Friday, January 13, 2006

Riemann Hypothesis: A Pure Love of Math

Mathematics Problem That Remains Elusive — And BeautifulBy Raymond Petersen
Dyson, one of the most highly-regarded scientists of his time, poignantly informed the young man that his findings into the distribution of prime numbers corresponded with the spacing and distribution of energy levels of a higher-ordered quantum state

Prime numbers are more than any assigned multitude of prime numbers.
Euclid Book IX Proposition 20

Mathematical ProblemsLecture delivered before the International Congress of Mathematicians at Paris in 1900 By Professor David Hilbert

8. Problems of prime numbers

Essential progress in the theory of the distribution of prime numbers has lately been made by Hadamard, de la Vallée-Poussin, Von Mangoldt and others. For the complete solution, however, of the problems set us by Riemann's paper "Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse," it still remains to prove the correctness of an exceedingly important statement of Riemann, viz., that the zero points of the function (s) defined by the series

Clay Mathematics Institute

The Riemann hypothesis asserts that all interesting solutions of the equation

ζ(s) = 0

lie on a certain vertical straight line. This has been checked for the first 1,500,000,000 solutions. A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers.

Over at Cosmic Variance, Clifford posted something in regards to this Hypothesis that I had been piece mealing over time trying to understand the implication of what is being offered here.

Prime Finding: Mathematicians mind the gapErica Klarreich

Goldston and Yildirim's novel idea was to examine the distribution not just of pairs of primes, but also of triples, quadruples, and larger groupings. Studying this wider question simplified the formulas estimating the spacing of primes, and to the team's surprise, the new result about smaller-than-average prime gaps fell out.

I learnt a lot by looking at the relationship of Gauss as the teacher of Riemann, and in this, itself is telling to a degree about the nature of how we move our thinking and our perception in a different way that we are not accustomed.

Primed to go

The algorithm at the centre of his method first checks whether the number in question is a perfect square, cube, or other power of a smaller number. If so, it is clearly not prime. If not, a sequence of tests using a type of mathematics called modular arithmetic is carried out. If certain conditions are satisfied, the number is definitely not prime. If they are not, then it is. Crucially, the time it takes to run the algorithm increases only slowly as the size of the number rises.

These tendencies are recognized in our consideration that were progressive, and Grossman's helping Einstein to think about the geometry taking place as you move to the consideration and the nature of GR and the curvatures that gravity implies.

On the Number of Prime Numbers less than a Given Quantity.Monatsberichte der Berliner Akademie, November 1859. by Bernhard Riemann

I believe that I can best convey my thanks for the honour which the Academy has to some degree conferred on me, through my admission as one of its correspondents, if I speedily make use of the permission thereby received
to communicate an investigation into the accumulation of the prime numbers; a topic which perhaps seems not wholly unworthy of such a communication, given the interest which Gauss and Dirichlet have themselves shown in it over a lengthy period.

Ulam's Spiral

Gauss saw this bright thinking, as a student gave his talk while he sat as a pround parent. What was he so proud of? I believe it is in the way that our vision had been changed from the confines of the natural world. Taking what we saw in all that it is.

Gaussian Primes

IMagine Einsteins youth and the meme's of compass instigated, that became modified in the request of science, as it progress's and propels the student along with the anomalies that one's perceptions had encountered.

Dynkin diagrams that if alloted to the way in which Baez talks about this, then how would such curvatures UV Gaussian coordinates or topologies, ever have been mapped from the 2d diagram to be viewed from these points drawn to a torus. Distances that would look so much different in how Gaussian coordinates are observed in relation to how these primes are aligned?

The Riemann Hypothesis in Song

Prime Obsession has an appendix containing the lyrics of Tom Apostol's song about the Riemann Hypothesis, with a full explanation of the lyrics. (Tom is Professor Emeritus of Mathematics at Caltech.)

