## number theory and fractality

J.H. Bruinier and K. Ono, "An algebraic formula for the partition function" (A.I.M. preprint, 01/2011)

[abstract:] "We derive a formula for the partition function $p(n)$ as a finite sum of algebraic numbers. The summands are discriminant $-24n + 1$ singular moduli for a special weak Maass form that we describe in terms of Dedekind's eta-function and Eisenstein series."

A. Folsom, Z.A. Kent and Ken Ono, "$l$-adic properties of the partition function" (A.I.M. preprint, 01/2011)

[excerpt:] "Our work brings completely new ideas to the problems," Ono says. "We prove that partition numbers are 'fractal' for every prime. These numbers, in a way we make precise, are self-similar in a shocking way. Our 'zooming' procedure resolves several open conjectures, and it will change how mathematicians study partitions."

L. Vepstas, The Modular Group and Fractals - An exposition of the relationship between fractals, the Riemann zeta function, the modular group, the Farey fractions and the Minkowski Question Mark.

S.C. Woon, "Riemann zeta function is a fractal"

"[We] infer three corollaries from Voronin's theorem [on the 'universality' of the Riemann zeta function]. The first is interesting, the second is a strange and amusing consequence, and the third is ludicrous and shocking (but a consequence nevertheless)."

"Computations of the Julia and Mandelbrot sets of the Riemann zeta function and observations of their properties are made.  In the appendix section, a corollary of Voronin's theorem is derived and a scale-invariant equation for the bounds in Goldbach conjecture is conjectured."

S.C. Woon, "Period-harmonic-tupling jumps to chaos and fractal-scaling in a class of series", Chaos, Solitons and Fractals 5 (1) (1995) 125.

[abstract:] "Series like Riemann zeta function are found to have large jumps at harmonic periods. the route to chaos for these series is via the cascade of period-harmonic-tupling jumps in contrast to the cascade of period-doubling bifurcations for unimodal iterative maps like the logistic map. The sets of oscillations between one jump and the next also exhibits the fractal-scaling property. This gives rise to the simplest of all approximations of Riemann zeta function..."

Hua Wu and D.W.L. Sprung, "Riemann zeta and a fractal potential", Physical Review E 48 (1993) 2595.

"The nontrivial Riemann zeros are reproduced using a one-dimensional local-potential model. A close look at the potential suggests that it has a fractal structure of dimension d = 1.5."

A. Ramani, B. Grammaticos, E. Caurier, "Fractal potentials from energy levels", Phys. Rev. E 51 (1995) 6323–6326

Hua Wu and D. W. L. Sprung, "Reply to "Fractal potentials from energy levels"", Phys. Rev. E 51 (1995) 6327

[abstract:] "We point out in this reply that Ramani, Grammaticos, and Caurier have made a useful technical improvement in the quantum inversion method we used, but the spectrum they worked with is not suitable for the purpose they claim."

D. Dominici, "Some remarks on the Wu-Sprung potential. Preliminary report" (preprint 10/05)

Wang Liang, Huang Yan and Dai Zhi-cheng, "Fractal in the statistics of Goldbach partition" (preprint 01/06)

[abstract:] "Some interesting chaos phenomena have been found in the difference of prime numbers. Here we discuss a theme about the sum of two prime numbers, Goldbach conjecture. This conjecture states that any even number could be expressed as the sum of two prime numbers. Goldbach partition r(n) is the number of representations of an even number n as the sum of two primes. This paper analyzes the statistics of series r(n) (n=4,6,8,...). The familiar 3 period oscillations in histogram of difference of consecutive primes appear in r(n).We also find r(n) series could be divided into different levels period oscillation series. The series in the same or different levels are all very similar, which presents the obvious fractal phenomenon. Moreover, symmetry between the statistics figure of sum and difference of two prime numbers are also described. We find the estimate of Hardy-Littlewood could precisely depict these phenomena. A rough analyzing for periodic behavior of r(n) is given by symbolic dynamics theory at last."

O. Shanker, "Zeroes of Riemann zeta function and Hurst exponent" (preprint 01/06)

[abstract:] "The theory underlying the location of the zeros of the Riemann zeta function is one of the most intriguing unsolved problems. It is interesting to physicists because of the Hilbert-Pólya Conjecture, that the non-trivial zeros of the zeta function correspond to the eigenvalues of some positive operator. Since there is no proof yet for this conjecture, it is important to study the properties of the locations of the zeroes empirically using a variety of methods. In this paper we use the rescaled range analysis to study the spacings between successive zeroes. We find that for large orders of the zeroes the spacings seem to have a Hurst exponent of about 0.095. This implies that the distribution has a high fractal dimension, and shows a lot of detailed structure. The distribution appears to be of the anti-persistent fractional Brownian motion type, with a significant degree of anti-persistence."

