Fourier analysis and number theory
[figure by Michael Rubinstein reproduced from Brian Conrey's survey
article on the Riemann hypothesis, Notices of the
AMS (2003), p. 346]
The above graph illustrates a Fouriertype duality relation between the
prime numbers and the zeros of the Riemann
zeta function. The function depicted is the Fourier transform of
the error term in the Prime Number Theorem, and
the visible spikes correspond to the imaginary parts of the zeta zeros.
This Fouriertype duality is most clearly expressed mathematically
by the RiemannWeil explicit formula.
What is most interesting, and least understood, about this situation
is the fact that the structure of the explicit formula is mirrored
by certain dynamical trace formulae.
The first instance of
this to be observed involved the Selberg trace formula
(discovered in the 1950's) which concerns the geodesic flow on a Riemann
surface, relating its periodic orbits and its energy
levels, i.e. eigenvalues of the LaplaceBeltrami operator. Here the orbits correspond to the primes
and the energy levels to the Riemann zeta zeros. The latter correspondence
lends credence to the spectral
interpretation of the Riemann zeta function, and the overall situation suggests the
existence of some kind of mysterious dynamical system underlying (or "lurking behind" as
N. Snaith put it in her Ph.D. thesis)
the distribution of prime numbers.
Note that the Selberg trace
formula can be thought of as a generalisation of the Poisson
summation formula, a very clear instance of Fouriertype duality
between functions on a torus.
The wider phenomenon of correspondence
between the explicit formulae
of number theory (of which the RiemannWeil formula is just one, important,
special case) and dynamical trace formulae points to some fundamental
issue of duality which is currently a great mystery, and may turn out
to be hugely significant in our understanding of both mathematical and
physical reality.
In this page we consider a range of material which involves Fourier analysis
in a number theoretical setting.
In H.M. Edwards' excellent book
Riemann's Zeta Function (Academic Press, 1974), while discussing in detail the
content of Riemann's groundbreaking
1859 paper on the distribution of primes, informs us in Chapter 1 that:
"Riemann was a master of Fourier analysis and his work in developing this theory must
certainly be counted among his greatest contributions to mathematics. It is not surprising,
therefore, that [in this paper] he immediately applies Fourier inversion to the formula
"
Chapter 10 of Edwards' book is called Fourier Analysis and begins:
"One of the basic ideas in Riemann's original paper is, as Chapter 1 shows, the idea
of Fourier analysis. This chapter is devoted to the formulation of a more modern approach
to Fourier analysis which I believe sheds some light on the meaning of the zeta function and
on its relation to the distribution of primes.
The approach to Fourier analysis I have in mind is that in which the fundamental
object of study is the algebra of invariant operators on the functions on a group. The group in
this instance is the multiplicative groups of positive real numbers..."
At the end of section 10.1:
"The technique by which Riemann derived his formula for [the primecounting function]
J(x) and von Mangoldt his
formula for [the logarithmic primecounting function]
is a technique of inversion, of going from the transform
to the operator, which might well be called Fourier synthesis  putting the operator
back together again when its effect on the invariant subspaces is known."
J.T. Tate, "Fourier analysis in number fields and Hecke's
zetafunctions" (1950 Princeton Ph.D. thesis)
"Tate's Thesis", as it is now called, is reproduced as Chapter 15
of Algebraic Number Theory by J.W.S. Cassels
and A. Fröhlich (Academic Press, 1967). Unfortunately this appears to be out of print.
The introduction explains: "This volume also contains...
Tate's doctoral thesis, which is for the first time published here after
it had over many years had a deep influence on the subject as a piece
of clandestine literature."
Tate proves the functional equations for a very general class of zeta
and Lfunctions by using a version of the Poisson
summation formula for idèle groups of algebraic number fields.
B.J. Green's notes on Fourier analysis and the
functional equation of the Riemann zeta function
J.F. Burnol, "On Fourier and zeta(s)"
[Abstract:] "We study some of the interactions between the Fourier Transform and
the Riemann zeta function (and DirichletDedekindHeckeTate Lfunctions)."
This involves indepth discussion of such physicsrelated topics as scattering,
causality, the Kramers–Wannier duality relation, and the Hilbert–Pólya idea.
[excerpt]
J.F. Burnol, "Sur certains espaces
de Hilbert de fonctions entieres, lies a la transformation de Fourier
et aux fonctions L de Dirichlet et de Riemann"
[Abstract:] "We construct in a Sonine space of entire functions a
subspace related to the Riemann zeta function and we show that the
quotient contains vectors intrinsically attached to the nontrivial
zeros and their multiplicities."
