### a selection of quotations
When reading these, take note of the profusion of emotional/poetic/ecstatic and religiously-oriented language
which is used throughout. The words *mystery, mysterious* and *secrets* appear numerous times,
but also *sense of wonder, strange, stunning, astonishing, baffling, bafflement, devilment, surprise, endless surprises, exasperating, perplexing, bedevilled, diabolical malice, teased, cruel and compelling,
stultifying, fascinating, (strange) fascination, yearning, obsession, mysterious attraction, breathtaking(ly), beautiful, most beautiful, incredibly beautiful, immense beauty, beautiful harmonies,
elegant, elegance, gorgeous, glamorous, enthralled, incredible, exalted, majestic, fantastic, miraculous, amazed, amazing, absolutely amazing, awed, impenetrable, impenetrability, tantalized, tantalizing,
tantalizingly, tantalizingly vulnerable, unveil, blazed...fearlessly, wreath its conqueror with glory, awesome vista, most ancient, cryptic, riddle, formidable enigmas, most enigmatic, strange conundrum, great white whale, quest, vast toil,
unthinkable complexity, utterly alien, secret source, profundity, profound mystery, great mystery, magic, alchemist, elixir, aesthetic appeal, works of art, poetry,
arcane music, secret harmony, Nature's gift, inexplicable secrets of creation, gem, gemstone, jewels, crown, heart, soul, cosmos, abyss(es),
divine, Holy Grail, Lucifer, Devil* and *God*.
William Blake or
John Milton might
feel at home with this. Mathematicians, however, are not ordinarily inclined to use such language so freely.
It is hard not to wonder what it is we are ultimately dealing with here.
*"...there is no apparent reason why one number is prime and another not. To the contrary, upon looking at these numbers one has the feeling of being in the presence of one of the inexplicable secrets of creation."*
D. Zagier from
"The first 50 million prime numbers", *The Mathematical Intelligencer*
**0** (1977) 8
*"I hope that...I have communicated a certain impression of the immense beauty of the prime numbers and the endless surprises which they have in store for us."*
D. Zagier from
"The first 50 million prime numbers", *The Mathematical Intelligencer*
**0** (1977) 16
*"As archetypes of our representation of the world, numbers form, in
the strongest sense, part of ourselves, to such an extent that it can
legitimately be asked whether the subject of study of arithmetic is not the
human mind itself. From this a strange fascination arises: how can it be
that these numbers, which lie so deeply within ourselves, also give rise
to such formidable enigmas? Among all these mysteries, that of the prime
numbers is undoubtedly the most ancient and most resistant."*
G. Tenenbaum and M. Mendès France, from
*The Prime
Numbers and Their Distribution* (AMS, 2000) page 1
*"The mystery that clings to numbers, the magic
of numbers, may spring from this very fact, that the intellect, in the
form of the number series, creates an infinite manifold of well distinguishable
individuals. Even we enlightened scientists can still feel it *e.g.* in the
impenetrable law of the distribution of prime numbers."*
H. Weyl from *Philosophy of Mathematics and Natural Science* (1927)
*"The theory of Numbers has always been regarded as one of the most obviously
useless branches of Pure Mathematics. The accusation is one against which there is no valid defence;
and it is never more just than when directed against the parts of the theory which are more
particularly concerned with primes. A science is said to be useful if its development tends to
accentuate the existing inequalities in the distribution of wealth, or more directly promotes the
destruction of human life. The theory of prime numbers satisfies no such criteria. Those who pursue it
will, if they are wise, make no attempt to justify their interest in a subject so trivial and so remote, and
will console themselves with the thought that the greatest mathematicians of all ages have found it in it
a mysterious attraction impossible to resist."*
G. H. Hardy from a 1915 lecture on prime numbers
*"To some extent the beauty of number theory seems to be related to the
contradiction between the simplicity of the integers and the complicated
structure of the primes, their building blocks. This has always attracted
people."*
A. Knauf from
"Number theory, dynamical systems and statistical
mechanics" (1998 lecture notes)
*"Prime numbers are the most basic objects in mathematics. They also
are among the most mysterious, for after centuries of study, the structure
of the set of prime numbers is still not well understood. Describing the
distribution of primes is at the heart of much mathematics..."*
A. Granville from
AMS press release, 5
December 1997
*"Mathematicians have tried in vain to this day to discover some order
in the sequence of prime numbers, and we have reason to believe that it is a
mystery into which the mind will never penetrate."*
Leonard Euler, in G. Simmons, *Calculus Gems*, McGraw-Hill, New York,
1992
*"...the prolific and immensely influential master mathematician Leonhard Euler
(1707-1783) expressed in 1751 his bafflement about the impenetrability of the primeland
thicket:
*
*. . . **
*Since primes are the basic building blocks of the number universe from which all the
other natural numbers are composed, each in its own unique combination, the perceived lack
of order among them looked like a perplexing discrepancy in the otherwise so rigorously
organized structure of the mathematical world.
