The RH has become a cornerstone for a significant portion of the mathematical edifice. There is no visible architectural substitute. Number theory would become chaotic for at least awhile if the RH were toppled. But even if it does not fail, we must contend with the slimness of its validity. It is as if the realm of numbers were being deliberately maintained at a critical point on the boundary between order and chaos.
[7/6]
Searching on "prime numbers" & chaos:
Quantumlike Chaos in Prime Number Distribution and in Turbulent Fluid Flows  A.M. Selvam (1999):
The model concepts enable to show that the continuum real number field contains unique structures, namely prime numbers which are analogous to the dominant eddies in the eddy continuum in turbulent fluid flows.
Finite Temperature Strings  Mark J. Bowick (1992)
We hope that the above exposition has given a flavor of the important connection between arithmetic gases and multiplicative number theory. Much remains to be understood for the case of statistical systems that arise from string theories and conformal field theories.
Continuing the speculation of Matthew Watkins from above: we could add something to the view of Leopold Kronecker that "God made the natural numbers; all else is the work of man." If God wished to make the integers wholesale, she could start with a quantum chaotic system with the approximate statistics of the zeta zeros and take the Fourier transform to produce the primes and then fill in the composite integers, making sure that the distributions of the zeros would iterate to produce the equally spaced natural numbers. Just a thought. Would it not be undignified for God to have to count that first time with her fingers and toes? This is just a fanciful way of restating the holographic principle that every bit of the world reflects and contains all of the information in its internal, essential relations with the rest.
[7/7]
A consistent idealism envisions a participatory universe, all aspects of which are essentially and internally related. The strong anthropic principle already pushes the envelope of what science can tolerate along these lines. We must go further. Idealism and immaterialism will necessarily contain strong elements pantheism, panpsychism and vitalism as part of their monistic repertoire. A materialist science cannot tolerate such elements. So much for materialism.
In a monistic and participatory universe there will necessarily be elements of parapsychology, alchemy and astrology. This is simply how an immaterial world manages to hang together. These are logically entailed by a holographic, holistic and, yes, a rational view of reality.
What I have been entertaining on these pages since at least last year is an idealist version of numerical participation. Critics will wish to label it 'numerology.' Well, it was not I who first pointed to the 'unreasonable effectiveness of mathematics.'
There is a possible conflict between this unreasonable effectiveness and the anthropic principle. Given the supposed inalterability of mathematics and the mathematical basis of physics, particularly w.r.t. exceptional structures such as the monster group, did God ever have a significant choice in Creation?
To put this point more succinctly, which came first: mind or math? If the materialists have all become Platonists in their embrace of mathematical physics, where does that leave us idealists? We have been elbowed out at the banquet table.
We will have to take a stand somewhere. The Riemann Hypothesis looms too large now for us not to have an opinion about it. Mathematicians and materialists both wish to see it proven expeditiously: business as usual. The looming presence of the RH could prove embarrassing if not otherwise proven. How so?
There is apparently a considerable resistance to treating the RH like the Continuum Hypothesis. In the case of the CH, it's validity was proven to be independent of the other axioms of set theory, and so we were free to take it or leave it. The clear consensus was to adopt the CH as an additional axiom.
With the RH, however, the feeling among mathematicians is that it ought to be provable. It is too closely integrated to the realm of number theory to be considered, or be proven, to be independent of that body of knowledge. Failure to prove it would be more like the albatross on the neck: not easy to ignore. It would raise issues of epistemology and ontology. It must be real, but we can't know it. We have to take it on faith! This recasts the Riemann Hypothesis into the mold of the God Hypothesis. This would not sit well with the materialists, one can imagine.
Here is another angle. You may have heard of the Skewes number, weighing in originally at 10^10^10^34. This is many more zeros than there are atoms in the universe. Even if you could turn the universe into a roll of leger paper, you would not be able to write it out. It has since been trimmed down to a very modest 10^371. This is the number before which the prime count must first exceed its asymptotic form as given by the prime number theorem, and given the RH. Even at this modest size, we might never have a computer big enough to actually calculate this first crossing point. There are known to be an infinite number of such crossings further out.