This includes a summary in English. The author has pointed out
that the article relates to the 'HilbertPólya
idea' for proving the Riemann Hypothesis.
J.F. Burnol on the RiemannWeil explicit formula
D.C. Brody and C.M. Bender, "Operatorvalued zeta functions and Fourier analysis" (preprint 10/2018)
[abstract:] "The Riemann zeta function $\zeta(s)$ is defined as the infinite sum $\sum_{n=1}^{\infty}n^{s}$, which converges when $\Re s>1$. The Riemann hypothesis asserts that the nontrivial zeros of $\zeta(s)$ lie on the line $\Re s = \frac{1}{2}$. Thus, to find these zeros it is necessary to perform an analytic continuation to a region of complex $s$ for which the defining sum does not converge. This analytic continuation is ordinarily performed by using a functional equation. In this paper it is argued that one can investigate some properties of the Riemann zeta function in the region $\Re s<1$ by allowing operatorvalued zeta functions to act on test functions. As an illustration, it is shown that the locations of the trivial zeros can be determined purely from a Fourier series, without relying on an explicit analytic continuation of the functional equation satisfied by $\zeta(s)$."
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 quasiperiodic versus the time n of the resonance and aperiodic
versus the order q of the resonance. New results arise from the
use of this RamanujanFourier 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,
"The
number theoretical spin chain and the Riemann zeroes", Comm. Math.
Phys. 196 (1998), no. 3, 703731
"It is an empirical observation that the Riemann zeta function can be
well approximated in its critical strip using the NumberTheoretical Spin
Chain. A proof of this would imply the Riemann Hypothesis.
Here we relate that question to the one of spectral radii of a family of
Markov chains...The general idea is to explain the pseudorandom features
of certain number theoretical functions by considering them as observables
of a spin chain of statistical mechanics."
[17] C. Castro and J. Mahecha, "Comments
on the Riemann conjecture and index theory on Cantorian fractal spacetime",
Chaos, Solitons and Fractals 13 (2002) 1407
The paper was inspired by
H. Gopalkrishna Gadiyar and R. Padma,
"RamanujanFourier series, the WienerKhintchine formula and the distribution
of prime pairs", Physica A 269 (1999) 503510.
[Abstract:] "The WienerKhintchine formula plays a central role in statistical
mechanics. It is shown that the problem of prime pairs is related to
autocorrelation and hence to a WienerKhintchine formula. "Experimental"
evidence is given for this."
More recently the same authors have produced this:
H. Gopalkrishna Gadiyar and R. Padma,
"Rota
meets Ramanujan: Probabilistic interpretation of RamanujanFourier series"
[Abstract:] "In this paper the ideas of Rota and Ramanujan are shown to be central to understanding
problems in additive number theory. The circle and sieve methods are two different facets of
the same theme of interplay between probability and Fourier series used to great advantage
by Wiener in engineering."
B.W. Ninham and S. Lidin, "Some remarks on quasicrystal structure",
Acta Crystallographica A 48 (1992) 640650
[abstract:] "The Fourier transform of skeleton delta function that characterizes the most
striking features of experimental quasicrystal diffraction patterns is evaluated. The
result plays a role analogous to the Poisson summation formula for periodic delta funcitons
that underlie classical crystallography. The realspace distribution can be interpreted in
terms of a backbone comprising a system of intersecting equiangular spirals into which are
inscribed (selfsimilar) gnomons of isosceles triangles with lengthtobase ratio the golden
mean...In addition to the vertices of these triangles, there is an infinite number of other
points that may tile space in two or three dimensions. Other mathematical formulae of
relevance are briefly discussed."
[from concluding remarks:] "Perhaps the most interesting feature is that our Fouriertransform
sum seems to have much in common with the distribution of the zeros of the Riemann zeta function...!
That indicates something of the depth of the problem. That the zeta function ought to come into the
scheme of things somehow is not surprising  the Poisson and related summation formulae are
special cases of the Jacobi theta function. [Indeed the Bravais lattices can be enumerated
systematically through an integral over all possible products and sums of products of any
three of the four theta functions in different combinations that automatically preserve
translational and rotational symmetries.] The thetafunction transformations are themselves
just another way of writing the [functional equation of the zeta
function]. Additionally, the properties of the zeta function are automatically connected
to the theory of prime numbers. So one expects that the RogersRamanujan relations must
play a central role in the scheme of things for quasicrystals."
a
proof of the prime number theorem involving Fourier transforms
(summarised by Jonas Wiklund)
N. Wiener's The Fourier Integral and Certain of its Applications, section 17,
"The PrimeNumber Theorem as a Tauberian Theorem" begins:
"The present section and the three following will be devoted to the application of
Tauberian theorems to the problem of the distribution of the primes. The theorem which we
shall eventually prove is the famous theorem of Hadamard and de la Vallée Poussin..."