. . .
**How can so much of the formal and systematic edifice of mathematics, the science of
pattern and rule and order ***per se*, rest on such a patternless, unruly, and
disorderly foundation? Or how can numbers regulate so many aspects of our physical world
and let us predict some of them when they themselves are so unpredictable and appear to
be governed by nothing but chance?"
H. Peter Aleff, from the 'e-book' *Prime Passages to Paradise*
*"The seeming absence of any ascertained organizing principle in the distribution
or the succession of the primes had bedevilled mathematicians for centuries and
given Number Theory much of its fascination. Here was a great mystery indeed, worthy
of the most exalted intelligence: since the primes are the bulding blocks of the
integers and the integers the basis of our logical understanding of the cosmos, how
is it possible that their form is not determined by law? Why isn't 'divine geometry'
apparent in their case?"*
A. Doxiadis, from the novel *Uncle Petros and Goldbach's Conjecture*,
p. 84 (Faber 2000)
*"The problem of distinguishing prime numbers from composite numbers and of resolving
the latter into their prime factors is known to be one of the most important and useful in
arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such
an extent that it would be superfluous to discuss the problem at length...Further, the dignity
of the science itself seems to require that every possible means be explored for the solution
of a problem so elegant and so celebrated."*
C.F. Gauss,
*Disquisitiones Arithmeticae*, article 329 (1801)
*"Prime numbers have always fascinated mathematicians, professional and amateur alike. They appear
among the integers, seemingly at random, and yet not quite: there seems to be some order or pattern, just a little below the surface, just a little
out of reach."*
Underwood Dudley, *Elementary Number Theory*
(Freeman, 1978) p.163
*"The primes have tantalized mathematicians since the Greeks, because they appear to be somewhat randomly distributed but not completely so."*
T. Gowers, *Mathematics: A Very Short Introduction* (Oxford University Press, 2002), p.118
* "Who would have imagined that something as straightforward as the natural numbers
(1, 2, 3, 4,...) could give birth to anything so baffling as the prime numbers (2, 3 ,5, 7, 11, ...)?"*
Ian Stewart, "Jumping Champions", *Scientific American*, December 2000
*
"Prime numbers belong to an exclusive world of intellectual conceptions. We speak of those marvellous
notions that enjoy simple, elegant description, yet lead to extreme - one might say unthinkable - complexity
in the details. The basic notion of primality can be accessible to a child, yet no human mind harbors anything
like a complete picture. In modern times, while theoreticians continue to grapple with the profundity of the
prime numbers, vast toil and resources have been directed toward the computational aspect, the task of
finding, characterizing, and applying the primes in other domains."*
R. Crandall and C. Pomerance,
*Prime
Numbers: A Computational Perspective* (Springer-Verlag, 2001)
*"[Primes] are full of surprises and very mysterious...They are like things
you can touch...In mathematics most things are abstract, but I have some feeling that
I can touch the primes, as if they are made of a really physical material. To me,
the integers as a whole are like physical particles."*
Y. Motohashi, quoted in K. Sabbagh's
*Dr.
