###
further quotes on the Riemann Hypothesis, the zeta zeros,
the spectral interpretation, and number theory in physics
*"...the idea that you can presumably correctly conjecture that infinitely many numbers are on a particular line, and you can't prove
it, is frustrating beyond any description. It's just unacceptable!"*
U.C. San Diego number theorist Harold
Stark on the Riemann hypothesis
*"If there *are* lots of zeros off the line - and there might be - the whole picture is
just horrible, horrible, very ugly. It's an Occam's razor sort of thing, you either have
absolutely beautiful behaviour of prime numbers, they behave just like you want them to
behave, or else it's *really* bad."*
S. Gonek, quoted in K. Sabbagh, *Dr.
Riemann's Zeros* (Atlantic, 2002), p.112
*"It is intriguing that any of the various new expansions and
associated observations relevant to the critical zeros arise from the
field of quantum theory, feeding back, as it were, into the study of
the Riemann zeta function. But the feedback of which we speak can
move in the other direction, as techniques attendant on the Riemann
zeta function apply to quantum studies."*
J. Borwein, D. Bradley and R. Crandall, "Computational strategies
for the Riemann zeta function", *J. Comp. App. Math.* **121** (2000)
*"The Riemann Hypothesis (RH) has been around for more than 140 years, and yet
now is arguably the most exciting time in its history to be working on RH. Recent years
have seen an explosion of research stemming from the confluence of several areas of
mathematics and physics."*
J. Brian Conrey, "The Riemann Hypothesis", *Notices
of the AMS* (March 2003)
*"Our purpose is to report on the development of
an analogy, in which three areas of mathematics and physics, usually regarded
as separate, are intimately connected. The analogy is tentative and tantalizing,
but nevertheless fruitful. The three areas are eigenvalue
asymptotics in wave (and particularly quantum) physics, dynamical chaos,
and prime number theory. At the heart of the analogy is a speculation concerning
the zeros of the Riemann zeta function (an infinite sequence of number encoding the primes): the Riemann zeros are related to the eigenvalues (vibrational frequencies or quantum energies) of some wave system, underlying which is a dynamical system whose rays
or trajectories are chaotic.*
Identification of this dynamical system would lead directly to a proof of the celebrated Riemann hypothesis. We do not know what the system is, but we do know many of its properties..."
M.V. Berry and J.P. Keating from "The
Riemann Zeros and Eigenvalue Asymptotics", *SIAM Review*
**41**, no. 2 (1999) 236–266
*"A classical mechanical system with a well-behaved
Hamiltonian is needed, which also has a good quantum-mechanical relative,
such that, on the one hand, the lengths of the periodic orbits are given
by the logarithms of the prime numbers, and on the other hand, the energy
levels are the zeros of the Riemann zeta function.*
M.C. Gutzwiller,
summarising the above approach to the RH, from *Chaos in Classical and
Quantum Mechanics *(Springer-Verlag, 1991), p. 309
*"It's a problem of extreme mathematical subtlety
and complexity. There's a lot that's understood, and a lot that's not understood.
One particular thing that's lacking is a good model which has the essence
of the general behaviour."*
M.V. Berry on quantum chaology, from "A
Prime Case of Chaos" by Barry Cipra
*"Riemann's
conjecture *[*has*]* a further significance: when (if)
the operator with
eigenvalues E*_{j} is found, it will surely be simple, and will
provide a paradigm for quantum chaology comparable with the harmonic oscillator
for quantum non-chaology."