B. van der Pol, "An
electromechanical investigation of the Riemann zeta function in
the critical strip", Bulletin of the AMS 53 (1947)
[commentary from Crandall and Pomerance's Prime Numbers: A Computational Perspective:]
"Peculiar as it may seem today, the scientist and engineer van der Pol did, in the 1940's,
exhibit tremendous courage in his 'analog' manifestation of an interesting Fourier decomposition.
An integral used by van der Pol was a special case ($\sigma$ = 1/2) of the following relation,
valid for $s = \sigma + it, \sigma \in (0,1)$:
$\zeta(s) = s\int_{\infty}^{\infty} e^{\sigma\omega}([e^{\omega}]e^{\omega})e^{i \omega t} d\omega$
Van der Pol actually built and tested an electronic circuit to carry out the requisite transform in
analog fashion for $\sigma=1/2$. In today's primarily digital world it yet remains an open
question whether the van der Pol approach can be effectively used with, say, a fast Fourier
transform to approximate this interesting integral. In an even more speculative tone, one notes that
in principle, at least, there could exist an analog device  say an extremely sophisticated circuit  that
sensed the prime numbers, or something about such numbers, in this fashion."
"Beyond the Riemann zeta function and special arithmetic functions
that arise in analytic number theory, there are other important
entities, the exponential sums. These sums generally contain
information  one might say "spectral" information  about special
functions and sets of numbers. Thus, exponential sums provide a
powerful bridge between complex Fourier analysis and number theory.
For a realvalued function, real t, and integers a < b, denote
E(f ;a,b,t) = Sum_{{a<n<=b}}
exp(2pi*itf(n))
Each term in such an exponential sum has absolute value 1, but the
terms can point in different directions in the complex plane. If the
various directions are "random" or "decorrelated" in an appropriate
sense, one would expect some cancellation of terms."
R. Crandall and C. Pomerance, Prime Numbers: A Compuational Perspective
(Springer, 2001) p. 40.
"The classical periodic orbits are a crucial stepping stone in the
understanding of quantum mechanics, in particular when then classical
system is chaotic. This situation is very satisfying when one thinks
of Poincaré who emphasized the importance of periodic orbits in
classical mechanics, but could not have had any idea of what they could
mean for quantum mechanics. The set of energy levels and the set
of periodic orbits are complementary to each other since they are
essentially related through a Fourier transform. Such a relation had
been found earlier by the mathematicians in the study of the Laplacian
operator on Riemannian surfaces with constant negative curvature. This
led to Selberg's trace formula in 1956 which has exactly the same form,
but happens to be exact. The mathematical proof, however, is based on
the high degree of symmetry of these surfaces which can be compared to
the sphere, although the negative curvature allows for many more
different shapes."
M.C. Gutzwiller,
"Chaos in Quantum Mechanics" (1998 lecture notes)
Extensive notes from Gutzwiller on
the Selberg Trace Formula and quantum chaos
Unfortunately there are few good references on the Selberg trace formula.
Peter Perry suggested the following:
H.P. McKean, "Selberg's trace formula as applied to a compact Riemannian
surface", Communications in Pure and Applied Mathematics 25
(1972) 225246. See also the relevant erratum in Comm. Pure Appl. Math.
27 (1974) p.134.
"This will give a somewhat breezy but in principal complete derivation of
Selberg's trace formula for a compact Riemann surface  where the Laplacian
has only eigenvalues and no continuous spectrum. There are generalizations
to cases where the underlying manifold is noncompact and the Laplacian has
continuous spectrum, but these are much more involved analytically.
A useful analogy is Poisson's summation formula for a torus, thought of as
a relation between the eigenvalues of the Laplacian (e.g.
4(pi)^{2}
n^{2}
on the circle viewed as [0,1) ) and the lengths of
closed geodesics (n for integers n). For a Schwarz class test
function the Poisson summation formula says that
and so relates the test function evaluated at lengths on the left to its
Fourier transform evaluation at square roots of eigenvalues on the right).