O. Shanker, "Generalised zeta functions and self-similarity of zero distributions" (preprint 01/06)

[abstract:] "There is growing evidence for a connection between Random Matrix Theories used in physics and the theory of L-functions. The theory underlying the locations of the zeroes of these generalised Zeta Functions is one of the key unsolved problems. Physicists are interested because of the Hilbert-Pólya Conjecture, that the non-trivial zeros of the zeta function correspond to the eigenvalues of some positive operator. To complement the continuing theoretical work, it would be useful to study empirically the locations of the zeroes by different complementary methods. In an earlier paper we found that the high order zeroes of the Riemann Zeta Function showed a remarkable self-similarity in their distribution, over ten orders of magnitude! This raises the possibility that other Zeta Functions may also show similar behaviour. In this paper we study the distribution of zeros for L-functions of conductors 3 and 4. While we do not have zeros for these functions over such a wide range, we find that the distribution (characterised by the Hurst exponent of rescaled range analysis) is similar to that of the Riemann Zeta Functions. Thus, the conclusions of the earlier paper seem to be valid not just for the Riemann Zeta Functions, but also for other L-functions. The remarkable implications of a low Hurst exponent which were discussed in our earlier work for the Riemann zeta function seem to hold for the Dirichlet L-functions also."

O. Shanker, "Hurst exponent for spectra of complex networks" (preprint 06/06)

[abstract:] "In this paper we use the rescaled range analysis to study the spacings between the eigenvalues of the adjacency matrices of different types of complex networks. The distribution seems to be of the persistent fractional Brownian motion type. The spacings have a Hurst exponent varying from 0.5 to 0.9 for the networks studied. This range implies a positive correlation between successive increments in the sequence of eigenvalues. For Hurst exponents at the lower end, a change in the parameters could lead to negative correlations."

[The concluding section contains a comparison with a related analysis of the Riemann zeta function.]

A. Le Méhauté, A. El Kaabouchi, L. Nivanen and Qiuping A. Wang, "Fractional dynamics, tiling equilibrium states and Riemann's zeta function" (preprint 07/09)

[abstract:] "It is argued that the generalisation of the mechanical principles to other variables than localisation, velocity and momentum leads to the laws of generalized dynamics under the condition of continuous and derivable space time. However, when the fractality arises, the mechanics principles may no more be extended especially because the time and space singularity appears on the boundary and creates curvature. There is no more equilibrium state, but only a horizon which might play a same role as equilibrium but does not close the problem - especially the problem of the invariance of the energy - which requires two complementary factors: a first one related to the closure in the dimensional space, and the second to scan dissymmetry stemming from the default of tiling the space time. A new discrete time arises from fractality. It leads irreversible thermodynamic properties. Space and time singularities lead to the relation between the above mentioned problematic and the Riemann zeta functions as well as its zeros."

B. Holdom, "Correlations, scale invariance and the Riemann Hypothesis" (preprint 03/2009)

[abstract:] "Negative correlations in the distribution of prime numbers are found to display a scale invariance. There are similarities and differences when compared to the scale invariant correlations of fractional Brownian motion. We conjecture that a violation of the Riemann hypothesis is equivalent to a breakdown of the scale invariance."

D. Schumayer, B.P. van Zyl and D.A.W. Hutchinson, "Quantum mechanical potentials related to the prime numbers and Riemann zeros" (preprint 11/2008)

[Abstract:] "Prime numbers are the building blocks of our arithmetic, however, their distribution still poses fundamental questions. Bernhard Riemann showed that the distribution of primes could be given explicitly if one knew the distribution of the non-trivial zeros of the Riemann $\zeta(s)$ function. According to the Hilbert–Pólya conjecture there exists a Hermitean operator of which the eigenvalues coincide with the real part of the non-trivial zeros of $\zeta(s)$. This idea encourages physicists to examine the properties of such possible operators, and they have found interesting connections between the distribution of zeros and the distribution of energy eigenvalues of quantum systems. We apply the Mar{\v{c}}henko approach to construct potentials with energy eigenvalues equal to the prime numbers and to the zeros of the $\zeta(s)$ function. We demonstrate the multifractal nature of these potentials by measuring the R{\'e}nyi dimension of their graphs. Our results offer hope for further analytical progress."

K. Iguchi, "Generalized Wigner lattices as a Riemann solid: Fractals in Hurwitz zeta function" (submitted to Modern Physics Letters B)

"We study the ground state configuration and the excitation energy gaps in the strong coupling limit of the extended Hubbard model with a long-range interaction in one dimension. As proved by Hubbard and Pokrovsky and Uimin, the ground state configuration is quasiperiodic and as proved by Bak and Bruinsma, the excitation energy has a finite gap which forms a devil's stair as a function of the density of particles in the system. We show that the quasiperiodicity and the fractal nature of the excitation energy come from the nature of the long-range interaction that is related to the fractal nature of the Hurwitz Zeta function and the Riemann Zeta function."

S. Ares and M. Castro, "Hidden structure in the randomness of the prime number sequence?", Physica A 360 (2006) 285

[abstract:] "We report a rigorous theory to show the origin of the unexpected periodic behavior seen in the consecutive differences between prime numbers. We also check numerically our findings to ensure that they hold for finite sequences of primes, that would eventually appear in applications. Finally, our theory allows us to link with three different but important topics: the Hardy-Littlewood conjecture, the statistical mechanics of spin systems, and the celebrated Sierpinski fractal."