M.V. Berry, from the "Quantum Chaology", The
Bakerian Lecture, 1987, *Proceedings of the Royal Society of London A*
**413** (1987) 183–198
* *[*If the Riemann
Hypothesis were disproven by a single counter-example, one question
would linger:*]* "How could it be that the Riemann
zeta function so convincingly mimics a quantum system without being
one?"*
M.V. Berry, from "A Prime Case of Chaos" by
Barry Cipra
*"Loosely speaking, the Riemann hypothesis
states that the primes have music in them", Berry says. But Berry is looking for more
than a musical analogy – he hopes to find the actual instrument behind the zeta function – a mathematical drum whose natural
frequencies line up with the zeroes of the zeta function. The answer, he thinks, lies
in quantum mechanics. "There are vibrations in classical physics too" he notes, "but
QM is a richer, more varied source of vibrating systems than any classical oscillators
that we know of." "*
from "A Prime Case of Chaos" by Barry Cipra
*"Last year *[*Alain*]*
Connes* *proved that his
prime-based quantum system
has energy levels corresponding to all the Riemann zeros that lie on the
critical line. He will win the fame and the
million-dollar
prize if he can make one last step: prove that there aren't any extra
zeros hanging around, unaccounted for by his energy levels.
*
*That last step is a formidable one. Has Connes simply replaced the Riemann
hypothesis with an equally difficult question? Some experts advise caution.
"I still think that some major new idea is need here", says
Bombieri.
*
*Berry, for his part, doesn't flinch at the mathematical peculiarity of
Connes's system. "I'm absolutely sure that if he's right, someone will find
a clever way to make it in the lab. Then you'll get the Riemann zeros out
just by observing its spectrum."*
E. Klarreich, from
"Prime Time" (*New Scientist*, 11/11/2000)
*
"We will. . .write
*
*
*r*
*_{n}
= 1/2* + i*g
_{n}
*
**. . .Nothing is known about the
*g*
*_{n},
but they are thought likely to be transcendental numbers, algebraically
independent of any reasonable numbers that have ever been considered."
A.M. Odlyzko
on the nontrivial zeros of the zeta
function, "Primes,
Quantum Chaos, and Computers"
*"*[*Odlyzko's
computations are*]* the first phenomenological insight that the zeroes
are absolutely, undoubtedly 'spectral'
in nature. Riemann
himself would be impressed"*
Princeton University mathematician
Peter Sarnak, from
"A Prime Case of Chaos" by Barry Cipra
*"Neither Hilbert
nor Pólya ever suggested a specific operator or even a specific space this
operator might act on. Still, their conjecture is regarded as perhaps the
most promising approach to proving the Riemann
hypothesis..."*
"What we do...is to assume that the operator associated to the
zeta function behaves in some way like a
random operator. This obviously cannot be completely correct, since
any operator associated to an object as special as the zeta function
cannot be random."
"The excellent fit between zeta function data
and the GUE does not prove the existence of
the Hilbert-Pólya operator. However, it is remarkable that the chain
of wild conjectures that were mentioned above leads to a prediction
that appears to be remarkably correct...The analogy with random
matrices should not be carried too far, because the zeros of
*zeta*(*s*) are after all very special."
A.M. Odlyzko,
"Primes,
Quantum Chaos, and Computers"
*"I was immediately led to the idea that somehow passing from the integers to the
primes is very similar to passing from quantum field theory, as we observe it, to the
elementary particles, whatever they are...
*
*I was invited to this [Seattle] meeting in 1996 and there I gave a talk,
but I was very struck by the fact that apart from the work on random matrices not much
was going on as far as the RH itself was concerned. So then I got really enticed to
go on and work on it, and when I came back from the conference, which was in the summer,
I pushed the analogy much further. The zeros didn't even show up in my previous work
but then, by going further, the zeros come up completely naturally in the sense that
one didn't have to define the Riemann zeta function to get the zeros - they were coming from a purely
spectral interpretation, and the basic new feature which came up can be explained very simply."
*
A. Connes, quoted in K. Sabbagh, *Dr.
Riemann's Zeros* (Atlantic, 2002), p.204.
*"So for all practical purposes, the Riemann zeta function does not show its true colours in the range available by numerical investigations. You should go up to the height 10*^{10,000}, then I would be much more convinced if things were still pointing strongly in the direction of the Riemann Hypothesis. So numerical calculations are certainly very impressive, and they are a triumph of computers and numerical analysis, but they are of limited capacity. The Riemann Hypothesis is a very delicate mechanism. It works as far as we know for all existing zeros, but we cannot, of course, verify numerically an infinity of zeros, so other theoretical ways of approach must be found, and for the time being they are insufficient to yield any positive conclusion."