Riemann's Zeros* (Atlantic, 2002), p.17
*"I sometimes have the feeling that the number system is comparable
with the universe that the astronomer is studying...The number system
is something like a cosmos."*
M. Jutila, quoted in K. Sabbagh, "Beautiful Maths", *Prospect*, January 2002.
*"The prime numbers are useful in analyzing problems concerning
divisibility, and also are interesting in themselves because of some
of the special properties which they possess as a class. These properties
have fascinated mathematicians and others since ancient times, and the
richness and beauty of the results of research in this field have been
astonishing."*
C.H. Denbow and V. Goedicke, *Foundations of
Mathematics* (Harper, 1959)
*"No branch of number theory is more saturated with
mystery than the study of prime numbers: those exasperating, unruly integers
that refuse to be divided evenly by any integers except themselves and 1. Some
problems concerning primes are so simple that a child can understand them and
yet so deep and far from solved that many mathematicians now suspect they
*have* no solution. Perhaps they are "undecideable". Perhaps number
theory, like quantum mechanics, has its own uncertainty principle that makes it
necessary, in certain areas, to abandon exactness for
probabilistic formulations."*
M. Gardner from "The remarkable lore of the prime numbers",
*Scientific American*, March 1964.
*"317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but *because it is so*, because mathematical reality is built that way."
*
G.H. Hardy, *A Mathematician's Apology*, (Cambridge University Press, 1940) p. 70.
*"There are two facts about the distribution of prime numbers which I
hope to convince you so overwhelmingly that they will be permanently engraved
in your hearts.*
*The first is that despite their simple definition and role as the
building blocks of the natural numbers, the prime numbers... grow like weeds
among the natural numbers, seeming to obey no other law than that of chance,
and nobody can predict where the next one will sprout.*
*The second fact is even more astonishing, for it states just the
opposite: that the prime numbers exhibit stunning regularity, that there are
laws governing their behaviour, and that they obey these laws with almost
military precision."*
D. Zagier from
"The first 50 million prime numbers", *The Mathematical Intelligencer*
**0** (1977) 7
*"To me, that the distribution of prime numbers
can be so accurately represented in a harmonic analysis is absolutely
amazing and incredibly beautiful. It tells of an arcane music and a
secret harmony composed by the prime numbers."*
E. Bombieri from "Prime
Territory" (*The Sciences*, Sept./Oct. 1992)
*"Addition and multiplication equip the set of positive natural numbers *{*1,2,3,. . .*}* with
a double structure of Abelian semigroup. The first is associated with a total order relation, and is
generated by the single number 1. The second, reflecting the partial order
of divisibility has an infinite number of generators: the prime numbers.
Defined since antiquity, this key concept has yet to deliver up all its
secrets – and there are plenty of them."*
G. Tenenbaum, *Introduction to Analytic and
Probabilistic Number Theory*, page 299
*"Already in this picture we can see that, despite small oscillations, *[*the number of primes*]*
grows quite regularly*
*But when I extend the domain of x values from a
hundred to fifty thousand, then this regularity becomes breathtakingly
clear, for the graph now looks like this*
*For me, the smoothness with which this curve climbs is one of
the most astonishing facts in mathematics."*
D. Zagier from
"The first 50 million prime numbers", *The Mathematical Intelligencer*
**0** (1977) 7–19.
*"Some order begins to emerge from this chaos when the primes are considered not
in their individuality but in the aggregate; one considers the social statistics of the
primes and not the eccentricities of the individuals."*
P.J. Davis and R. Hersh,
*The Mathematical Experience*, Chapter 5
*"It [is] possible to predict with rather good accuracy the number of primes smaller than
*N* (especially when *N* is large); on the other hand, the distribution of primes in short intervals
shows a kind of built-in randomness. This combination of 'randomness' and 'predictability'
yields at the same time an orderly arrangement and an element of surprise in the distribution of
primes. According to Schroeder (1984), in his intriguing book *Number Theory in Science and
Communication*, these are basic ingredients of works of art. Many mathematicians will readily
agree that this topic has a great aesthetic appeal."*
P. Ribenboim,The Book of Prime Number Records, 2nd ed. (Springer-Verlag, 1989), p.153
*"*[Tschebycheff]
*was the only man ever able to cope with the
refractory character and erratic flow of prime numbers and to confine the
stream of their progression with algebraic limits, building up, if I may so say,
banks on either side which that stream, devious and irregular as are its windings,
can never overflow."*
J.J.
