## The 'Golden Mean' in number theory

[abstract:] "Beginning with the most general fractal strings/sprays construction recently expounded in the book by Lapidus and Frankenhuysen, it is shown how the complexified extension of El Naschie's Cantorian-Fractal spacetime model belongs to a very special class of families of fractal strings/sprays whose scaling ratios are given by suitable pinary (pinary, p prime) powers of the Golden Mean. We then proceed to show why the logarithmic periodicity laws in Nature are direct physical consequences of the complex dimensions associated with these fractal strings/sprays. We proceed with a discussion on quasi-crystals with p-adic internal symmetries, von Neumann's Continuous Geometry, the role of wild topology in fractal strings/sprays, the Banach-Tarski paradox, tesselations of the hyperbolic plane, quark confinement and the Mersenne-prime hierarchy of bit-string physics in determining the fundamental physical constants in Nature."

Castro's observation possibly linking the 'Golden String' to a function central to the behaviour of certain eigenvalues in random matrix theory (which in turn appears to be deeply linked to the behaviour of the nontrivial zeros of the Riemann zeta function).

P. Cvitanovic, "Circle Maps: Irrationally Winding" from Number Theory and Physics, eds. C. Itzykson, et. al. (Springer, 1992)

"We shall start by briefly summarizing the results of the 'local' renormalization theory for transitions from quasiperiodicity to chaos. In experimental tests of this theory one adjusts the external frequency to make the frequency ratio as far as possible from being mode-locked. this is most readily attained by tuning the ratio to the 'golden mean' (51/2 - 1)/2. The choice of the golden mean is dictated by number theory: the golden mean is the irrational number for which it is hardest to give good rational approximants. As experimental measurements have limited accuracy, physicists usually do not expect that such number-theoretic subtleties as how irrational a number is should be of any physical interest. However, in the dynamical systems theory to chaos the starting point is the enumeration of asymptotic motions of a dynamical system, and through this enumeration number theory enters and comes to play a central role."

B.W. Ninham and S. Lidin, "Some remarks on quasi-crystal structure", Acta Crystallographica A 48 (1992) 640-650

[abstract:] "The Fourier transform of skeleton delta function that characterizes the most striking features of experimental quasi-crystal diffraction patterns is evaluated. The result plays a role analogous to the Poisson summation formula for periodic delta functions that underlie classical crystallography. The real-space distribution can be interpreted in terms of a backbone comprising a system of intersecting equiangular spirals into which are inscribed (self-similar) gnomons of isoceles triangles with length-to-base ratio the golden mean...In addition to the vertices of these triangles, there is an infinite number of other points that may tile space in two or three dimensions. Other mathematical formulae of relevance are briefly discussed."

[from concluding remarks:] "Perhaps the most interesting feature is that our Fourier-transform sum seems to have much in common with the distribution of the zeros of the Riemann zeta function...! That indicates something of the depth of the problem. That the zeta function ought to come into the scheme of things somehow is not surprising - the Poisson and related summation formulae are special cases of the Jacobi theta function. [Indeed the Bravais lattices can be enumerated systematically through an integral over all possible products and sums of products of any three of the four theta functions in different combinations that automatically preserve translational and rotational symmetries.] The theta-function transformations are themselves just another way of writing the [functional equation of the zeta function]. Additionally, the properties of the zeta function are automatically connected to the theory of prime numbers. So one expects that the Rogers-Ramanujan relations must play a central role in the scheme of things for quasi-crystals."

V. Dimitrov, T. Cooklev and B. Donevsky, "Number theoretic transforms over the golden section quadratic field.", IEEE Trans. Sig. Proc. 43 (1995) 1790-1797

V. Dimitrov,G. Jullien, and W. Miller, "A residue number system implementation of real orthogonal transforms", IEEE Trans. Sig. Proc. 46 (1998) 563-570.

M.L. Lapidus and M. van Frankenhuysen, "A prime orbit theorem for self-similar flows and Diophantine approximation", Contemporary Mathematics volume 290 (AMS 2001) 113-138.

"EXAMPLE 2.23 (The Golden flow). We consider the nonlattice flow GF with weights w1 = log 2 and w2 = \phi log2, where \phi = (1 + 51/2)/2 is the golden ratio. We call this flow the golden flow. Its dynamical zeta function is

\zetaGF(s) = 1/(1 - 2-s - 2-\phis)"

C. Bonanno and M.S. Mega, "Toward a dynamical model for prime numbers" Chaos, Solitons and Fractals 20 (2004) 107-118

[abstract:] "We show one possible dynamical approach to the study of the distribution of prime numbers. Our approach is based on two complexity methods, the Computable Information Content and the Entropy Information Gain, looking for analogies between the prime numbers and intermittency."

