InterJournal Complex Systems, 2136 Status: Accepted |
Manuscript Number: [2136] Submission Date: 20080226 |
Riemann’s zeta function and the prime series display a biotic pattern of diversification, novelty, and complexity |
Subject(s): CX.0
Category: Article
Abstract:
We find biotic behavior in series generated from the Riemann’s zeta function. The series are generated by taking real and imaginary parts of the zeta function evaluated along vertical lines in the complex plane including the half line ½ + s i. That biotic patterns occur in the zeta function is interesting in relation to studying the prime numbers since the Riemann’s zeta function is a re-encoding of the structure of the prime numbers. We also find biotic patterns in the series of prime numbers itself including the series of prime differences and the series obtained by A(n+1) = A(n) + sin(p(n)) where p(n) is the nth prime. Furthermore we see biotic patterns in approximations to the primes by sieving out only a finite number of prime multiples from the natural numbers and one can see these approximation approach the properties of the actual series for the primes. Bios is a causally-generated pattern characterized by diversification, novelty and complexity. For references on Bios see Kauffman and Sabelli, Cybernetics and Systems, 1998; Sabelli and Kauffman Cybernetics and Systems 1999; Kauffman, and Sabelli. 2003. Mathematical Bios. Kybernetes 2003. The main tool for measuring bios is the use of recurrence plot and quantification of isometries compared to those for shuffled versions of the series using the Bios Data Analyzer (Sabelli, Sugerman, Kovacevic, Kauffman, Carlson-Sabelli, Patel, and Konecki. Nonlinear Dynamics, Psychology and the Life Sciences, 2005).
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