matrix vs the complex plane

[AntonioLao]

If a matrix of infinite size with positive integer elements can be shown with a topological equivalence to the 1st quadrant of the complex plane then the positive half of the Riemann zeta function is conformally mapped onto a sieve of Diophantus. Although the non-trivial zeros of the Riemann zeta function appear randomly distributed along the critical line, their corresponding transformed images on the sieve of Diophantus are arranged into ordered group patterns of 3 by 2 submatrices. Since conformal mappings preserve angles among curves, the angles preserved, between the sieve of Diophantus and the complex plane, are all equal to 90°. It is a true orthogonal transformation applicable both locally and globally. This provides a basis for a complete commutative Abelian gauge theory. However, gauge invariance remains embedded within any 2 by 2 reciprocal or inverse submatrix as integrable multiples of second order square symmetric Hadamard matrices. Since both the sieve of Diophantus and the Riemann zeta function are mathematically two dimensional, their transformations never traverse to lower or higher space-time dimensions. At the most it is a Q² « Q² but at the least a Z² « Z² transformation, which indicates that Q² « R² and Z² « R² transformations are unnecessary added complications.

[SteveA]

I see a lot of correlations with concepts I've had along similar lines as well.

You have an additive/sum as well as multiplicative/product form of the Riemann Zeta function.

In the product form it's similar to filtering a broadband energy - basically we have "notches" similar to resonances along a serial string that absorb, block (or redirect) energy.

In the summation form, it's similar to a transformation from a serial obstructive/blocking form into a parallel representation where orthogonal energy components are summed instead from a state of zero/no energy. (So you could see it similar to rotating this string and looking at its resonance energies being redirected outside a linear resonance - the spectrums are complimentary and the two views are similar to the reciprocal forms conductance and resistance in electronics)

In a sense we could also see this similar to a central chaotic state (the primes) sandwiched between two forms of "energy" as pure logic (no energy) and complete randomness (any relationship is equally (in)valid - which in many ways is impossible) and one representation of this intermediate chaotic state is looking from the origin of a 2-D grid/matrix and seeing a 1-D angular sweep of nearest (lowest reduced ratios) grid points. In many ways the two views are similar to an inside and outside view of this chaotic/fractal boundary.

A couple times I've wondered if there might not be a way to tile a 2-D pi^2/6 area using these two forms and find a common structure between these two representations.

I don't completely understand your comments, but I do see many correlations between them and some of my thoughts as well.

As a sidenote, I don't think that a perfectly uniform spectrum can exist and any random samples of something would appear to never be able to fulfill all expectations of randomness simultaineously - trade offs in various random characteristics should occur. There may be something in the relationships between these forms of trade offs that give a more realistic representation of how "randomness" would truly need to exist as something with detectable properties. (A perfect randomness would be inexplicable - a continuous constant amplitude of energy over a spectrum is also impossible - it would require infinitely more time to approach a uniform amplitude spectrum - some of these tradeoffs would also likely possess the equivalent of probabilistic windows of interactions, like that seen in quantum mechanics, because altering a single element in a sequence could appear to influence that randomness or predictable of many components in the sequence, hence we have an equivalent of fields of interactions (and equivalent forces) between these components)

Anyway, I know you're already familiar with many of these forms of ideas, but maybe something will give you that novel view you've been trying to find that makes it click.

[AntonioLao]

a lot of correlations with concepts I've had along similar lines as well.

I would like to send you the PDF manuscript that was rejected by the American Mathematical Society. I need your help making it publishable.

[SteveA]

I would like to send you the PDF manuscript that was rejected by the American Mathematical Society. I need your help making it publishable.

I haven't published any papers nor do I have a formal education in mathematics so translating things into more conventional descriptions would not be one my fortes.

Though I might be able to help you construct algorithms to model some of the characteristics of these ideas or manners to map the ideas into simpler representations.

I'll send you my e-mail address.

[AntonioLao]

algorithms to model some of the characteristics

I do need this algorithm and that is for a given large whole number M find the integer solutions for both x and y using the rational hyperbolic function y+2xy+3x-M=0. It seems that parallel processings can be applied to speed up the computations.

[SteveA]

I do need this algorithm and that is for a given large whole number M find the integer solutions for both x and y using the rational hyperbolic function y+2xy+3x-M=0. It seems that parallel processings can be applied to speed up the computations.

Ok, I gave you my e-mail. BTW, your name sounds familiar with regard to some work on primes. Have you given any lectures with regard to prime numbers before?

(Yes, parallel processing optimizations are something I'm not bad at . That's definitely a possibility for something I could help with)

In fact there are ways of using reconfigurable electronic hardware that could greatly speed this up as well.

[AntonioLao]

Have you given any lectures with regard to prime numbers before?

No, I have not. But it would be nice to think that it is the future you see. Back to the function y+2xy+3x-M=0, let M=11 then let x=2 substitute these give y+4y+6-11=0, solving for y gives y=1 showing both x and y as integers. However, for very large number for M, how can an algorithm finds all the integer values of both x and y? If no integer solutions then M represents the sieve row index of a prime number.

[SteveA]

I began looking at some ideas for this and there are at least a couple things that could work well for efficient hardware implementations (probably 100 times better per dollar than using PCs).

I'm hoping to hit on an epiphany that really optimizes much of it, but as an example of something more immediately realistic, there are programmable/reconfigurable electronic circuits (field programmable gate arrays) that pack a lot of computational power into a relatively inexpensive power, but they're typically a bit limited on memory (then again there are ways of optimizing memory usage also).

http://www.xilinx.com/products/devices.htm

But they use very low level operations and primarily operate off simply boolean logic and truth tables, with small memory. If we got rid of the multiplication and iterated the computation, for constant x:

f(x,y(t))=y+2xy+3x

Where y(t+1)=y(t)+1

We could also offset the base of the range for M by 3x to remove the 3x component (electronically this could also occur as a shifting of the range of M by 3 memory units every time x isincremented ... so we could have a window for M that continually shifts out results in blocks of 3).

Then f(x,y(t)) increments by a delta of (y+1)-y+2x(y+1)-2xy+3x-3x=2x+1, which is extremely simple and can be heavily "pipelined" in electronics (basically you can push the clock to very high frequencies and use techniques similar to a bucket brigade to handle many computations flowing in parallel). If values of x can be optimized to be sparely represented as powers of 2, though I can't think of anything immediately that would allow that (just something to consider - another interesting idea could be to optimize representations for x using different bases so that x has a sparse representation with mostly 0 digits - with a reconfigurable array of computational elements, that's a possibility and then you can trim out all the unused overhead for the 0 digits).

But I'm trying to think of a better way to optimize things on a higher level and avoid redundancy. Increases in brute for computation can help a lot but ultimately the best gains, especially for large values and ranges of M are algorithmic optimizations.

I'm still looking at your paper and I enjoy seeing some of your ideas as they help stimulates thoughts for myself as well. I wanted to let you know I hadn't forgotten about your ideas here.

Thanks,
Steve

[AntonioLao]

Steve, I'll email you another version of this rational test for the prime suspect.