M. Shlesinger, "On the Riemann hypothesis: a fractal random walk approach", Physica A 138 (1986) 310-319

[abstract:] "In his investigation of the distribution of prime numbers Riemann, in 1859, introduced the zeta function with a complex argument. His analysis led him to hypothesize that all the complex zeros of the zeta function lie on a vertical line in the complex plane. The proof or disproof of this hypothesis has been a famous outstanding problem in mathematics. We are able to recast Riemann's Hypothesis into a probabilistic framework connected to the fractal behavior of a lattice random walk. Fractal random walks were introduced by P. Levy, and in the continuum are called Levy flights. For one particular lattice version of a Levy flight we show the connection to Weierstrass' continuous but nowhere differentiable function. For a different lattice version, using a Mellin transform analysis, we show how the zeroes of the zeta function become the singularities of a complex integrand which governs the behavior of a fractal random walk. The laws of probability place restrictions on the locations of the zeroes of the zeta function. No inconsistencies with probability theory are found if the Riemann Hypothesis is false."

I.I. Iliev, "Riemann zeta function and hydrogen spectrum", Electronic Journal of Theoretical Physics 10 (2013) 111–134

[abstract:] "Significant analytic and numerical evidence, as well as conjectures and ideas connect the Riemann zeta function with energy-related concepts. The present paper is devoted to further extension of this subject. The problem is analyzed from the point of view of geometry and physics as wavelengths of hydrogen spectrum are found to be in one-to-one correspondence with complex-valued positions. A Zeta Rule for the definition of the hydrogen spectrum is derived from well-known models and experimental evidence concerning the hydrogen atom. The Rydberg formula and Bohr's semiclassical quantization rule are modified. The real and the complex versions of the zeta function are developed on that basis. The real zeta is associated with a set of quantum harmonic oscillators with the help of relational and inversive geometric concepts. The zeta complex version is described to represent continuous rotation and parallel transport of this set within the plane. In both cases we derive the same wavelengths of hydrogen spectral series subject to certain requirements for quantization. The fractal structure of a specific set associated with $\zeta(s)$ is revealed to be represented by a unique box-counting dimension."

C. King, "Experimental observations on the Riemann hypothesis, and the Collatz conjecture" (preprint 05/2010)

[abstract:] "This paper seeks to explore whether the Riemann hypothesis falls into a class of putatively unprovable mathematical conjectures, which arise as a result of unpredictable irregularity. It also seeks to provide an experimental basis to discover some of the mathematical enigmas surrounding these conjectures, by providing Matlab and C programs which the reader can use to explore and better understand these systems."

[This includes an exploration of the Julia and Mandelbrot sets associated with the Riemann zeta function.]

M. Wolf, "Multifractality of prime numbers", Physica A 160 (1989) 24-42.

M. Wolf, "1/f noise in the distribution of prime numbers", Physica A 241 (1997), 493-499.

M. Wolf, "Random walk on the prime numbers", Physica A 250 (1998) 335.

Chung-Ming Ko, "Distribution of the units digit of primes", Chaos, Solitons and Fractals 13 (2002) 1295-1302

[abstract:] "A sequence is formed by the units digit of consecutive prime numbers. The sequence is not random. To visualize the non-randomness of the sequence, we utilize a method put forward by Hao et. al. [Chaos, Solitons and Fractals 11 (2000) 825]. A fractal-like structure is observed."

[Note to sceptics: The author certainly addresses the issue of base-representation!]

I. Antoniou and Z. Suchanecki, "Quantum systems with fractal spectra", Chaos, Solitons and Fractals 14, (2002) 799-807

[abstract:] "We study Hamiltonians with singular spectra of Cantor type with a constant ratio of dissection and show strict connections between the decay properties of the states in the singular subspace and the algebraic number theory. More specifically, we study the decay properties of free n-particle systems and the computability of decaying and non-decaying states in the singular continuous subspace."

M.L. Lapidus, "The sound of fractal strings and the Riemann Hypothesis" (preprint 05/2015)

"We give an overview of the intimate connections between natural direct and inverse spectral problems for fractal strings, on the one hand, and the Riemann zeta function and the Riemann hypothesis, on the other hand (in joint works of the author with Carl Pomerance and Helmut Maier, respectively). We also briefly discuss closely related developments, including the theory of (fractal) complex dimensions (by the author and many of his collaborators, including especially Machiel van Frankenhuijsen), quantized number theory and the spectral operator (jointly with Hafedh Herichi), and some other works of the author (and several of his collaborators)."

M. Lapidus, In Search of the Riemann Zeros (AMS, 2008)

[from publisher's description:] "In this book, the author proposes a new approach to understand and possibly solve the Riemann Hypothesis. His reformulation builds upon earlier (joint) work on complex fractal dimensions and the vibrations of fractal strings, combined with string theory and noncommutative geometry. Accordingly, it relies on the new notion of a fractal membrane or quantized fractal string, along with the modular flow on the associated modui space of fractal membranes. Conjecturally, under the action of the modular flow, the spacetime geometries become increasingly symmetric and crystal-like, hence, arithmetic. Correspondingly, the zeros of the associated zeta functions eventually condense onto the critical line, towards which they are attracted, thereby explaining why the Riemann Hypothesis must be true.

Written with a diverse audience in mind, this unique book is suitable for graduate students, experts and nonexperts alike, with an interest in number theory, analysis, dynamical systems, arithmetic, fractal or noncommutative geometry, and mathematical and mathematical or theoretical physics."