Sylvester, quoted in E. Kramer, *The Nature and
Growth of Mathematics* (Princeton University Press, 1970)
*"The zeta
function is probably the most challenging and mysterious object of
modern mathematics, in spite of its utter simplicity. . . The main
interest comes from trying to improve the Prime Number
Theorem, i.e. getting better estimates for the distribution of the
prime numbers. The secret to the success is assumed to lie in proving a conjecture
which Riemann stated in 1859 without much fanfare, and whose proof has
since then become the single most desirable achievement for a
mathematician."*
M.C. Gutzwiller, *Chaos in Classical
and Quantum Mechanics*, page 308
*"Let us now pursue an apparently tangential path. We wish to consider one of the most
fascinating and glamorous functions of analysis, the Riemann zeta function..."*
R. Bellman, *A Brief Introduction of Theta Functions* (Holt, 1961) p. 30
*"We may - paraphrasing the famous sentence of
George Orwell - say that "all mathematics is beautiful, yet some is more
beautiful than the other." But the most beautiful in all mathematics is
the zeta function. There is no doubt about it."*
Polish cosmologist Krzysztof
Maslanka
*"It's remarkable how the Riemann zeta function
seems to be trying *intentionally* to deceive us!"*
Warren D. Smith, "Cruel and unusual behavior of the Riemann zeta
function"
*"These ideas are then utilized to unveil a new image of the zeta-function...
revealing it as the main gem of a necklace composed of all automorphic *L*-functions."
*
From Cambridge University Press description
of Y. Motohashi's *Spectral Theory of the Riemann-Zeta Function*
*"It's a whole beautiful subject and the Riemann zeta function is just the first one
of these, but it's just the tip of the iceberg. They are just the most amazing objects, these
*L*-functions - the fact that they exist, and have these incredible properties are tied up with
all these arithmetical things - and it's just a beautiful subject. Discovering these things is
like discovering a gemstone or something. You're amazed that this thing exists, has these
properties and can do this."
*
B. Conrey, quoted in K. Sabbagh's *Dr.
Riemann's Zeros* (Atlantic, 2002), p.166
*"At this point, it is not possible to remain silent on what is probably the most
intriguing unsolved problem in the theory of the zeta function and actually in all of number
theory - and most likely even one of the most important unsolved problems in contemporary
mathematics, namely the famous Riemann hypothesis...Still, the problem is open and
fascinates and teases the best contemporary minds."
*
E. Grosswald, *Topics in the Theory of Numbers* (MacMillan, 1966) p.137
*"It has long been (and still is in some quarters) the fashion to think of the primes as somehow lawless and devilish. It was indeed
this very thought that stopped Littlewood, in 1907, from continuing his attempt to prove Riemann's conjecture. He...gave up after six days
in the...belief that the 'devilment' in the primes would make his task impossible."*
George Spencer-Brown, 2006
* "In 1859, a German mathematician called Bernhard Riemann, a 'timid diffident soul with a horror of
attracting attention to himself,' published a paper that drew more attention to him than to almost any other
mathematician in the 19th century,. In it he made an important statement: *the non-trivial zeros of the Riemann
zeta function all have real part equal to 1/2*. That is the Riemann Hypothesis: 15 words encapsulating a
mystery at the heart of our number system."
*
K. Sabbagh, "Beautiful Maths", *Prospect*, January 2002
*"Riemann showed the importance of study of [the zeta] function for a range of
problems in number theory centering around the distribution of prime numbers, and he
further demonstrated that many of these problems could be settled if one knew the location
of the zeros of this function. In spite of continued assaults and much progress since
Riemann's initial investigations this tantalizing question remains one of the major
unsolved problems in mathematics." *
D. Reed, *Figures of Thought* (Routledge, New York, 1995) p.123
*"Hilbert
included the problem of proving the Riemann
hypothesis in his list of the most important unsolved problems which
confronted mathematics in 1900, and the attempt to solve this problem has
occupied the best efforts of many of the best mathematicians of the
twentieth century. It is now unquestionably the most celebrated problem in
mathematics and it continues to attract the attention of the best
mathematicians, not only because it has gone unsolved for so long but also
because it appears tantalizingly vulnerable and because its solution would
probably bring to light new techniques of far-reaching
importance."*
H.M. Edwards, *Riemann's Zeta Function*,
page 6
*"In *[his
1859 paper]*, Riemann made an incidental remark - a guess, a hypothesis.
