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Computation 2015, 3(4), 528-540;

A Scale Invariant Distribution of the Prime Numbers

Division of Radiation Oncology, University of Ottawa, 501 Smyth Road, Ottawa, ON K1H 8L6, Canada
Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, Odense M DK-5230, Denmark
These authors contributed equally to this work.
Author to whom correspondence should be addressed.
Academic Editor: Demos T. Tsahalis
Received: 15 September 2015 / Revised: 12 October 2015 / Accepted: 21 October 2015 / Published: 28 October 2015
(This article belongs to the Section Computational Engineering)
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The irregular distribution of prime numbers amongst the integers has found multiple uses, from engineering applications of cryptography to quantum theory. The degree to which this distribution can be predicted thus has become a subject of current interest. Here, we present a computational analysis of the deviations between the actual positions of the prime numbers and their predicted positions from Riemann’s counting formula, focused on the variance function of these deviations from sequential enumerative bins. We show empirically that these deviations can be described by a class of probabilistic models known as the Tweedie exponential dispersion models that are characterized by a power law relationship between the variance and the mean, known by biologists as Taylor’s power law and by engineers as fluctuation scaling. This power law behavior of the prime number deviations is remarkable in that the same behavior has been found within the distribution of genes and single nucleotide polymorphisms (SNPs) within the human genome, the distribution of animals and plants within their habitats, as well as within many other biological and physical processes. We explain the common features of this behavior through a statistical convergence effect related to the central limit theorem that also generates 1/f noise. View Full-Text
Keywords: exponential dispersion models; 1/f noise; fractal scaling; self-organized criticality exponential dispersion models; 1/f noise; fractal scaling; self-organized criticality

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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Kendal, W.S.; Jørgensen, B. A Scale Invariant Distribution of the Prime Numbers. Computation 2015, 3, 528-540.

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