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1/f noise, signal processing and number theory

M. Planat,
"1/f frequency noise in a communication receiver and the Riemann
Hypothesis", from *Noise, Oscillators and Algebraic Randomness:
From Noise in Communication Systems to Number Theory*
(conference proceedings - Chapelle des Bois, France, April 5-10, 1999),
M. Planat, ed. (Springer-Verlag, 2000)

M. Planat,
"1/*f* noise, the measurement of time and number theory",
*Fluctuation and Noise Letters* **0** (2001).

The following is a brief review paper:

M. Planat, "The impact of prime number theory
on frequency metrology", to appear in Proceedings of 6th Symposium on
Frequency Standards and Metrology, St. Andrews, Scotland, 9-14 September
2001 (World Scientific)

M. Planat and E. Henry, "The arithmetic of 1/*f* noise in a
phase locked loop", *Applied Physics Letters* (April 2002)

"A phase lock loop [is] characterized...leading to a possible
relationship between 1/*f* noise close to baseband and
arithmetical sequences of prime number theory."

M. Planat and C. Eckert, "On the frequency and amplitude
spectrum and fluctuations at the output of a communications receiver"

The following is to be published in the proceedings of the recent
conference "The
Nature of Time: Geometry, Physics & Perception" (Tatranská Lomnica,
Slovak Republic, May 2002):

M. Planat, "Time measurements, 1/*f* noise of the
oscillators and algebraic numbers"

"Many complex systems from physics, biology, society...exhibit a
1/*f* power spectrum in their time variability so that it is tempting
to regard 1/*f* noise as a unifying principle in the study of time.
The principle may be useful in reconciling two opposite views of time,
the cyclic and the linear one, the philosophic view of eternity as opposed
to that of time and death. The temporal experience of such complex
systems may only be obtained thanks to clocks which are continuously or
occasionally slaved. Here time is discrete with a unit equal to the
averaging time of each experience. Its structure is reflected into the
measured arithmetical sequence. They are resets in the frequencies and
couplings of the clocks, like in any human made calendar. The statistics
of the resets shows about constant variability whatever the averaging time:
this is characteristic of the flicker (1/*f*) noise. In a number
of electronic experiments we related the variability in the oscillators
to number theory, and time to prime numbers. In such a context, time
(and 1/*f* noise) has to do with Riemann's hypothesis that all zeros
of the Riemann zeta function are located on the critical line, a mathematical
conjecture still open after 150 years."

The following extends the work in some very interesting directions:

M. Planat, H. Rosu and S. Perrine, "Ramanujan sums for signal processing
of low frequency noise" (submitted to *Physical Review E*

[Abstract:] "An aperiodic (low frequency) spectrum may originate from
the error term in the mean value of an arithmetical function such as
Möbius function or Mangoldt function, which are coding sequences for
prime numbers. In the discrete Fourier transform the analyzing wave is
periodic and not well suited to represent the low frequency regime. In
place we introduce a new signal processing tool based on the Ramanujan
sum *c*_{q}(*n*), well adapted to the analysis of
arithmetical sequences with many resonances *p*/*q*. The sums
are quasi-periodic versus the time *n* of the resonance and aperiodic
versus the order *q* of the resonance. New results arise from the
use of this Ramanujan-Fourier transform (RFT) in the context of arithmetical
and experimental signals."

The final paragraph:

"The other challenge behind Ramanujan sums relates to prime number
theory. We just focused our interest to the relation between 1/*f*
noise in communication circuits and the still unproved Riemann hypothesis
[13]. The mean value of the modified Mangoldt function *b*(*n*),
introduced in (29), links Riemann zeros to the 1/*f*^{2\alpha}
noise and to the Möbius function. This should follow from generic
properties of the modular group *SL*(2,*Z*), the group of 2
by 2 matrices of determinant 1 with integer coefficients [15], and to
the statistical physics of Farey spin chains [16]. See also the link to
the theory of Cantorian fractal spacetime [17]."