Now the piece mealing makes it hard for someone to see anything of significance if one only seen the minute part of the process, and in context of what math would mean to the underlying basis of reality. Pure thought, and pure math, which would not refute the way in which we dress up reality and hide this pure thought under the forms that we do.

Andrey Kravstov

If you held the view of supersymmetry at the beginning of Andrey's image, then how would you discern the lengths of lines held under these gravitational perspectives? While we are often treated to time clocks and such, Gaussian coordinates change the way we can see these lines. That's part of the change in perception.

If one would have seen the gravitational collapse in context of the temperature values increased as this collapse, degrees in the boundaries of the blackhole then what have we been taken back too, instead of the singularities that are talked about in the production of the Princess's Pea, now, we see the superfluids?

I would have to explain myself as I do that in terms of what has happened with how I see what Conformal theory and temperature valuation(Bekenstein Bound and the 5d recognition of what goes on the horizon) might have meant when looking at the blackhole horizon. How such quantum mechanical interpretations, would increase the "probabilities of things" happening in regards to those temperatures (entropic increase to expansive considerations), that we had contained the whole system, within these views, as it cooled.

So if we thought of each collision process and all the scattering that went on in how we look at this in the Calorimeters, how well the concepts are considered along side of the energy valuation and primes?

Boltzman, Pascal, and the basis of the ordering of the selection of these numbered systems and how they make themself known? While issuing from a such a pure state of 5d considerations what transpired to see pascal's triangle would/should include primes?

IN binomial series such expressions are raised and probabilty characteristics, that would define position and momentum, as we see correlations to particle ejection formed from such collisions?

One would have to know why this particular numbering system? What pattern is there?

             1   2   3   4   5   6   7   8   9  10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100

.....and so on.

Primes to 500

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,
53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127,
131, 137, 139,149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197,
199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277,
281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367,
373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449,
457, 461, 463, 467, 479, 487, 491, 499, ...

Eratosthenes (275-194 B.C., Greece) devised a 'sieve' to discover prime numbers. A sieve is like a strainer that you drain spaghetti through when it is done cooking. The water drains out, leaving your spaghetti behind. Eratosthenes's sieve drains out composite numbers and leaves prime numbers behind.

How do such things make there way into reality and these prime numbers as signatures of the atoms and ways in which they would relate themself to this elemental table for viewing, and something then shifts in my perception. I don't know why?

If you turn probabilities above all the things held to entropic design, held in context of this universe, then such probabilities would have had to been recognized?

Does this bring us any closer to understanding of what issued from the mind of all probabilties, and in this, the idea of Bell's curve, or a bose condensate, or as a soliton form. In this situation, what said "this prime" to be associated to the probable outcome?

Ramanujan Modular Forms

Modular functions are used in the mathematical analysis of Riemann surfaces. Riemann surface theory is relevant to describing the behavior of strings as they move through space-time. When strings move they maintain a kind of symmetry called "conformal invariance"

Conformal invariance (also called "scale invariance") is related to the fact that points on the surface of a string's world sheet need not be evaluated in a particular order. As long as all points on the surface are taken into account in any consistent way, the physics should not change. Equations of how strings must behave when moving involve the Ramanujan function.

Did one ever figure out the value of the pitch? So you see, the universe is a concert as well:) You remember Pascal's triangle? The probabilistic valuation assigned to the marble drop? Well I created another triangle, but it is a little different model, and does not use numbers for mathematic discretion as a emergent property of first principal. Although mine is distinctive of these characteristics the universe is being applied in sound relation.

Riemann's Hypothesis was always held in my opinion in context of a 5D consideration as sounds analogy) of billiards balls making that clicking sound as they collide, or, as how Wayne Hu might look at the universe in it's hill and valleys.

So looking at Heaven's ephemeral qualities, I could 't but help think of Ramanujan here, and the exercise of Hardy and the Taxi Cab.

What was the pattern incited here that would be transfered to how we see probabilistic outcome reduced, from 5D considerations and higher. It had to be reduced, but how?

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