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)

popularly accessible blog piece on the remarkable discoveries described in the above two papers

[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)."

S.C. Woon, "Fractals
of the Julia and Mandelbrot sets of the Riemann zeta function"

"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..."

Woon's
images of the Julia and Mandelbrot sets associated with the 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

[abstract:] "We analyze the reconstruction by Wu and Sprung [*Phys. Rev.
E* **48**, 2595 (1993)] of a fractal one-dimensional potential, the quantum spectrum of which reproduces the first
500 nontrivial zeros of the Riemann $\zeta$ function. Our construction is based on a spectrum with Gaussian unitary ensemble
statistics as far as the nearest-neighbor spacing distribution is concerned. Our results show that a reliable estimate of
the fractal dimension of the potential necessitates a very large number of levels."

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."

M.L. Lapidus and M. van Frankenhuijsen, "Fractality, self-similarity and complex
dimensions", to appear in *Proceedings of Symposia of Pure Mathematics*, title:
"Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot"

[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."

N. Lal and Michel L. Lapidus, "Higher-dimensional complex dynamics and spectral zeta functions of fractal differential Sturm–Liouville operators" (preprint 02/2012)

[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."

M.L. Lapidus, "An overview of complex fractal dimensions: From fractal strings to fractal drums, and back" (preprint 03/2018)

[abstract:] "Our main goal in this long survey article is to provide an overview of the theory of complex fractal dimensions and of the associated geometric or fractal zeta functions, first in the case of fractal strings (one-dimensional drums with fractal boundary), and then in the higher-dimensional case of relative fractal drums and, in particular, of arbitrary bounded subsets of Euclidean space of RN, for any integer $N=1$. Special attention is paid to discussing a variety of examples illustrating the general theory rather than to providing complete statements of the results and their proofs. Finally, in an epilogue, entitled "From quantized number theory to fractal cohomology", we briefly survey aspects of related work (motivated in part by the theory of complex fractal dimensions) of the author with H. Herichi (in the real case), and with T. Cobler (in the complex case), respectively, as well as in the latter part of a book in preparation by the author."

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
lambda*_{n}. 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 *p*inary (*p*inary, *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."

K.V. Shajesh, I. Brevik, I. Cavero-Peláez and P. Parashar, "Self-similar plates: Casimir energies" (preprint 07/2016)

[abstract:] "We construct various self-similar configurations using parallel $\delta$-function plates and show that it is possible to evaluate the Casimir interaction energy of these configurations using the idea of self-similarity alone. We restrict our analysis to interactions mediated by a scalar field, but the extension to electromagnetic field is immediate. Our work unveils an easy and powerful method that can be easily employed to calculate the Casimir energies of a class of self-similar configurations. As a highlight, in an example, we determine the Casimir interaction energy of a stack of parallel plates constructed by positioning $\delta$-function plates at the points constituting the Cantor set, a prototype of a fractal. This, to our knowledge, is the first time that the Casimir energy of a fractal configuration has been reported. Remarkably, the Casimir energy of some of the configurations we consider turn out to be positive, and a few even have zero Casimir energy. For the case of positive Casimir energy that is monotonically decreasing as the stacking parameter increases the interpretation is that the pressure of vacuum tends to inflate the infinite stack of plates. We further support our results, derived using the idea of self-similarity alone, by rederiving them using the Green's function formalism. These expositions gives us insight into the **connections between the regularization methods used in quantum field theories and regularized sums of divergent series in number theory**."

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

W. da Cruz,
"Fractal statistics, fractal
index, and fractons"

"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."

W. da Cruz, "A note of Farey sequences and 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."

W. da Cruz, "The
Hausdorff dimension of fractal sets and fractional quantum Hall effect"

"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."

This article involves the Ising model
and concludes:

"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."

*MathWorld*
notes on the Devil's Staircase

physics preprints involving the Devil's staircase

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

D. Chistyakov, "Fractal
geometry for images Of continuous map Of *p*-adic numbers
and *p*-adic solenoids into Euclidean spaces"

[abstract:] "Explicit formulas are obtained for a family of continuous mappings of
*p*-adic numbers **Q**_{p} and solenoids **T**_{p}
into the complex plane **C** and the
space **R**^{3}, respectively. Accordingly, this family includes the mappings for
which the Cantor set and the Sierpinski triangle are images of the unit balls in **Q**_{2}
and **Q**_{3} . 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 **Q**_{p} coincides with the Haar measure on **Q**_{p}.
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."

D. Chistyakov, "Fractal
measures, *p*-adic numbers and continuous transition between dimensions"

[abstract:] "Fractal measures of images of continuous maps from the set of *p*-adic
numbers **Q**_{p} 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 **R**^{n}, the formula for the change of variables
for GHM is obtained. The family of continuous maps **Q**_{p} 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>Q_{p} is **C**
(real axis in **C**.); 2) the fractal measures coincide with the images of the
Haar measure in **Q**_{p}, 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 **Q**_{p} 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

**
**