values.

**Phase Gee Two**. The phase of the Weierstrass ellipic function
invariant g_{2} as a function of the square of the nome q=exp (i pi tau).
The phase runs from -pi in the black areas, 0 in the green areas,
and yellow-read in 0 to +pi regions. Note that the zeros of this
function are easily located, as being located at the points where the
colors wrap completely around.

**Really Discriminating**. The real part of the Modular Discriminant
(Dedekind eta to the 24'th), as a function of the square of the nome
q=exp (i pi tau).
The real part is negative in the black areas, and positive in the
colored areas. Blue denotes smaller values, passing through red for the
larger values.

**Imagine Discriminating**. The imaginary part of the Modular Discriminant
(Dedekind eta to the 24'th), as a function of the square of the
nome q=exp (i pi tau).
The imaginary part is negative in the black areas, and positive in the
colored areas. Blue denotes smaller values, passing through red for the
larger values.

**Jah Really**. The real part of Klein's J-invariant
g_2^3/delta.

**Jah Really**. The phase of Klein's J-invariant
g_2^3/delta. As usual, black = -pi, green=0, and red=+pi.

**Really Moebius Three**. Amazingly, this one is studied
by undergraduates enrolled in first-semester introductory number
theory classes. Since books on number theory never seem to have
graphics in them, I wonder if the students have any idea that this is
what they're dealing with. I wonder if the professors have any idea ...

**Euler's Gear**. Another good name for this one might
**Really Totiently Toiling Away**, searching for the definitive
totative answer, trying to be as Dirichlet productive as possible.

**Really a More Thue Investigation**.

**Really Random**.

Moral of the story: explore random functions, and we will almost invariably get hit by the modular group symmetry in the proceedings. The Dirichlet series associated with randomly generated arithmetic functions have fascinating analytic properties. Literally, *any* random function will do ...

I think the general statement is that "almost all analytic functions are hyperbolic". The general program is this: pick some random arithmetic series (possibly not convergent). The Maclaurin sum for the series is an analytic function, that is, a harmonic function, that is, a two-dimensional function with vanishing laplacian. By classical electrostatics, it has equipotential lines that are closed loops, and 'electric field lines' running perpendicular to the equipotentials. Pick an equipotential. By the uniformization theorem, that equipotential maps to either the 2-sphere, the plane, or the Poincare disk (+1,0,-1 curvature). One can say "almost all" random series conformally map to the Poincare disk; almost all series are strongly hyperbolic. Equipotentials aren't the only route: these are "trivial" genus-0 Riemann surfaces; they have a metric, they have geodesics. They have zeros and they have cuts. Is there a multi-sheeted "flat connection" that we can give these functions? i.e. the analog of the no-curvature form of the tetratorus?

Next, we know that the hyperbolic conformal mappings are going to be determined by some Fuchsian group. So the question really is: given any random arithmetic series, and given any fixed equipotential line on its Maclaurin series, what is the Fuchsian group for the uniformization for that equipotential? The equipotential lines provide a continuous, differentiable parameter for the set of Fuchsian groups. The whole she-bang can inherit a metric in several different ways; and so we ask, what is the topology of the resulting set?

This all seems to be a rich area for exploration, and damned if I know of any classical treatments for this stuff. I mean, this is so basic, why didn't I get this as an undergrad, or a grad student? It seems that the pros just launch into Kac-Moody algebras and Wess-Zumino models, while I'm stuck trying to figure out what the heck an analytic function is. Even simple analytic functions, such as those obtained from the iterated tent map, seem to have an incredibly rich structure ... Sigh. Life's too short.

Copyright (c) 2005 Linas Vepstas

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