If the RH fails, there will be that first zeta zero which falls off the critical 1/2 line. The first 440 billion zeros @ ZetaGrid as of today are right on the money. And it took several days to calculate the first fifteen zeros back in 1903. It does not appear that the BruijnNewman constant is in jeopardy of being forced positive by these new results, but I would like to see a plot of its more recent values based on the close zeros data. The record for closeness is well over a year old now. There ought to be an alarm bell for the next record. This effort might be compared with SETI@home. Which is more likely to have a positive result?
Although SETI presently has garnered much greater interest on the Internet, in the BPW the ZetaGrid ought to be the way to go. It could be more ambiguous and thus more insidious, but the eschatological significance would ultimately point to the same Omega. It has taken awhile for me to get back to the page topic, but there it is. Think about it.
There would be considerable irony in such a Y2C event. Jesus was supposed to come back in white robes on a flying saucer, shades of SETI. With ZetaGrid, Jesus comes back on a Pentium (R) chip, so to speak. It would be a much more subtle and less disturbing advent. But how could we link JC to Lehmer's Phenomenon? Well, that's one way!
The first nonRiemannian zero would be the most significant number in mathematics, just short of pi, perhaps. We could make something of it. And, unlike pi, its provenance would be much more openended. There is not a closed formula to calculate any of the zeros. In holographic fashion, each zero contains information about every prime number. There are various computing tricks to obtain approximate information about their locations.
THE RIEMANNSIEGEL FORMULA AND LARGE SCALE COMPUTATIONS OF THE RIEMANN ZETA FUNCTION by GLENDON RALPH PUGH B.Sc., University of New Brunswick, 1992
Here it gets a little tricky. Find theoretically or computationally the first nonconforming zero and start computing its deviation from the critical line. This computation will be an open ended process requiring additional theoretical progress. I would guess that in this extended process additional coincidences such as that of e & pi would show up. We would gradually be lead to a more organic or vitalistic view of mathematics, a perspective from which we could understand how the existence and physical role of the Monster Group could be compatible with the Anthropic Principle. In effect, we would see that mathematics in all its Platonic glory is not exempt from the teleology that governs our macro and microcosm.
This increase in our understanding of the coherence of the world could come about without the intervention of a deviant zero, but let us afford ourselves just this minimal bit of drama or external stimulus. If there is an eschatological teleology of mind over matter, there will have to be some such minimal numerological component. To increase the level of the drama, it could probably be arranged for there to be a more conventional, or SETI style, message somehow embedded within the numerical coincidences. This much to placate Hollywood.
One might as easily arrange for there to be a message embedded in the positions of the galaxies, say, and meant to be viewed by the Hubble telescope. Perhaps, but this would have nothing to say about astronomy per se. It would necessarily be more epiphenomenal than what might be possible in exploiting a mathematical medium. It would be too much like just exploiting the SETI medium. God is known to be arcane.
[7/8]
Numerology is an almost lost art. It is due for revival. Ramanujan was the last great numerologist. He had a direct knowledge of numbers that was unencumbered by layers of analytical education. Numbers have been disenchanted to become mere labels, no longer having intrinsic meaning. We turn away from the e & pi coincidence as we might from seeing a ghost. Mathematical physics has demonstrated that numbers are bound up in the atoms. How do they leap from the atoms to the mind of a Srinivasa? Such genius is possible only if the world is a hologram, waiting for us to comprehend the coherence of the micro and macrocosm.