A. Ivic, quoted in K. Sabbagh, *Dr.
Riemann's Zeros* (Atlantic, 2002), p.122
*"...the best upper bound for the abscissa of convergence
of Z*_{M}
(1/zeta*)
has not decreased since
Riemann.
This is the sign of a nasty phenomenon at Re(*s*) = 1!"*
B. L. Julia, "Statistical Theory of Numbers", from *Number Theory and Physics* (Luck, Moussa, Itzykson, eds., Springer-Verlag, (1990)
*"The idea that number theory can be related to the properties of Landau
levels of a 2-dimensional electron system will probably seem arcane to
most scientists, and should astonish most mathematicians. However, the
idea has rich and intriguing possibilities."*
J.C. Phillips, "Microscopic
origin of collective exponentially small resistance states" (preprint, 03/03)
*"The primes by themselves are just a set; there is nothing we can say about them as such. Interesting
problems about the primes arise only when we throw them into other sets...
*
*...we remain almost ignorant about the most basic problem of the distribution of primes within the continuum
of real numbers ***R**^{+}. The prime number theorem...is equivalent to the
analytic statement that ζ(1+it) [is nonzero]*. However we have no control over the error term π(x) – Li(x);
the fact that this is of size O(x*^{1/2 + ε}), for any ε > 0 ... is equivalent to the Riemann hypothesis...
The Riemann hypothesis has numerous formulations...for example, the equidistribution of the rationals **Q**^{+}
within **R**^{+}. The problem is thus not with the primes, but with their interaction with the real
numbers, and, as we shall see, this is intimately connected with the mysteries of the
real prime."
M.J. Shai Haran, *The
Mysteries of the Real Prime* (OUP, 2001) vi–vii.
*"Typically for the Riemann hypothesis, closing the gap in the above proof could be a formidable
task. However, as the way to a solution often yields more fruit than the proof itself, we can already
see clear evidence of close connections between the Riemann hypothesis and the uncertainty principle,
with its singular expressions over the real numbers."*
M.J. Shai Haran, *The
Mysteries of the Real Prime* (OUP, 2001) p225.
*"One is faced with an infinite series of axioms which can be extended further and further, without
any end being visible...It is true that in the mathematics of today the higher levels of this hierarchy are practically
never used...it is not altogether unlikely that this character of present-day mathematics may have something to
do with its inability to prove certain fundamental theorems, such as, for example, Riemann's Hypothesis."*
from M. du Sautoy, *The Music of the
Primes* (Fourth Estate, 2003)
*"A simple criterion is derived in order that a
number sequence is a permitted spectrum of a quantised system. The sequence
of prime numbers fulfils the criterion. . .The existence of such a potential
implies that...primality testing can in principle be resolved by the sole
use of physical laws."*
G.
Mussardo, "The Quantum Mechanical Potential for the Prime Numbers" (preprint, 1997)
*"I have recently explored the possibility to design
a physical experiment that provides not only a test of primality (as done
in my previous paper) but also the factorization into prime numbers when
the number is composite. This is indeed possible."*
G. Mussardo, personal communication (October
1999)
*
"The evolution of the power spectrum and
Liapunov exponents are
the most elementary tests to be applied to a series of numbers generated
by some unknown dynamics in order to search for some hidden regularity.