[13] M. Planat, "Noise, oscillators and algebraic randomness", *Lecture
Notes in Physics* **550** (Springer, Berlin, 2000)

[15] S. Perrine, *La theorie de Markoff et ses developpements*,
ed. T. Ashpool (Chantilly, 2000)

[16] A. Knauf, *Communications in Mathematical Physics* **3**
(1998) 703

[17] C. Castro and J. Mahecha, arXiv: hep-th/0009014v3 (8 Apr. 2001)

The paper was inspired by

H. Gopalkrishna Gadiyar and R. Padma,
"Ramanujan-Fourier series, the Wiener-Khintchine formula and the distribution
of prime pairs"

The ideas are developed further here:

M. Planat, "Modular
functions and Ramanujan sums for the analysis of 1/*f* noise
in electronic circuits", to appear in the proceedings of a WSEAS
International Conference on Mathematical Methods & Computational
Techniques in Electrical Engineering, Athens, Dec. 29-31 (2002)

[abstract:] "A number theoretical model of 1/*f* noise found in phase
locked loops is developed. Oscillators from the input of the non linear and low
pass filtering stage are shown to lock their frequencies from continued fraction
expansions of their frequency ratio, and to lock their phases from modular functions
found in the hyperbolic geometry of the half plane. A cornerstone of the analysis
is the Ramanujan sums expansion of arithmetical functions found in prime number
theory, and their link to Riemann hypothesis."

M. Planat, H. Rosu, S. Perrine, "Arithmetical
chaology and the signatures of 1/*f* noise" (presented at International Conference on Theoretical Physics - Paris, UNESCO, 22-27
July 2002)

M. Planat, "Class numbers
in the imaginary quadratic field and the 1/f noise of an electron gas" (preprint 07/03)

[abstract:] "Partition functions $Z(x)$ of statistical mechanics are generally approximated
by integrals. The approximation fails in small cavities or at very low temperature, when the
ratio $x$ between the energy quantum and thermal energy is larger or equal to unity. In
addition, the exact calculation, which is based on number theoretical concepts, shows excess
low frequency noise in thermodynamical quantities, that the continuous approximation fails to
predict. It is first shown that Riemann zeta function is essentially the Mellin transform of
the partition function $Z(x)$ of the non degenerate (one dimensional) perfect gas. Inverting
the transform leads to the conventional perfect gas law. The degeneracy has two aspects. One
is related to the wave nature of particles: this is accounted for from quantum statistics,
when the de Broglie wavelength exceeds the mean distance between particles. We emphasize here
the second aspect which is related to the degeneracy of energy levels. It is given by the
number of solutions $r_3(p)$ of the three squares diophantine equation, a highly discontinuous
arithmetical function. In the conventional approach the density of states is proportional to
the square root of energy, that is $r_3(p)\simeq 2 \pi p^{1/2}$. We found that the exact
density of states relates to the class number in the quadratic field $Q(\sqrt{-p})$. One finds
$1/f$ noise around the mean value."

M. Planat, "Quantum 1/*f* noise
in equilibrium: from Planck to Ramanujan", *Physica* A
**318** (2003) 371

[abstract:] "We describe a new model of massless thermal bosons
which predicts an hyperbolic fluctuation spectrum at low frequencies.
It is found that the partition function per mode is the Euler
generating function for unrestricted partitions *p*(*n*).
Thermodynamical quantities carry a strong arithmetical structure: they
are given by series with Fourier coefficients equal to summatory
functions $\sigma_k(n)$ of the power of divisors, with *k* = -1
for the free energy, *k* = 0 for the number of particles and
*k* = 1 for the internal energy. Low frequency contributions are
calculated using Mellin transform methods. In particular the internal
energy per mode diverges as $\frac{\tilde{E}}{kT}=\frac{\pi^2}{6 x}$
with $x=\frac{h \nu}{kT}$ in contrast to the Planck energy
$\tilde{E}=kT$. The theory is applied to calculate corrections in
black body radiation and in the Debye solid. Fractional energy
fluctuations are found to show a $1/\nu$ power spectrum in the low
frequency range. A satisfactory model of frequency fluctuations in a
quartz crystal resonator follows. A sketch of the whole
Ramanujan-Rademacher theory of partitions is reminded as well."

M. Planat, "Invitation to the 'spooky' quantum phase-locking effect and its link
to 1/*f* fluctuations", submitted to *Fluctuation and Noise Letters* (10/03)

[abstract:] "An overview of the concept of phase locking at the nonlinear, geometric
and quantum level is attempted, in relation to finite resolution measurements in a
communication receiver and its 1/*f* noise. Sine functions, automorphic functions
and cyclotomic arithmetic are respectively used as the relevant trigonometric tools.
The common point of the three topics is found to be the Mangoldt function of prime number
theory as the generator of low frequency noise in the coupling coefficient, the scattering
coefficient and its quantum critical statistical states. Huyghens coupled pendulums, the
Adler equation, the Arnold map, continued fraction expansions, discrete Möbius
transformations, Ford circles, coherent and squeezed phase states, Ramanujan sums, the
Riemann zeta function and Bost and Connes' KMS states are but a few concepts which are
used synchronously in the paper."

