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Riemann Hypothesis in j-space 4
^{th} August 2014 Let us consider an infinite series Z(s) represented as,
The function Z(s) also known as Euler series, is divergent for s less than or equal to 1. The Euler series can also be written in terms of prime numbers as, Using the method of analytic continuation we can write the Riemann Zeta function as, where s can be a complex number. We note that following properties, (i) The Riemann Zeta function is undefined for s=1. (ii) If RZ function does not diverge, it is equal to Z(s). (iii) Non-trivial zeros exist for s = -2, -4, -6, ...... (iv) For |s| < 1 (unit circle in complex-plane), non-trivial zeros exist in the complex plane such that, s = 1/2 + i r, where r is an irrational number. We will focus our attention on the analytic continuation first. We note that analytic continuation allows the existence of the multi-valued functions. In a discrete measurement space or j-space no two measurements can be alike (hence no multi-valued functions) due to entropy associated with the finite-capability macroscopic observer Obs _{M}. However we can assume the measurements to be almost identical in Obs_{M}
measurement space by invoking VTS symmetry and then results from
analytic continuation can be applied to explain observations in terms
of a series expansion in j-space. Furthermore we note the
presence of irrational numbers in the imaginary component of the
non-trivial zeros. Usually we encounter irrational numbers such as 'pi'
and 'e' while describing an indeterminate phenomenon in a discrete
measurement space. Thus we have two important issues to consider,
(i) the presence of a boundary at |s| = 1, or the unit circle in a
complex plane, and (ii) the irrational numbers which indicate the
presence of j-space which is likely to have a quantum nature. It will be worthwhile to point out a footnote (p. 526) in Adler's Quaternionic Quantum Mechanics and Quantum Fields, in which the work of Sir Michael Berry based on Odlyzko's numerical analysis of primes, is referred to. It is indicated that the Riemann Zeros are eigenvalues of a Hamiltonian of a classical dynamical system of chaotic orbits which is quantized with no time reversal symmetry (possibly the effect of spin is not considered). We had discussed the observer's capability in terms of particle-in-a-box problem. Rather than changing the complexity of the problem we consider observers with varying capabilities solving the same problem. For an observer of an infinite capability his measurements are described by a shallow-well. Whereas for an observer, Obs _{M}, of
infinitesimal capability, his measurements are described by
particle-in-a-box. The Riemann Zeros are likely to describe a system
based on the measurements of Obs_{M}.One of the benefits of the string theory is to establish the necessity of the abstract thought before the mathematical description is developed. So when would a curve would cross a line or when it would not becomes an important question to answer, so is the question whether a sphere can be divided into exact eight parts or not. Note: The article "A Friendly Introduction to The Riemann Hypothesis" by Thomas Wright is highly recommended along with John Derbyshire's Prime Obsession. Information on www.ijspace.org is licensed under a Creative Commons Attribution 4.0 International License. |
Previous Blogs: Spiral Galaxies Infinite Fulcrum Cauchy and Gaussian Distributions Discrete Space, b-field and lower mass bound Incompleteness II The supersymmetry The cat in box The initial state and symmetries Incompleteness I Discrete measurement space The frog in well |