If at least one of the *[*Liupanov exponents*]* is positive,
we know the underlying
dynamics is chaotic. . .we studied the succession coming from the difference
between the
prime counting function and its analytic
approximation*
*R(x) = Li(x) – Li(x*^{1/2})/2 - Li(x^{1/3})/3
– ^{...}
*...the largest Liupanov exponent...is unequivocally
positive in all the range. This...was calculated by using the method of Eckmann,
et al. This method gives also a minimal embedding dimension of 4 for the
unknown subjacent *[*underlying*]* classical dynamics...Also, from a physical
point-of-view, we can say that any physical system whose dynamics is unknown
but isomorphic to the prime number distribution has a chaotic behaviour."*
Z. Gamba, J. Hernando, L. Romanelli, "Are the Prime Numbers Regularly Ordered?", *Physics Letters A.* **145**, no. 2,3 (1990) 106–108
*
"In the past there were a lot of papers discussing
self-organized criticality. The abundance
of natural phenomena where the power spectrum displays the 1/f noise
was attempted to be explained by means of the self-organized critical models.
They describe systems without finite characteristic length scale. Primes
indeed seem to be distributed among all natural numbers with all possible
gaps between them and there is no natural length scale. We end this short
paper by asking the question: Are the prime numbers in a self-organized
critical state?"*
M.
Wolf, "1/f noise in the distribution of prime numbers", *Physica A*,
**241** (1997) 493–499
*"It seems strange that on one hand the most practical of disciplines,
namely, physics has connections with the most arcane of disciplines,
namely, number theory. However, surprising connections have appeared
betwen number theory and physics...The work of Ramanujan in particular
has had surprising connections with string theory, conformal field
theory and statistical physics."*
R. Padma and H. Gopalkrishna Gadiyar, "Renormalisation
and the density of prime pairs" (preprint 06/98)
*"It is a pleasant surprise that the Wiener-Khintchine
formula which normally occurs in practical problems of brownian motion,
electrical engineering and other applied areas of technology and statistical
physics has a role in the behaviour of prime numbers which are studied
by pure mathematicians."*
H. Gopalkrishna Gadiyar and R. Padma,
"Ramanujan–Fourier series, the Wiener–Khintchine formula and the
distribution of prime pairs", *Physica A* **269** (1999)
503–510
*
"Consider the integers divisible by both *p* and *q
[both primes]*. To be divisible by *p* and *q* is
equivalent to being divisible by *pq* and consequently the
density of the new set is 1/*pq*. Now, 1/*pq* =
1/*p* * 1/*q*, and we can interpret this by saying that the
"events" of being divisible by *p* and *q* are independent.
This holds, of course, for any number of primes, and we can say using
a picturesque but not very precise language, that the primes play a
game of chance! This simple, nearly trivial, observation is the
beginning of a new development which links in a significant way number
theory on the one hand and probability theory on the other."*
M. Kac "Statistical Independence in Probability, Analysis, and Number Theory" – the Carus Mathematical Monograph no. 12 (MAA, 1964)
*"Delicate questions concerning the distribution of prime numbers still
seem to be very mysterious. It may be that by taking into account divisibility
by small primes we can obtain a very accurate picture; or it may be that there
are other phenomena disturbing the equi-distribution of primes, that await
discovery."*
A. Granville,
"Unexpected irregularities in the distribution of primes numbers"
*"If you choose a number *n* and ask how many prime numbers there are less than *n* it turns out that the answer closely approximates
the formula: *n*/log *n*. The formula is not exact, though: sometimes it is
a little high and sometimes it is a little low. Riemann looked at these deviations and saw that they contained periodicities. Berry likens these to musical harmonics: "The question is what are the harmonics in
the music of the primes? Amazingly, these harmonics or magic numbers behave
exactly like the energy levels in quantum systems that classically would
be chaotic."
This correspondence emerges from statistical correlations between the spacing of the Riemann numbers and the spacing of the energy levels. Berry and his collaborator Jon Keating used them to show how techniques in number theory can be applied to problems in quantum chaos and vice versa. In itself such a connection is very tantalising. Although sometimes described as the Queen of mathematics, number theory is often thought of as pretty useless, so this deep connection with physics is quite astonishing. Berry is also convinced that there must be a particular chaotic system which when quantised would have energy levels
that exactly duplicate the Riemann numbers. "Finding this system could be the discovery of the century," he says. it would become a model system for describing chaotic systems in the same way that the simple harmonic oscillator is used as a model for all kinds of complicated oscillators. It could play a fundamental role in describing all kinds of chaos. The search for this model system could be the holy grail of chaos...