The main idea here is that the Manneville map Tz exhibits a phase transition at z = 2, at which point the mean Algorithmic Information Content of the associated symbolic dynamics is n/log n. n is a kind of iteration number. For this to work, the domain of Tz [0,1] must be partitioned as [0,0.618...] U [0.618...,1] where 1.618... is the golden mean.

The authors attempt to exploit the resemblance to the approximating function in the Prime Number Theorem, and in some sense model the distribution of primes in dynamical terms, i.e. relate the prime number series (as a binary string) to the orbits of the Manneville map T2. Certain refinements of this are then explored.

The Phyllotaxis project's notes on the Farey Tree and the Golden Mean

Selvam's attempts to link the Riemann zeta function to fluid flow, atmospheric turbulence, etc. (the Golden Mean appearing as a winding number)

I have discovered a particularly simple Beurling generalised-prime configuration wherein the associated zeta function has a 'fixed point' at the Golden Ratio (i.e. zeta(1.618...) = 1.618...   Notes will be added here in due course.

J. Dudon, "The golden scale", Pitch I/2 (1987) 1-7.

"The Golden scale is a unique unequal temperament based on the Golden number. The equal temperaments most used, 5, 7, 12, 19, 31, 50, etc. are crystallizations through the numbers of the Fibonacci series, of the same universal Golden scale, based on a geometry of intervals related in Golden proportion. The author provides the ratios and dimensions of its intervals and explains the specific intonation interest of such a cycle of Golden fifths, unfolding into microtonal coincidences with the first five significant prime numbers ratio intervals (3:5:7:11:13)." [Note that here the Fibonacci sequence mentioned differs slightly from, but is closely related to, the usual one.]

[abstract:] "The present paper is a review, a thesis of some very important contributes of E. Witten, C. Beasley, R. Ricci, B. Basso et al. regarding various applications concerning the Jones polynomials, the Wilson loops and the cusp anomaly and integrability from string theory. In this work, in Section 1, we have described some equations concerning the knot polynomials, the Chern–Simons from four dimensions, the D3-NS5 system with a theta-angle, the Wick rotation, the comparison to topological field theory, the Wilson loops, the localization and the boundary formula. We have described also some equations concerning electric-magnetic duality to $N = 4$ super Yang-Mills theory, the gravitational coupling and the framing anomaly for knots. Furthermore, we have described some equations concerning the gauge theory description, relation to Morse theory and the action. In Section 2, we have described some equations concerning the applications of non-abelian localization to analyze the Chern–Simons path integral including Wilson loop insertions. In the Section 3, we have described some equations concerning the cusp anomaly and integrability from string theory and some equations concerning the cusp anomalous dimension in the transition regime from strong to weak coupling. In Section 4, we have described also some equations concerning the "fractal" behaviour of the partition function.

Also here, we have described some mathematical connections between various equation described in the paper and (i) the Ramanujan's modular equations regarding the physical vibrations of the bosonic strings and the superstrings, thence the relationship with the Palumbo-Nardelli model, (ii) the mathematical connections with the Ramanujan's equations concerning $\pi$ and, in conclusion, (iii) the mathematical connections with the golden ratio $\phi$ and with $1.375$ that is the mean real value for the number of partitions $p(n)$."

In their paper "The golden mean as clock cycle of brain waves" (Chaos, Solitons and Fractals 18 No. 4 (2003) 643-652, Harald and Volkmar Weiss acknowledge this website as one of several "...without which our work would be impossible", and in a subsequent email, Volkmar Weiss wrote "Your site was very helpful to us in an extraordinary way."

Although the article has no explicit number theoretical content, it relates closely to quite a few different areas of research which are relevant to this archive.

[abstract:] "The principle of information coding by the brain seems to be based on the golden mean. Since decades psychologists have claimed memory span to be the missing link between psychometric intelligence and cognition. By applying Bose-Einstein-statistics to learning experiments, Pascual-Leone obtained a fit between predicted and tested span. Multiplying span by mental speed (bits processed per unit time) and using the entropy formula for bosons, we obtain the same result. If we understand span as the quantum number n of a harmonic oscillator, we obtain this result from the EEG. The metric of brain waves can always be understood as a superposition of n harmonics times $2\Phi$, where half of the fundamental is the golden mean $\Phi$ (= 1.618) as the point of resonance. Such wave packets scaled in powers of the golden mean have to be understood as numbers with directions, where bifurcations occur at the edge of chaos, i.e. $2\Phi = 3 + \phi^3$. Similarities with El Naschie's theory for high energy particle's physics are also discussed."

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