M. Planat, "On the cyclotomic quantum algebra of time
perception" (preprint 03/04)

[abstract:] "I develop the idea that time perception is the quantum counterpart to time measurement. Phase-locking
and prime number theory were proposed as the unifying concepts for understanding the optimal synchronization of
clocks and their 1/*f* frequency noise. Time perception is shown to depend on the thermodynamics of a quantum
algebra of number and phase operators already proposed for quantum computational tasks, and to evolve according to
a Hamiltonian mimicking Fechner's law. The mathematics is Bost and Connes quantum model for prime numbers.
The picture that emerges is a unique perception state above a critical temperature and plenty of them allowed below,
which are parametrized by the symmetry group for the primitive roots of unity. Squeezing of phase fluctuations close to
the phase transition temperature may play a role in memory encoding and conscious activity."

M. Planat, "Huyghens, Bohr, Riemann and Galois: Phase-Locking"
(written in relation to the ICSSUR'05 conference held in Besancon, France - to be published at a special issue of *IJMPB*)

[abstract:] "Several mathematical views of phase-locking are developed. The classical Huyghens approach is
generalized to include all harmonic and subharmonic resonances and is found to be connected to 1/*f* noise and
prime number theory. Two types of quantum phase-locking operators are defined, one acting on the rational
numbers, the other on the elements of a Galois field. In both cases we analyse in detail the phase properties
and find them related respectively to the Riemann zeta function and to incomplete Gauss sums."

S. Perrine, "Recherches
autour de la theorie de Markoff" (compact version of a text published as *La Theorie
de Markoff et ses Developements* (Tessier & Ashpool, 2002))

[abstract:] "The text deals with generalizations of the Markoff equation in number theory,
arising from continued fractions. It gives the method for the complete resolution of such new
equations, and their interpretation in algebra and algebraic geometry. This algebraic approach
is completed with an analytical development concerning fuchsian groups. The link with the
Teichmuller theory for punctured toruses is completely described, giving their classification
with a reduction theory. More general considerations about Riemann surfaces, geodesics and
their hamiltonian study are quoted, together with applications in physics,
1/*f* noise and zeta function. Ideas about important conjectures are presented. Reasons
why the Markoff theory appears in different geometrical contexts are given, thanks to
decomposition results in the group GL(2,*Z*)."

[personal note from author:] "My text deals in fact with trees which are similar to the
Farey tree. With my trees I build generalizations of the Markoff
equation (my generalizations) which are in fact trace formulas (I explained how). These
equations are known to have interpretations on some bundles (that is to say sometimes
particles...). With Michel [Planat], we tried to understand the link that he discovered
with his oscillators, hence with 1/*f* noise, the idea being that it could be of
arithmetical origin."

M. Planat, M. Minarovjech and M. Saniga, "Ramanujan sums analysis of long-period sequences
and 1/f noise" (preprint 12/2008)

[abstract:] "Ramanujan sums are exponential sums with exponent defined over the
irreducible fractions. Until now, they have been used to provide converging expansions to
some arithmetical functions appearing in the context of number theory. In this paper, we
provide an application of Ramanujan sum expansions to periodic, quasiperiodic and
complex time series, as a vital alternative to the Fourier transform. The
Ramanujan-Fourier spectrum of the Dow Jones index over 13 years and of the coronal index
of solar activity over 69 years are taken as illustrative examples. Distinct long periods
may be discriminated in place of the $1/f^{\alpha}$ spectra of the Fourier transform."

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

K.K.
Nambiar, "Information-theoretic equivalent of Riemann
Hypothesis"

[abstract:] "Riemann Hypothesis is viewed as a statement about the capaicity of a
communication channel as defined by Shannon."

Y.V. Fyodorov, G.A. Hiary and J.P. Keating, "Freezing transition, characteristic polynomials of random matrices, and the Riemann zeta-function" (preprint 02/2012)

[abstract:] "We argue that the freezing transition scenario, previously explored in the statistical mechanics of $1/f$-noise random energy models, also determines the value distribution of the maximum of the modulus of the characteristic polynomials of large $N \times N$ random unitary (CUE) matrices. We postulate that our results extend to the extreme values taken by the Riemann zeta-function $\zeta(s)$ over sections of the critical line $s=1/2+it$ of constant length and present the results of numerical computations in support. Our main purpose is to draw attention to possible connections between the statistical mechanics of random energy landscapes, random matrix theory, and the theory of the Riemann zeta function."