M.L. Lapidus and M. van Frankenhuysen, Fractal Geometry and Number Theory: Fractal Strings and Zeros of Zeta Functions (Birkhauser, 2000)

M.L. Lapidus and C. He, Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings and the Riemann-Zeta-Function (AMS, 1997)

"Studies the effect of nonpower-like irregularities of the geometry of the fractal boundary on the spectrum of fractal drums and especially of fractal strings. The authors use the notion of generalized Minkowski content, which is defined through some suitable gauge functions other than the power functions. By so doing, they obtain more precise estimates in the situation in which the power function is not the natural gauge function."

[abstract:] "We present an overview of a theory of complex dimensions of self-similar fractal strings, and compare this theory to the theory of varieties over a finite field from the geometric and the dynamical point of view. Then we combine the several strands to discuss a possible approach to establishing a cohomological interpretation of the complex dimensions."

[abstract:] "We investigate the spectral zeta function of a self-similar Sturm–Liouville operator associated with a fractal self-similar measure on the half-line and C. Sabot's work connecting the spectrum of this operator with the iteration of a rational map of several complex variables. We obtain a factorization of the spectral zeta function expressed in terms of the zeta function associated with the dynamics of the corresponding renormalization map, viewed as a rational function on the complex projective plane. The result generalizes to several complex variables and to the case of fractal Sturm–Liouville operators a factorization formula obtained by the second author for the spectral zeta function of a fractal string and later extended to the Sierpinski gasket and some other decimable fractals by A. Teplyaev. As a corollary, in the very special case when the underlying self-similar measure is Lebesgue measure on $[0,1]$, we obtain a representation of the Riemann zeta function in terms of the dynamics of a certain polynomial in the complex projective plane, thereby extending to several variables an analogous result by A. Teplyaev. The above fractal Hamiltonians and their spectra are relevant to the study of diffusions on fractals and to aspects of condensed matters physics, including to the key notion of density of states."

A. Teplyaev, "Spectral zeta functions of fractals and the complex dynamics of polynomials" (preprint 05/05)

[abstract:] "We obtain formulas for the spectral zeta function of the Laplacian on symmetric finitely ramified fractals, such as the Sierpinski gasket, and a fractal Laplacian on the interval. These formulas contain a new type of zeta function associated with a polynomial. It is proved that this zeta function has a meromorphic continuation to a half plain with poles contained in an arithmetic progression. It is shown as an example that the Riemann zeta function is the zeta functions of a quadratic polynomial, which is associated with the Laplacian on an interval. The spectral zeta function of the Sierpinski gasket is a product of the zeta function of a polynomial and a geometric part; the poles of the former are canceled by the zeros of the latter. A similar product structure was discovered by M.L. Lapidus for self-similar fractal strings."

Hong Deng and Gongwen Peng, "Eigenvalues for high-order elliptic operators in a fractal string", Fractals 7 No. 3 (1999) 267-275

[This involves the Weyl-Berry conjecture and the Riemann zeta function.]

M. Levitin and D. Vassiliev, "Spectral asymptotics, renewal theorem, and the Berry conjecture for a class of fractals", Proceedings of the London Mathematical Society (3) 72 (1996) 188-214.

C. Castro, "On two strategies towards the Riemann Hypothesis: Fractal Supersymmetric QM and a trace formula" (preprint 06/06)

[abstract:] "The Riemann Hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form $s_n =1/2+i lambda_n$. An improvement of our previous construction to prove the RH is presented by implementing the Hilbert-Pólya proposal and furnishing the Fractal Supersymmetric Quantum Mechanical (SUSY-QM) model whose spectrum reproduces the imaginary parts of the zeta zeros. We model the fractal fluctuations of the smooth Wu-Sprung potential (that capture the average level density of zeros) by recurring to a weighted superposition of Weierstrass functions $W(x,p,D)$ and where the summation has to be performed over all primes $p$ in order to recapture the connection between the distribution of zeta zeros and prime numbers. We proceed next with the construction of a smooth version of the fractal QM wave equation by writing an ordinary Schrödinger equation whose fluctuating potential (relative to the smooth Wu-Sprung potential) has the same functional form as the fluctuating part of the level density of zeros. The second approach to prove the RH relies on the existence of a continuous family of scaling-like operators involving the Gauss-Jacobi theta series. An explicit trace formula related to a superposition of eigenfunctions of these scaling-like operators is defined. If the trace relation is satisfied this could be another test of the Riemann Hypothesis."

C. Castro and J. Mahecha, "A fractal SUSY-QM model and the Riemann hypothesis" (preprint 06/03)

[abstract:] "The Riemann hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form $s = 1/2 + i\lambda_n$. Hilbert-Polya argued that if a Hermitian operator exists whose eigenvalues are the imaginary parts of the zeta zeros, $\lambda_n$, then the RH is true. In this paper a fractal supersymmetric quantum mechanical (SUSY-QM) model is proposed to prove the RH. It is based on a quantum inverse scattering method related to a fractal potential given by a Weierstrass function (continuous but nowhere differentiable) that is present in the fractal analog of the CBC (Comtet, Bandrauk, Campbell) formula in SUSY QM. It requires using suitable fractal derivatives and integrals of irrational order whose parameter $\beta$ is one-half the fractal dimension of the Weierstrass function. An ordinary SUSY-QM oscillator is constructed whose eigenvalues are of the form $\lambda_n = n\pi$, and which coincide with the imaginary parts of the zeros of the funciton sin(iz). This sine function obeys a trivial analog of the RH. A review of our earlier proof of the RH based on a SUSY QM model whose potential is related ot the Gauss-Jacobi theta series is also included. The spectrum is given by s(1 - s) which is real in the critical line (location of the nontrivial zeros) and in the real axis (location of the trivial zeros)."