Riemann's zeta function and the monster group are just two slices of that hologram. Each number comes alive inside every other number as part of the universal resonance. That resonance manifests most dramatically as our Omega. 'e' and pi taken separately are just labels. Taken together they possess a harmony that even Srinivasa could not completely decipher. We can and will cultivate that genius once we understand its natural necessity. It will take something like the drama of a 'misplaced' zeta 0 to trigger our deeper curiosity. That 'misplaced' zero would act as the irritant forcing us to bend our minds toward the reestablishment of the larger harmony. If this is not sufficient to set us on course to the Omega, then something else will be.
We understand the significance of 'little' alpha, the fine structure constant, with a value of ~ 1/137. We know that if it differed by more than one or two percent, carbon based life would have been impossible in this universe. This is just a piece of Anthropics. We don't know how it came to have this magical value. Some suppose that it is merely a random number in each of an infinite set of universes. It is what it is because we are here to measure it. That is weak anthropics. I am a strong anthropicist. There is just this one BPW in which everything is held together in optimal coherence, this despite the apparent and superficial preMillennial incoherence.
I suggest that Srinivasa II will find a 'formula' for alpha in terms of e & pi, and then will be able to explain it to the rest of us, along with the meaning of the misplaced zeta 0. Such a 'formula' would summarize the holographic involution of mind and matter. No small order, this.
[7/10]
I'm perusing the class number problem. This relates to the near integral value of the Ramanujan number: e^(pi*sqrt(163)). This might shed some light on e^pi ~= 20 + pi, or it might not.
Uniform Distribution of Heegner Points  V. Vatsal, 2001.
HEEGNER ZEROS OF THETA FUNCTIONS  Jorge JimenezUrroz and Tonghai Yang
[7/11]
Rational points: 9800 hits.
It is clear that mathematicians are into rationalization in a big way. The study of rational points on elliptic curves is major. But I can't tell you what it is about. I may be dense, however, there is a dearth of background explanation as the mathematicians go busily about their work. Are they concerned about rational points just because of the intrinsic challenge, or is there an ulterior motive?
Why do I care about rational points? It goes back to the natural and venerable impulse of wanting to square the circle. It is a desire for closure. That is the point of reason, not to play on words. The materialist impulse has been taking us precisely in the opposite direction: embracing the Apeiron that was anciently abhorred. Is it too late to put the Apeiron back in the tube? Certainly that would be the consensus view if anyone bothered to ask. Yet there is this residual interest, whatever its provenance.
For instance, take a look here: Rational Points on Elliptic Curves. It is as though someone were trying to refute Fermat's last theorem: A^3 + B^3 /= C^3. But no word of explanation: quite peculiar. Maybe I should mind my own business and not poke my nose in other folks playpens.
But then go here: Overview of "Mathematician's Secret Room". I am being reminded that number theory is about numbers, after all, not about those nasty irrationals (sic). It was a terrible day for the Platonists and Pythagoreans in 500 B.C. when Hippasus of Metapontum (Hyperbridge?)
used geometric methods to demonstrate that the hypotenuse of an isosceles triangle with legs of length one (i.e., sqrt 2, sometimes called Pythagoras's constant) cannot be expressed as a ratio of integers. A number of this type is now called an irrational number. Legend has it that Hippasus made his discovery at sea and was thrown overboard by fanatic Pythagoreans.
And can you blame them? Ever since, there has been a tension in mathematics between the rationalists and irrationalists. The rationalists remain stranded in number theory while their irrational colleagues help to explore the, no longer rational, heavens. Leonhard Euler attempted to bridge this chasm back in the 18th century with his invention of analytic number theory. His goal remains elusive.
One answer to the above question is simply: the computer or the information explosion. This is the commercial motive behind the revival of number theory and the renewed, but still in the closet, interest in squaring the circle. Yes, Pythagoras dream of the harmony of the spheres was reawakened first by digital music. We often refer to mathematical physicists as Pythagoreans, but the true Pythagoreans would have thrown them overboard right along with Hippasus. Only Roger Penrose might have been spared from among them.