What he tossed out to the assembled mathematicians that day has proven to be
almost cruelly compelling to countless scholars in the ensuing years...
*
*...it is that incidental remark - the Riemann Hypothesis - that is
the truly astonishing legacy of his 1859 paper. Because Riemann
was able to see beyond the pattern of the primes to discern traces
of something mysterious and mathematically elegant at work - subtle
variations in the distribution of those prime numbers. Brilliant
for its clarity, astounding for its potential consequences, the
Hypothesis took on enormous importance in mathematics. Indeed, the
successful solution to this puzzle would herald a revolution in
prime number theory. Proving or disproving it became the greatest
challenge of the age...
*
*It has become clear that the Riemann Hypothesis, whose resolution
seems to hang tantalizingly just beyond our grasp holds the key to a
variety of scientific and mathematical investigations. The making and breaking of modern codes,
which depend on the properties of the prime numbers, have roots in the
Hypothesis. In a series of extraordinary developments during the 1970s, it
emerged that even the physics of the atomic nucleus is connected in ways not yet
fully understood to this strange conundrum. ...Hunting down
the solution to the Riemann Hypothesis has become an obsession for
many - the veritable 'great white whale' of mathematical research.
Yet despite determined efforts by
generations of mathematicians, the Riemann Hypothesis defies resolution.""*
J. Derbyshire, from the dustjacket description of
*Prime Obsession* (John Henry Press, 2003)
*"Whoever proves or disproves* [the Riemann Hypothesis] *will cover himself in glory..."
*
Eric Temple Bell, 1937
*"The Riemann hypothesis...is still widely considered to be one of the greatest unsolved problems in mathematics, sure to wreath its conqueror with glory."*
B. Schechter, "143-year-old problem still has mathematicians guessing" (*New York Times*, 2 July 2002).
*"So if you could be the Devil and offer a mathematician to sell his soul for the proof
of one theorem - what theorem would most mathematicians ask for? I think it would be the
Riemann Hypothesis."*
H. Montgomery, quoted in K. Sabbagh's
*Dr.
Riemann's Zeros* (Atlantic, 2002), p.29
*"...the Riemann hypothesis remains one of the outstanding challenges of
mathematics, a prize which has tantalized and eluded some of the most brilliant
mathematicians of this century...Hilbert is reputed to have said that the first
comment he would make after waking at the end of a thousand year sleep would be,
'Is the Riemann hypothesis established yet?'"*
R. Bellman, *A Brief Introduction of Theta Functions* (Holt, 1961) p. 33-34
*"The Riemann Hypothesis is a precise statement, and in one sense what it means is
clear, but what it's connected with, what it implies, where it comes from, can be very
unobvious."
*
M. Huxley, quoted in K. Sabbagh's
*Dr.
Riemann's Zeros* (Atlantic, 2002), p.186
*"The Riemann Hypothesis is the central problem and it implies many, many things. One thing that makes
it rather unusual in mathematics today is that there must be over five hundred papers -
somebody should go and count - which start "Assume the Riemann Hypothesis", and the conclusion is fantastic.
And those [conclusions] would then become theorems...With this one solution you would have
proven five hundred theorems or more at once."
*
P. Sarnak, quoted in K. Sabbagh's
*Dr.
Riemann's Zeros* (Atlantic, 2002), p.188
*"Sometimes I think that we essentially have a complete proof of the Riemann Hypothesis except for a
gap. The problem is, the gap occurs right at the beginning, and so it's hard to fill that gap
because you don't see what's on the other side of it."
*
H. Montgomery, quoted in K. Sabbagh's
*Dr.
Riemann's Zeros* (Atlantic, 2002), p.227
*"[The Riemann Hypothesis is] no longer just analytic number theorists involved, but
all mathematicians know about the problem, and many realize that they may have useful insights
to offer. As far as I can see, a solution is as likely to come from a probabilist,
geometer or mathematical physicist, as from a number theorist."
*
R. Heath-Brown, quoted in K. Sabbagh's
*Dr.