C. Castro, A. Granik, and J. Mahecha, "On SUSY-QM, fractal strings and steps towards a proof of the Riemann hypothesis" (preprint 07/01)

(abstract) "The steps towards a proof of Riemann's conjecture using spectral analysis are rigorously provided. We prove that the only zeroes of the Rieamnn zeta-function are of the form s = 1/2 + i lambdan. A supersymmetric quantum mechanical model is proposed as an alternative way to prove the Riemann conjecture, inspired in the Hilbert-Pólya proposal; it uses an inverse scattering approach associated with a system of p-adic harmonic oscillators. An interpretation of the Riemann's fundamental relation Z(s) = Z(1 - s) as a duality relation, from one fractal string L to another dual fractal string L' is proposed."

C. Castro, "Fractal strings as the basis of Cantorian-Fractal spacetime and the fine structure constant" (preprint 03/02)

[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."

C. Castro and J. Mahecha, "Fractal supersymmetric quantum mechanics, geometric probability and the Riemann Hypothesis", International Journal of Geometric Methods in Modern Physics 1 no. 6 (2004) 751-793

[abstract:] "The Riemann Hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form $s = 1/2 + i\lambda_{n}$. Earlier work on the RH based on Supersymmetric QM, whose potential was related to the Gauss-Jacobi theta series, allows to provide the proper framework to construct the well defined algorithm to compute the probability to find a zero (an infinity of zeros) in the critical line. Geometric Probability Theory furnishes the answer to the very difficult question whether the probability that the RH is true is indeed equal to unity or not. To test the validity of this Geometric Probabilistic framework to compute the probability if the RH is true, we apply it directly to the hyperbolic sine function sinh(s) case which obeys a trivial analog of the RH. Its zeros are equally spaced in the imaginary axis $s_n = 0 + in\pi$. The Geometric Probability to find a zero (and an infinity of zeros) in the imaginary axis is exactly unity. We proceed with a fractal supersymmetric quantum mechanical (SUSY-QM) model to implement the Hilbert-Pólya proposal to prove the RH by postulating a Hermitian operator that reproduces all the $\lambda_n$'s for its spectrum. Quantum inverse scattering methods related to a fractal potential given by a Weierstrass function (continuous but nowhere differentiable) are applied to the analog of the fractal analog of the CBC (Comtet-Bandrauk-Campbell) formula in SUSY QM. It requires using suitable fractal derivatives and integrals of irrational order whose parameter $\beta$ is one-half the fractal dimension (D = 1.5) of the Weierstrass function. An ordinary SUSY-QM oscillator is also constructed whose eigenvalues are of the form $\lambda_n = n\pi$ and which coincide which the imaginary parts of the zeros of the function sinh(s). Finally, we discuss the relationship to the theory of 1/f noise."

P.B. Slater, "A numerical examination of the Castro-Mahecha supersymmetric model of the Riemann zeros" (preprint 11/05)

[abstract:] "The unknown parameters of the recently-proposed (Int J. Geom. Meth. Mod. Phys. 1, 751 [2004]) Castro-Mahecha model of the imaginary parts (lambda_{j}) of the nontrivial Riemann zeros are the phases (alpha_{k}) and the frequency parameter (gamma) of the Weierstrass function of fractal dimension D=3/2 and the turning points (x_{j}) of the supersymmetric potential-squared Phi^2(x) - which incorporates the smooth Wu-Sprung potential (Phys. Rev. E 48, 2595 [1993]), giving the average level density of the Riemann zeros. We conduct numerical investigations to estimate/determine these parameters - as well as a parameter (sigma) we introduce to scale the fractal contribution. Our primary analyses involve two sets of coupled equations: one set being of the form Phi^{2}(x_{j}) = lambda_{j}, and the other set corresponding to the fractal extension - according to an ansatz of Castro and Mahecha - of the Comtet-Bandrauk-Campbell (CBC) quasi-classical quantization conditions for good supersymmetry. Our analyses suggest the possibility strongly that gamma converges to its theoretical lower bound of 1, and the possibility that all the phases (alpha_{k}) should be set to zero."