Selberg's trace formula is the spiritual ancestor
of the celebrated DuistermaatGuillemin trace formula. See:
J. Duistermaat and V. Guillemin "The spectrum of positive elliptic operators
and periodic bicharacteristics", Invent. Math. 29 (1975), no. 1,
3979."
S. Albeverio, A. Khrennikov, and R. Cianci, "On the Fourier transform and
the spectral properties of the padic momentum and Schrodinger
operators", Journal of Physics A 30 (1997) 57675784.
A. Khrennikov and R. Cianci, "On the Fourier transform and the
padic momentum and Schrodinger operators", General Relativity and
gravitational physics, Rome, 1996 (World Scientific Publications,
1997) 279285.
N. De GrandeDe Kimpe, A. Khrennikov, and L. Van Hamme, "The Fourier
transform for padic smooth distributions", Lecture Notes in Pure
and Applied Mathematics 207 (Dekker, 1999) 97112.
V. Dimitrov, T. Cooklev and B. Donevsky,
"Number
theoretic transforms over the golden section quadratic field.", IEEE Trans. Sig. Proc. 43 (1995) 17901797
V. Dimitrov,G. Jullien, and W. Miller, "A residue number system
implementation of real orthogonal transforms", IEEE Trans. Sig.
Proc. 46 (1998) 563570.
J. Main, V.A. Mandelshtam, G. Wunner and H.S. Taylor, "Riemann
zeros and periodic orbit quantization by harmonic inversion"
[abstract:] "In formal analogy with Gutzwiller's semiclassical trace formula, the density of
zeros of the Riemann zeta function zeta(z=1/2iw) can be written as a nonconvergent series
rho(w)=pi^{1} sum_p sum_{m=1}^infty ln(p)p^{m/2} cos(wm ln(p)) with p running over the prime
numbers. We obtain zeros and poles of the zeta function by harmonic inversion of the time
signal which is a Fourier transform of rho(w). More than 2500 zeros have been calculated to
about 12 digit precision as eigenvalues of small matrices using the method of
filterdiagonalization. Due to formal analogy of the zeta function with Gutzwiller's periodic
orbit trace formula, the method can be applied to the latter to accurately calculate individual
semiclassical eigenenergies and resonance poles for classically chaotic systems. The periodic
orbit quantization is demonstrated on the three disk scattering system as a physical example."
E. Doron, "Do
spectral trace formulae converge?"
[abstract:] "We evaluate the Gutzwiller trace formula for the level density of classically
chaotic systems by considering the level density in a bounded energy range and truncating its
Fourier integral. This results in a limiting procedure which comprises a convergent
semiclassical approximation to a well defined spectral quantity at each stage. We test this
result on the spectrum of zeros of the Riemann zeta function, obtaining increasingly good
approximations to the level density. The Fourier approach also explains the origin of the
convergence problems encountered by the orbit truncation scheme."
Y. Wei, "Dirichlet
multiplication and easilygenerated function analysis", Computers and Mathematics with
Applications 39 (2000) 173199
[abstract:] "Following sinecosine functions, sawtooth wave, square wave, triangular wave,
and trapezoidal wave become new easilygenerated periodic functions. Can a signal be considered
to be a superposition of easilygenerated functions with different frequencies? In order to
answer this question, we generalize Fourier analysis to easilygenerated function analysis
including easilygenerated function series, easilygenerated function transformation, and
discrete transformation for easilygenerated functions. The results in this paper make it
possible to represent a signal by use of easilygenerated functions. A lot of techniques based
on sinecosine functions can be translated into techniques based on easilygenerated functions.
Because Dirichlet multiplication in number theory plays a basic role in easilygenerated
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, easilygenerated function series, easilygenerated
function transformation, discrete transformation for easilygenerated functions, and techniques
of easilygenerated function analysis."
P. Flajolet, P. Grabner, P. Kirschenhofer, H. Prodinger, and R. Tichy,
"Mellin
transforms and asymptotics: digital sums", Theoretical Computer Science 123
(1994) 291314
[abstract:] "Arithmetic functions related to number representation systems exhibit various
periodicity phenomena. For instance, a wellknown theorem of Delange expresses the total number
of ones in the binary representations of the first n integers in terms of a periodic
fractal function.
We show that such periodicity phenomena can be analyzed rather systematically using
classical tools from analytic number theory, namely the MellinPerron formulae. This approach
yields naturally the Fourier series involved in the expansions of a variety of digital sums
related to number representation systems."
C.F. Woodcock, "Special padic analytic functions
and Fourier transforms", Journal of Number Theory, 60 (2) (1996) 393408
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