Riemann's Zeros* (Atlantic, 2002), p.228
*"...the Riemann Hypothesis will be settled without any fundamental changes in our
mathematical thoughts, namely, all tools are ready to attack it but just a penetrating idea is
missing."
*
Y. Motohashi, quoted in K. Sabbagh's
*Dr.
Riemann's Zeros* (Atlantic, 2002), p.228
*"Mother Nature has such beautiful harmonies, so you couldn't say that something like
[the Riemann Hypothesis] is false."*
H. Iwaniec, quoted in K. Sabbagh's
*Dr.
Riemann's Zeros* (Atlantic, 2002), p.288
*"...I don't believe or disbelieve the Riemann Hypothesis. I have a certain amount of
data and a certain amount of facts. These facts tell me definitely that the thing has not been
settled. Until it's been settled it's a hypothesis, that's all. I would like the Riemann Hypothesis
to be true, like any decent mathematician, because it's a thing of beauty, a thing of elegance,
a thing that would simplify many proofs and so forth, but that's all."*
A. Ivic, quoted in K. Sabbagh's
*Dr.
Riemann's Zeros* (Atlantic, 2002), p.228
*"The consequences [of the Riemann Hypothesis] are fantastic: the distribution of primes, these
elementary objects of arithmetic. And to have tools to study the distribution of these of objects."
*
H. Iwaniec, quoted in K. Sabbagh's
*Dr.
Riemann's Zeros* (Atlantic, 2002), p.30
*"If [the Riemann Hypothesis is] not true, then the world is a very different place.
The whole structure of integers and prime numbers would be very different to what we could
imagine. In a way, it would be more interesting if it were false, but it would be a disaster
because we've built so much round assuming its truth."
*
P. Sarnak, quoted in K. Sabbagh's
*Dr.
Riemann's Zeros* (Atlantic, 2002), p.30
*"In 1985 there was a flurry of publicity for an announced proof
of the Riemann Hypothesis...This announcement was premature and the zeta
function retains its secrets. Fame, fortune and many sleepless nights await
whoever uncovers them." *
I. Stewart, *The Problems of Mathematics* (1987) p.164
*"Now, fifty years after the publication of Riemann's great
paper "On the number of prime numbers less than a given quantity",
we have only just begun to understand and absorb what Riemann's
supremely creative imagination produced. Progress along the path that
Riemann blazed so fearlessly has been hesitant and slow; and the
justly famous hypothesis that lies at the kernel of that thesis has
resisted all efforts at proof."*
[Allegedly, E.
Landau, 1909, although I have been unable to find any confimation of this. If you know anything about the origin of this quotation, please get in touch.]
*"Ask any professional mathematician what the single most
important open problem in the entire field is and you are almost certain
to receive the answer "the Riemann Hypothesis"."*
K. Devlin, from *Mathematics: The New Golden Age* (1999)
*"It remains unresolved but, if true, the Riemann
Hypothesis will go to the heart of what makes so much of mathematics tick:
the prime numbers. These indivisible numbers are the atoms of arithmetic.
Every number can be built by multiplying prime numbers together. The
primes have fascinated generations of mathematicians and
non-mathematicians alike, yet their properties remain deeply mysterious.
Whoever proves or disproves the Riemann Hypothesis will discover the key
to many of their secrets and this is why it ranks above Fermat as the
theorem for whose proof mathematicians would trade their soul with
Mephistopheles.*
*Although the Riemann Hypothesis has never quite
caught on in the public imagination as Mathematics' Holy Grail, prime
numbers themselves do periodically make headline news...But for
mathematicians, such news is of only passing interest...Rather
mathematicians like to look for patterns, and the primes probably offer
the ultimate challenge. When you look at a list of them stretching off to
infinity, they look chaotic, like weeds growing through an expanse of
grass representing all numbers. For centuries mathematicians have striven
to find rhyme and reason amongst this jumble. Is there any music that we
can hear in this random noise? Is there a fast way to spot that a
particular number is prime? Once you have one prime, how much further must
you count before you find the next one on the list? These are the sort of
questions that have tantalised generations."*
M. du
Sautoy, "The Music
of the Primes", *Science Spectra* **11**
(1998)
* "For many mathematicians working on it, $1m
is less important than the satisfaction that would come from finding a proof. Throughout my researches among
the mathematicians' tribe (I have interviewed 30 in the past year), Riemann's Hypotheis was often described
to me in awed terms. Hugh Montgomery of the University of Michigan said this was the proof for which a
mathematician might sell his soul. Henryk Iwaniec, a Polish-American mathematician, sounded as if he were
already discussing terms with Lucifer"
* *'I would trade everything I know in mathematics for the proof
of the Riemann Hypothesis. It's gorgeous stuff. I'm only worried
that I'll be unable to understand it. That would be the worst...'"