P. Slater, "Fractal fits to Riemann zeros" (preprint 05/2006)

[abstract:] "Wu and Sprung (1993) reproduced the first 500 nontrivial Riemann zeros, using a one-dimensional local potential model. They concluded - and similarly van Zyl and Hutchinson (2003) - that the potential possesses a \em{fractal} structure of dimension $d = 3/2$. We model the nonsmooth fluctuating part of the potential by the alternating-sign sine series fractal of Berry and Lewis $A(x,\gamma)$. Setting $d=3/2$, we estimate the frequency parameter $(\gamma)$, plus an overall scaling parameter $(\sigma)$ we introduce. We search for that pair of parameters $(\gamma,\sigma)$ which \em{minimizes} the least-square fit $S_n(\gamma,\sigma)$ of the lowest $n$ eigenvalues - obtained by solving the one-dimensional stationary (non-fractal) Schrödinger equation with the trial potential (smooth \em{plus} nonsmooth parts) - to the lowest $n$ Riemann zeros for $n = 25$. For the additional case we study, $n = 50$, we simply set $\sigma = 1$. The fits obtained are compared to those gotten by using just the \em{smooth} part of the Wu-Sprung potential \em{without} any fractal supplementation. Some limited improvement - 5.7261 \em{vs.} 6.39207 ($n = 25$) and 11.2672 \em{vs.} 11.7002 ($n = 50$) - is found in our (non-optimized, computationally-bound) search procedures. The improvements are relatively strong in the vicinities of $\gamma = 3$ and (its \em{square}) 9. Further, we extend the Wu-Sprung semiclassical framework to include \em{higher-order} corrections from the Riemann-von Mangoldt formula (beyond the leading, dominant term) into the smooth potential."

P. Slater, "Extended fractal fits to Riemann zeros" (preprint 05/2007)

[abstract:] "We extend to the first 300 Riemann zeros, the form of analysis reported by us in arXiv:math-ph/0606005, in which the largest study had involved the first 75 zeros. Again, we model the nonsmooth fluctuating part of the Wu-Sprung potential, which reproduces the Riemann zeros, by the alternating-sign sine series fractal of Berry and Lewis A(x,g). Setting the fractal dimension equal to 3/2. we estimate the frequency parameter (g), plus an overall scaling parameter (s) introduced. We search for that pair of parameters (g,s) which minimizes the least-squares fit of the lowest 300 eigenvalues -- obtained by solving the one-dimensional stationary (non-fractal) Schrodinger equation with the trial potential (smooth plus nonsmooth parts) -- to the first 300 Riemann zeros. We randomly sample values within the rectangle 0 < s < 3, 0 < g < 25. The fits obtained are compared to those gotten by using simply the smooth part of the Wu-Sprung potential without any fractal supplementation. Some limited improvement is again found. There are two (primary and secondary) quite distinct subdomains, in which the values giving improvements in fit are concentrated."

A.M. Selvam, "Universal Characteristics of Fractal Fluctuations in Prime Number Distribution" (preprint 11/2008)

[Abstract:] "The frequency of occurrence of prime numbers at unit number spacing intervals exhibits selfsimilar fractal fluctuations concomitant with inverse power law form for power spectrum generic to dynamical systems in nature such as fluid flows, stock market fluctuations, population dynamics, etc. The physics of long-range correlations exhibited by fractals is not yet identified. A recently developed general systems theory visualises the eddy continuum underlying fractals to result from the growth of large eddies as the integrated mean of enclosed small scale eddies, thereby generating a hierarchy of eddy circulations, or an inter-connected network with associated long-range correlations. The model predictions are as follows: (i) The probability distribution and power spectrum of fractals follow the same inverse power law which is a function of the golden mean. The predicted inverse power law distribution is very close to the statistical normal distribution for fluctuations within two standard deviations from the mean of the distribution. (ii) Fractals signify quantumlike chaos since variance spectrum represents probability density distribution, a characteristic of quantum systems such as electron or photon. (ii) Fractal fluctuations of frequency distribution of prime numbers signify spontaneous organisation of underlying continuum number field into the ordered pattern of the quasiperiodic Penrose tiling pattern. The model predictions are in agreement with the probability distributions and power spectra for different sets of frequency of occurrence of prime numbers at unit number interval for successive 1000 numbers. Prime numbers in the first 10 million numbers were used for the study."

A.M. Selvam, "Cantorian fractal patterns, quantum-like chaos and prime numbers in atmospheric flows" (preprint 10/1998)

"The quantum-like chaos in atmospheric flows can be quantified in terms of the following mathematical functions/concepts: (1) The fractal structure of the flow pattern is resolved into an overall logarithmic spiral trajectory with the quasiperiodic Penrose tiling pattern for the internal structure and is equivalent to a hierarchy of vortices. The incorporation of Fibonacci mathematical series, representative of ramified bifurcations, indicates ordered growth of fractal patterns. (2) The steady state emergence of progressively larger fractal structures incorporates unique primary perturbation domains of progressively increasing number equal to z/ln z where z, the length step growth stage is equal to the length scale ratio of large eddy to turbulent eddy. In number theory, z/ln z gives the number of primes less than z. The model also predicts that z/ln z represents the normalised cumulative variance spectrum of the eddies and which follows statistical normal distribution. The important result of the study is that the prime number spectrum is the same as the eddy energy spectrum for quantum-like chaos in atmospheric flows."

S.A. Oprisal, The classical gases in the Tsallis statistics using the generalized Riemann zeta functions", J. Phys. I France 7 (July 1997) 853-862.

(Abstract) "In the last few years an increasing interest has been paid to fractal inspired statistics. Our aim is to describe some new insight obtained using Tsallis statistics. In the framework of the generalized statistics we described some properties of the Maxwell-Boltzmann gases. The behavior of the occupation numbers with respect to the temperature indicates similarities with Fermi gases. Using the Nernst theorem we also determine the fractal index of statistics."

