Friday, September 30, 2016

Riemannian Distribution Patterns

Special Note: Today, it is now and finally live. I believe this interpretation of the Riemann hypothesis prime distribution to be not only entertaining but true, notwithstanding the irrational amount of patterns that may be drawn from any given initial distribution of natural numbers. These are the excerpts from my paper which is available on the Free Internet Archive in its entirety. I hope you enjoy them! 8*]

The Riemann Zeta equation has been tested and proven to be correct for the first 10 trillion non-trivial solutions. It is known that the mathematical solutions that satisfy the Riemann Zeta equation take the trivial form of either negative even integers or the non-trivial form of ½ + it. The Riemann hypothesis thereby states that all non-trivial solutions can be found to be on a straight line critical path with a varying imaginary form and a static real part. Through the use of base pattern distribution analysis and parametric resonance (isolver), one is able to reasonably conclude enough evidence to support Riemann's hypothesis by establishing a strong framework for order and sustainability within emergent prime number distribution patterns and their numerical bases.

Following the logic and relationships needed to explore one kind of pattern type, the first seventy-three natural numbers can now be considered a superset of distribution groups containing at least three different kinds of base pattern types known as reflective, wave/orthogonal, and polar digit sets. One repeating pattern can definitely be observed within the first set of one-hundred and four natural numbers continuously and that is the wave/orthogonal pattern. When we begin to understand that that these kinds of patterns appear to exist, it becomes easier to develop a general concept about pattern repetition, supersets, and type recognition. In the selected prime and composite examples, each iteration of a composite number, while every prime along with unity is explored.

There is a number sequence that can be gathered from the On-Line Encyclopedia of Integer Sequences that closely reflects the base pattern [8, 9, 3, 3, 3, 2...] for the first seventy-three integers. This is the decimal expansion of the imaginary part of 4th non-trivial zero of the Riemann Zeta equation, archived as pattern number A065453 in the OEIS. This pattern type also coincidentally begins on the seventy-third digit of the imaginary part of the non-trivial solution itself and will be considered an emergent polar base pattern for the sake of completeness in this analysis. A comparison of this sequence to a similar one of [8, 9, 3, 3, 2, 1...], its equality, its derivative, and its anti-derivative, shows potentially unique pattern sequence changes for a similar base pattern distribution drawn from the polynomial “decimal expansion of least x>0 satisfying x^2+4*x*cos(x)=4*sin(x),” archived as pattern number A199619 in the OEIS.

To know whether or not the base pattern frequencies observable in the prime distribution really are representative of a statistically accurate natural number analysis which provides a plausible link to every non-trivial solution to the Riemann hypothesis is an arduous endeavor. However, finding a solid and reliable framework to arrange the primes within the first one-hundred and four integers is not only an alluring process, but also intriguing from the point of view of consistency as it applies to searching for a concrete way to model a simple resolution towards a viably sound Riemann's hypothesis proof. From this analysis we can see that the initially random distribution of background patterns observable does correlate closely with a base pattern frequency distribution that can eventually be traced back to the Zeta function and at least one non-trivial solution which emerges on the critical line as the fourth. This implies that the initial appearance of patterns always harbors a definite order where the imaginary part of an early non-trivial zero is traced back to a specific characteristic, which may in fact only pertain to some and not all non-trivial solutions. However, due to the natural distribution of prime numbers, which appears to become more uniform only as they extend toward infinity, and the initial order that the identifiable base pattern groups tend to emerge from the superset, there is enough evidence to conclude that all patterns, while they do not emerge equally, harbor a common phenomenon which is their reliance upon a certain, as opposed to truthfully random structure.

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