[We] cannot be sure of its properties, but Berry believes the system is likely to be rather simple, and expects it to lead to totally new physics. It
is a tantalising thought."*
Julian
Brown, "Where Two Worlds Meet", *New Scientist,* 16 May 1996
*
"We shall in fact bring a large chunk of theoretical physics technology to bear on important mathematical problems and conversely one hopes to learn from a century of analytical number theory to understand better several issues of modern physics like the quark-gluon plasma transition and the Hagedorn critical temperature."*
B. Julia, "Theories statistique et thermodynamique
des nombres", in: Conference de Strasbourg en l'honneur de P. Cartier, Proc.
IRMA-RCP25, Vol. 44 (1993).
*
"Our main point here is that one could have been led to the main outline of the
proof of the prime number theorem by using the physical
interpretation of the Laplace transforms provided by statistical mechanics.
In particular, the function —**ζ*'(*x*)/*ζ*(*x*)
whose representation as a
Dirichlet series (Laplace transform with discrete measure) plays a central role
in the proof has a direct physical interpretation as the internal energy function."
G.W. Mackey, *Unitary Group Representations in
Physics, Probability and Number Theory* (Benjamin, 1978), p.300
*"We..wish to emphasize that statistical mechanics provides a motivation for
a rather detailed study of the relationship between a measure and its Laplace transform
and provides physical interpretations for theorems in the subject. This is especially
interesting because number theory also provides such a motivation - indeed much of the
theory of the Laplace transform...was developed in response to the needs of number theory.
Combining these two connections, one is led to physical interpretations of results in
number theory as well as applications of number theoretical results to physics."
*
G.W. Mackey, *Unitary Group Representations in
Physics, Probability and Number Theory* (Benjamin, 1978), p.297
*"On the fundamental level our world is neither real nor *p*-adic;
it is adelic. For some reasons, reflecting the physical nature of our kind of
living matter (*e.g.* the fact that we are built of massive particles), we
tend to project the adelic picture onto its real side. We can equally well
spiritually project it upon its non-Archimediean side and calculate most important
things arithmetically.
*
*The relations between "real" and "arithmetical" pictures of the world is that of
complementarity, like the relation between conjugate observables in quantum
mechanics."*
Y. Manin, in *Conformal Invariance and String Theory*, (Academic Press, 1989)
p.293-303
*"This paper finds its roots in the conviction that the Riemann
Hypothesis has a lot to do with (suitably envisioned) Quantum Fields.
The belief in a possible link between the Riemann Hypothesis and Quantum
Mechanics seems to be widespread and is a modern formulation of the
Hilbert-Pólya operator approach. I believe
that techniques and philosophy more organic to Quantum Fields will be
most relevant. As this point of view has not so far led to success,
I will conclude here with extracts from William Blake's "The Marriage of
Heaven and Hell" which, to my mind remarkably illustrate some salient
points:
*
*". . .But first the notion that man has a body distinct from his soul
is to be expunged; this I shall do, by printing in the infernal method,
by corrosives, which in Hell are salutary and medicinal, melting
apparent surfaces away, and displaying the infinite which was hid.
*
*If the doors of perception were cleansed every thing would appear to
man as it is, infinite. For man has closed himself up, till he sees all
things tho' narrow chinks of his cavern...
*
*By degrees we beheld the infinite Abyss, fiery as the smoke of a
burning city; beneath us at an immense distance, was the sun, black
but shining; round it were fiery tracks on which revolv'd vast spiders,
crawling after their prey; which flew, or rather swum, in the infinite deep,
in the most terrific shapes of animals sprung from corruption; and the
air was full of them, and seem'd composed of them: these are Devils,
and are called Powers of the air. I now asked my companion which was my
eternal lot? He said, 'between the black and white spiders.'....
*
*Opposition is true Friendship. . .""*
J.-F. Burnol,
"The Explicit Formula
and a Propagator" (preprint, 1998)
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