The following all relate (number theoretical) Farey sequences to fractality in some way:

"The concept of fractal index is introduced in connection with the idea of universal class $h$ of particles or quasiparticles, termed fractons, which obey fractal statistics. We show the relation between fractons and conformal field theory(CFT)-quasiparticles taking into account the central charge $c[\nu]$ and the particle-hole duality $\nu\longleftrightarrow\frac{1}{\nu}$, for integer-value $\nu$ of the statistical parameter. The Hausdorff dimension $h$ which labelled the universal classes of particles and the conformal anomaly are therefore related. We also establish a connection between Rogers dilogarithm function, Farey series of rational numbers and the Hausdorff dimension."

"We prove that the Farey sequences can be express into equivalence classes labeled by a fractal parameter which looks like a Hausdorff dimension h defined within the interval 1 < h < 2. The classes h satisfy the same properties of the Farey series and for each value of h there exists an algebraic equation."

"We consider Farey series of rational numbers in terms of fractal sets labeled by the Hausdorff dimension with values defined in the interval 1 < h < 2 and associated with fractal curves. Our results come from the observation that the fractional quantum Hall effect-FQHE occurs in pairs of dual topological quantum numbers, the filling factors. These quantum numbers obey some properties of the Farey series and so we obtain that the universality classes of the quantum Hall transitions are classified in terms of h. The connection between Number Theory and Physics appears naturally in this context."

W. da Cruz, "A quantum-geometrical description of the statistical laws of nature", talk given at the 2nd International Londrina Winter School: Mathematical Methods in Physics, August 26-30 (2002),

[abstract:] "We consider the fractal characteristic of the quantum mechanical paths and we obtain for any universal class of fractons labelled by the Hausdorff dimension defined within the interval 1 < h < 2, a fractal distribution function associated with a fractal von Newmann entropy. Fractons are charge-flux systems defined in two-dimensional multiply connected space and they carry rational or irrational values of spin.

This formulation can be considered in the context of the fractional quantum Hall effect-FQHE, where we discovered that the quantization of the Hall resistance occurs in pairs of dual topological quantum numbers, the filling factors. In this way, these quantum numbers get their topological character from the Hausdorff dimension associated with the fractal quantum path of such particles termed fractons. On the other hand, the universality class of the quantum Hall transitions can be classified in terms of h. Another consequence of our approach, which is supported by symmetry principles, is the prediction of the FQHE. The connection between Physics and Number Theory appears naturally in this context."

W. da Cruz, "Fractal sets of dual topological quantum numbers" (preprint, 06/03)

[abstract:] "The universality classes of the quantum Hall transitions are considered in terms of fractal sets of dual topological quantum numbers filling factors, labelled by a fractal or Hausdorff dimension defined into the interval 1 < h < 2 and associated with fractal curves. We show that our approach to the fractional quantum Hall effect-FQHE is free of any empirical formula and this characteristic appears as a crucial insight for our understanding of the FQHE. According to our formulation, the FQHE gets a fractal structure from the connection between the filling factors and the Hausdoff dimension of the quantum paths of particles termed fractons which obey a fractal distribution function associated with a fractal von Neumann entropy. This way, the quantum Hall transitions satisfy some properties related to the Farey sequences of rational numbers and so our theoretical description of the FQHE establishes a connection between physics, fractal geometry and number theory. The FQHE as a convenient physical system for a possible prove of the Riemann hypothesis is suggested."

M. Piacquadio Losada and E. Cestaratto, "Multifractal spectrum and thermodynamic formalism of the Farey tree"

"The task of comparing the Hausdorff spectrum, the computational spectrum, and the Legendre spectrum of a fractal set endowed with a probability measure, was tackled by several authors - Cawley and Mauldin, Riedi and Mandelbrot, among others. For self-similar measures all three spectra coincide. We compare these spectra for the hyperbolic measure (inducing the Farey Tree partition), fundamentally different from the self-similar one."

S. Grynberg and M. Piacquadio, "Self-similarity of Farey staircases" (preprint 06/03)

[abstract:] "We study Cantor Staircases in physics that have the Farey-Brocot arrangement for the Q/P rational heights of stability intervals I(Q/P), and such that the length of I(Q/P) is a convex function of 1/P. Circle map staircases and the magnetization function fall in this category. We show that the fractal sets $\Omega$ underlying these staircases are connected with key sets in Number Theory via their $(\alpha, f(\alpha))$ multifractal decomposition spectra. It follows that such sets $\Omega$ are self similar when the usual (Euclidean) measure is replaced by the hyperbolic measure induced by the Farey-Brocot partition."

"In order to study Cantor staircases in physics – forced pendulums, magnetization, etc. – showing the Farey–Brocot arrangement for intervals I(Q/P), a natural connection with Number Theory, precisely due to the ubiquitous presence of the Farey-Brocot partition. But when closely examining the behaviour of These staircases, we were forced to considerably refine the $J_{\beta}$ nested classes into the $G_{\beta}$ disjoint ones.

We are saying that problems in empirical Physics produced a refinement of key tools in Number theory

The properties of these $G_{\beta} allowed us to extract theoretical and practical information about the multifractal spectrum of such cantordusts$\Omega\$ underlying Cantor staircases in physics, and about the nature of the self-similarity of said staircases."