All that we rationalists are asking for is an ouroboric closure. We will even grant you the Apeiron, but it must be the best possible Apeiron, and is must serve the Ouroboros.
Number theorists continue to strive mightily to contain the inherent wildness of the primes. Theirs is truly a mission unto the wilderness. Among other treasures they bring back is the EulerMascheroni constant (~0.111^1/4). Imagine if it turned out rational. That possibility could well be related to the disposal of the RH.
New Math Formulas Discovered With Supercomputers: (p.7)
In April 1993, Enrico AuYeung, an undergraduate at the University of Waterloo, brought to the attention of Jonathan Borwein, his professor, the curious fact that:
[a double sum over inverse integers ~= 17*pi^4/360]
based on a computation to 500,000 terms. Borwein was skeptical of this finding  if there was such an identity, why hadn't the theory behind it been discovered by mathematicians centuries ago? Borwein tried computing this sum to a higher level of precision in order to demonstrate to the student that this conjecture really did not precisely hold. Surprisingly, in subsequent computations by Borwein to 30 digits and by myself to over 100 decimal digits, this relation was upheld. Needless to say, it is rather unlikely that a mathematical relation could hold to such high precision and yet not really be true.
Then see IntegerRelations.
My objection to this approach is that it sacrifices rationality for precision, which is what got us into our analytic fix in the first place. This approach would miss the near integers, for instance. Heegner's insight would be lost.
[713]
Integer relations (632 hits):
Experimental Mathematics and Integer Relations by Jonathan M. Borwein, 2002:
Mathematicians increasingly use symbolic and numeric computation, visualisation tools, simulation and data mining. This is both problematic and challenging. For example, we mathematicians care more about the reliability of our literature than other sciences. These new developments, however, have led to the role of proof in mathematics now being under siege.
[...] Many of my favourite examples originate in between mathematical physics and number theory/analysis/knot theory and involve the ubiquitous Zeta Function, of Riemann hypothesis fame.
One thing they do at CECM is attempt to distinguish between algebraic and transcendental numbers. On a related issue see Is Visualization Struggling under the Myth of Objectivity? Also see Euler sums in Math World, and Visible Structures in Number theory  Peter Borwein.
Let us note again the jfunction. It is responsible for the near integers associated with our e&pi syzygy, and for the structure of the MG. This is taking us back to moonshine (and here).
week173  John Baez:
The dimensions of the irreducible representations of the Monster are closely connected to the coefficients of an important function in complex analysis, called the jfunction  this connection is known as Monstrous Moonshine.
week66:
[...] what interests me most about this stuff is the whole idea of "exceptional structures"  symmetrical structures that don't fit into the neat regular families in classification theorems. The remarkable fact is that many of them are deeply related. As Geoffrey Dixon put it to me, they seem to have a "holographic" quality, meaning that each one contains in encoded form some of the information needed to construct all the rest! It thus seems pointless to hope that one is "the explanation" for the rest, but I would still like some conceptual "explanation" for the whole collection of them  though I have no idea what it should be.
[7/14]
Modular Form  from MathWorld:
Modular forms satisfy rather spectacular and special properties resulting from their surprising array of internal symmetries.
Langlands program.
Modular Forms and Elliptic Curves: TaniyamaShimura
For a particular elliptic curve, the number of integer solutions in each clock arithmetic forms an Lseries for that curve.
A modular form is defined by two axes, x and y, but EACH axis has a real and imaginary part. In effect it is four dimensional (xr, xi, yr, yi) where xr means real part of x, xi means imaginary part of x, and similarly with yr and yi. The fourdimensional space is called hyperbolic space. The interesting thing about modular forms is that they exhibit infinite symmetry under [modular] transformations....
Richard E. Borcherds by C. S. Rajan
Even before the monster group was proved to exist, hints of its intricate connections with the theory of modular functions began to appear. It was observed by Ogg that in a certain naturally occurring sequence {Sn} of modular curves, Sp has genus 0 (namely it is the Riemann sphere) for a prime p if and only if p divides the order of the monster group. McKay and Thompson found interesting connections between dimensions of vector spaces with irreducible representations of the monster, with the coefficients of the Fourier series expansion of the elliptic modular jfunction.