D. Klusch, "The
sampling theorem, Dirichlet series and Hankel transforms", *Journal of Computational and
Applied Mathematics* **44** (1992) 261-273

[abstract:] "Some very surprising relations
between fundamental theorems and formulas of signal analysis, of analytic number theory and of
applied analysis are presented. It is shown that generalized forms of the classical
Whittaker-Kotelnikov-Shannon sampling theorem as well as of the Brown-Butzer-SplettstöBer
approximate sampling expansion for non-band-limited signal functions can be deduced via the
theory of Dirichlet series with functional equations from a new summation formula for Hankel
transforms. This counterpart to Poisson's summation formula is shown to be essentially
'equivalent' to the famous functional equation of Riemann's zeta-function, to the 'modular
relation' of the theta-function, to the Nielsen-Doetsch summation formula for Bessel functions
and to the partial fraction expansion of the periodic Hilbert kernel."

S. Yonezawa, "A
deterministic phase shifter for holographic memory devices", *Optics Communications* **19**
(1976) 370-373

[abstract:] "A family of deterministic phase shifters for the holographic memory is presented. It is designed
by applying the gaussian sum, which is known in the field of algebraic number theory. The phase shifter, which
has a few phase quantized levels, was made in the form of an optical device and experiments have been
performed. It satisfactorily disperses the signal light on the hologram plane."

Y. Wei, "Dirichlet
multiplication and easily-generated function analysis", *Computers and Mathematics with
Applications* **39** (2000) 173-199

[abstract:] "Following sine-cosine functions, sawtooth wave, square wave, triangular wave,
and trapezoidal wave become new easily-generated periodic functions. Can a signal be considered
to be a superposition of easily-generated functions with different frequencies? In order to
answer this question, we generalize Fourier analysis to easily-generated function analysis
including easily-generated function series, easily-generated function transformation, and
discrete transformation for easily-generated functions. The results in this paper make it
possible to represent a signal by use of easily-generated functions. A lot of techniques based
on sine-cosine functions can be translated into techniques based on easily-generated functions.
Because Dirichlet multiplication in number theory plays a basic role in easily-generated
function analysis, we briefly introduce this concept and present a related formula. The main
contents of this paper include dirichlet multiplication and a related formula, relations
between basic waveforms in electronics, easily-generated function series, easily-generated
function transformation, discrete transformation for easily-generated functions, and techniques
of easily-generated function analysis."

A. Bershadskii, "Hidden periodicity in the sequence of prime numbers" (preprint 02/2011)

[abstract] "Logarithmic gaps have been used in order to find a periodic component of the sequence of prime numbers, hidden by a random noise (stochastic or chaotic). It is shown that multiplicative nature of the noise is the main reason for the successful application of the logarithmic gaps transforming the multiplicative noise into an additive one. The recovered period for the sequence of the first 6000 prime numbers is equal to $8 \pm 1$ (with a logarithmic precision, subject to the prime number theorem)."

M. Pitkänen,
"Quantum criticality and 1/*f* noise"
(submitted for publication in *Fluctuation and Noise Letters* **0**)

Wentian Li's extensive
1/f Noise Bibliography

A. Rifat, "A Physics-based Explanation of the Riemann Hypothesis and its Relationship to Signal Processing"

"As a hobbyist mathematician, my interest in RH stems from a seminal paper presented by Professor Berry, at the Royal Society
in the late 80's, in which he shows an amazing agreement between the statistics
of the spacings distributions of the roots of zeta on the critical line and chaotic
billiards."

[from introduction] "My own investigation has been to derive a physical mechanism that produces the
characteristic statistics of the Riemann Zeta function, which no one appears to have done. They have evolved
through discussion and argument with experts in this field including Professor Sir Michael Berry who has taken
a great deal of trouble to look and sanitise my ideas up to a point where he fundamentally disagrees with my
approach. From this point, these observations are a result of culmination of my ideas. I believe I have taken a
very open view without becoming fixated in a particular mathematical theory or purely a non-physical
mathematical world-view, nor trying to bias results or fit data from the start. My own limited knowledge of
Riemann Hypothesis calls up facts from my background in sonar and signal processing that I know to be true."

Sergei Maslov's homepage

Maslov on 1/f noise

Maslov on self-organized criticality

Maslov's 1/f noise 'sandpile' applet

B. Mandelbrot,
*Multifractals and 1/f Noise (Wild Self-Affinity
in Physics)* (Springer-Verlag, New York, 1998)

notes on the controversy
surrounding Bak, Tang and Wiesenfeld's purported link between
1/*f* noise and self-organised criticality

Chao Tang's homepage

A few reviews of Per
Bak's 1997 book
*How Nature Works: The Science of Self-Organized Criticality*

*Sadly, Per Bak died in October 2002 at the age of 54 (1948-2002).*