J. Fiala, P. Kleban, A. Özlük, "The phase transition in statistical models defined on Farey fractions" (accepted for publication, J. Stat. Physics)

[abstract:] "We consider several statistical models defined on the Farey fractions. Two of these models may be regarded as "spin chains", with long-range interactions, while another arises in the study of multifractals associated with chaotic maps exhibiting intermittency. We prove that these models all have the same free energy. Their thermodynamic behavior is determined by the spectrum of the transfer operator (Ruelle-Perron-Frobenius operator), which is defined using the maps (presentation functions) generating the Farey "tree". The spectrum of this operator was completely determined by Prellberg. It follows that these models have a second-order phase transition with a specific heat divergence of the form [t (ln t)^2]^(-1). The spin chain models are also rigorously known to have a discontinuity in the magnetization at the phase transition."

B. Barbanis, H. Varvoglis and Ch. L. Vozikis, "Imperfect fractal repellers and irregular families of periodic orbits in a 3-D model potential"

[abstract:] "A model, plane symmetric, 3-D potential, which preserves some features of galactic problems,is used in order to examine the phase space structure through the study of the properties of orbits crossing perpendicularly the plane of symmetry. It is found that the lines formed by periodic orbits, belonging to Farey sequences, are not smooth neither continuous. Instead they are deformed and broken in regions characterised by high Lyapunov Characteristic Numbers (LCN's). It is suggested that these lines are an incomplete form of a fractal repeller, as discussed by Gaspard and Baras (1995), and are thus closely associated to the 'quasi-barriers' discussed by Varvoglis et al. (1997). There are numerical indications that the contour lines of constant LCN's possess fractal properties. Finally it is shown numerically that some of the periodic orbits -members of the lines- belong to true irregular families. It is argued that the fractal properties of the phase space should affect the transport of trajectories in phase or action space and,therefore, play a certain role in the chaotic motion of stars in more realistic galactic potentials."

N. Cornish and J. Levin, "The mixmaster universe: A chaotic Farey tale"

[abstract:]"When gravitational fields are at their strongest, the evolution of spacetime is thought to be highly erratic. Over the past decade debate has raged over whether this evolution can be classified as chaotic. The debate has centered on the homogeneous but anisotropic mixmaster universe. A definite resolution has been lacking as the techniques used to study the mixmaster dynamics yield observer dependent answers. Here we resolve the conflict by using observer independent, fractal methods. We prove the mixmaster universe is chaotic by exposing the fractal strange repellor that characterizes the dynamics. The repellor is laid bare in both the 6-dimensional minisuperspace of the full Einstein equations, and in a 2-dimensional discretisation of the dynamics. The chaos is encoded in a special set of numbers that form the irrational Farey tree. We quantify the chaos by calculating the strange repellor's Lyapunov dimension, topological entropy and multifractal dimensions. As all of these quantities are coordinate, or gauge independent, there is no longer any ambiguity-the mixmaster universe is indeed chaotic."

The Farey Room (L. Vepstas's Farey map graphics)

B. Devaney, "The Mandelbrot Set and the Farey Tree", Amer. Math. Monthly 106 (1999) 289-302

[abstract:] "Explicit formulas are obtained for a family of continuous mappings of p-adic numbers Qp and solenoids Tp into the complex plane C and the space R3, respectively. Accordingly, this family includes the mappings for which the Cantor set and the Sierpinski triangle are images of the unit balls in Q2 and Q3 . In each of the families, the subset of the embeddings is found. For these embeddings, the Hausdorff dimensions are calculated and it is shown that the fractal measure on the image of Qp coincides with the Haar measure on Qp. It is proved that under certain conditions, the image of the p-adic solenoid is an invariant set of fractional dimension for a dynamic system. Computer drawings of some fractal images are presented."

[abstract:] "Fractal measures of images of continuous maps from the set of p-adic numbers Qp into complex plane C are analyzed. Examples of 'anomalous' fractals, i.e. the sets where the D-dimensional Hausdorff measures (HM) are trivial, i.e. either zero, or sigma-infinite (D is the Hausdorff dimension (HD) of this set) are presented. Using the Caratheodory construction, the generalized scale-covariant HM (GHM) being non-trivial on such fractals are constructed. In particular, we present an example of 0-fractal, the continuum with HD=0 and nontrivial GHM invariant w.r.t. the group of all diffeomorphisms C. For conformal transformations of domains in Rn, the formula for the change of variables for GHM is obtained. The family of continuous maps Qp in C continuously dependent on "complex dimension" d in C is obtained. This family is such that: 1) if d = 2(1), then the image of b>Qp is C (real axis in C.); 2) the fractal measures coincide with the images of the Haar measure in Qp, and at d = 2(1) they also coincide with the flat (linear) Lebesgue measure; 3) integrals of entire functions over the fractal measures of images for any compact set in Qp are holomorphic in d, similarly to the dimensional regularization method in QFT."

S.R. Dahmen, S.D.Prado and T.Stuermer-Daitx, "Similarity in the Statistics of Prime Numbers", Physica A 296 (2001) 523-528

[abstract:] "We present numerical evidence for regularities in the distribution of gaps between primes when these are divided into congruence families (in Dirichlet's classification). The histograms for the distribution of gaps of families are scale invariant."

an idea to be explored - speculative notes on possible phenomenon relating number theory, fractal geometry, Notalle's scale invariance, Renormalisation Group, etc.

graphically-inspired speculations about possible fractality in the distribution of primes

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