[...] An example of a vertex operator algebra is given by the Fock space of a string propagating on a torus. The moonshine module is obtained by combining a twisted as well as an untwisted vertex operator module associated to the Leech lattice, and amounts to a theory of a string propagating on an orbifold that is not a torus.
week 95  John Baez
[...] Well, in dimension 24, there are 24 even unimodular lattices, which were classified by Niemeier. A few of these are obvious, like E8 + E8 + E8 and E8 + D16+, but the coolest one is the "Leech lattice", which is the only one having no vectors of length 2. This is related to a whole WORLD of bizarre and perversely fascinating mathematics, like the "Monster group", the largest sporadic finite simple group  and also to string theory. I said a bit about this stuff in "week66", and I will say more in the future, but for now let me just describe how to get the Leech lattice.
[...] However, Conway doesn't seem to explain *why* the Weyl vectors have this ascending form. So I'm afraid I really don't understand how all the pieces fit together. All I can say is that for some reason the Weyl vectors have this ascending form, and the fact that the Weyl vector is also lightlike makes a lot of magic happen in 26 dimensions. For example, it turns out that in 26 dimensions there are *infinitely many* fundamental roots, unlike in the two lower dimensional cases.
Just to add mystery upon mystery, Conway notes that in higher dimensions there is no vector v for which all the fundamental roots r have r.v equal to some constant. So the pattern above does not continue.
I find this stuff fascinating, but it would drive me nuts to try to work on it. It's as if God had a day off and was seeing how many strange features he could build into mathematics without actually making it inconsistent.
Re The Monster and the Leech  John Baez
In dimension 26 we get infinitely many fundamental roots forming a Dynkin diagram with one node for each point in the Leech lattice, and edges in a pattern that depends only on the distances between these points in the Leech lattice.
[7/15]
Information in the Holographic Universe: This article from the current issue of Scientific American points to a problem with most scientific speculation. In short, it is too little, too late, too superficial and too literal. It contrasts unfavorably with the more meaning based, metaphorical 'holographic' speculation of John Baez excerpted above. It would be nice if there were a clear path from physics to metaphysics via quantum information, but there is not. Quantum information is much too atomistic. Meaning, on the other hand, is fundamentally irreducible. The path from materialism to immaterialism must pass through coherence and monism. It must first recognize the irreducible unity of all being. Short of that realization, scientific speculation is likely to be counterproductive, as is true in this case. Cosmic intelligence is to be neither quantized nor quantified.
It is this problem that has motivated me to look to mathematics as the soft underbelly of materialism. The physical quantum in this context is largely a red herring. It is a symptom of the underlying, overlying immaterialism, not its source. At best it is a doorway from physics into the realm of math. The Baezian 'holographic' unity of math is much closer to the cosmic 'holography' of the BPW than is the quantum of physics.
What is the meaning of 'exceptional beauty'? At this point, no one really knows, but we can take a stab at it. The crux of the matter lies here with the interdependence of the exceptional structures. Naively we might suppose that mathematical complexity is epiphenomenal, that it only lies on the periphery of that domain. This is the impression given by chaos theory, where a very few commonalities are overwhelmed by the innumerable profusion of possibilities. Such is not the case with exceptionality. Exceptional complexity is truly sparse, and rather than inhabiting the periphery of the subject, it resides at its core, or so we are being lead to believe. How do we account for this surprising development? What does it mean, and how might it relate to the BPW hypothesis?
These exceptional structures are what appear to account for the organicity of mathematics. These form the vertebrae of the subject, held together by an underlying 'functoriality'.
Representation theory  its rise and its role in number theory  Robert Langlands (1989). This review is a bit more accessible from a physics background.