*
K. Sabbagh, "Beautiful Mathematics", *Prospect*,
January 2002.
*"The failure of the Riemann hypothesis would
create havoc in the distribution of prime numbers. This fact alone
singles out the Riemann hypothesis as the main open question of prime
number theory."*
E. Bombieri, from "Prime Territory" (*The Sciences*, Sept./Oct. 1992)
*". . .there have been very few attempts at
proving the Riemann hypothesis, because, simply, no one has ever had any
really good idea for how to go about it."*
Atle
Selberg
*"Despite the stunning advances linking
Riemann's zeta function to 20th century physics, no one is predicting
an imminent proof of the Riemann hypothesis. Odlyzko's
numerical experiments and evidence amassed by physicists have convinced
everyone that a spectral
interpretation of the zeta zeros is the way to go, but number
theorists say they are at least one "big idea" away from even the
beginnings of a proof. Mathematicians aren't yet sure what to aim at, says
[Princeton University mathematician Peter] Sarnak"*
Barry Cipra, "A Prime Case of Chaos"
*"Indicative of the depth of mathematics lurking behind physicists'
conjectures is that fact that the properties that one would like to
establish about the renormalization theory of critical circle maps
might turn out to be related to number-theoretic abysses such as
the Riemann conjecture..."*
P. Cvitanovic,
"Circle Maps: Irrationally Winding" from *Number Theory and Physics*,
*eds.* C. Itzykson, *et. al.* (Springer, 1992)
*"...in one of those unexpected connections that make theoretical physics so delightful, the quantum chaology
of spectra turns out to be deeply connected to the arithmetic of prime numbers, through the celebrated zeros of the
Riemann zeta function: the zeros mimic quantum energy levels of a classically chaotic system. The connections is not only
deep but also tantalizing, since its basis is still obscure - though it has been fruitful for both mathematics and physics."*
M. V. Berry from "Chaos and the semiclassical limit of Quantum mechanics
(is the moon there when somebody looks?)",
in *Quantum Mechanics: Scientific perspectives on divine action* (eds: R. J. Russell, P. Clayton, K. Wegter-McNelly
and J. Polkinghorne), Vatican Observatory CTNS publications, pp 41-54 (2001).
*"If the Riemann Hypothesis is true. . .the function f(u) constructed from the primes
has discrete spectrum; that is, the support of its Fourier transform is
discrete. If the Riemann Hypothesis is false this is not the case. The
frequencies t*_{n} are reminiscent of the decomposition of a
musical sound into its consituent harmonics. Therefore there is a sense in
which we can give a one-line non technical statement of the Riemann
hypothesis: "The primes have music in them".
M.V.Berry and J.P.Keating from "The Riemann
Zeros and Eigenvalue Asymptotics" (*SIAM Review* **41**,
no.2 (1999), page238.)
*"Berry isn't speaking in metaphors. "I've
tried to play this music by putting a few thousand primes into my
computer," he says "but it's just a horrible cacophony. You'd actually
need billions or trillions someone with a more powerful machine should
do it.""*
E. Klarreich from "Prime Time" (*New
Scientist*, 11/11/00)
*"*[It has been]* said that the zeros *[of the Riemann zeta function]* weren't real, nobody measured them. They are
as real as anything you will measure in a laboratory - this has to be the way we look at
the world."*
P. Sarnak from 1999 MSRI lecture
"Random matrix
theory and zeroes of zeta functions - a survey"
*"I am firmly convinced that the most important
unsolved problem in mathematics today is the truth or falsity of a
conjecture about the zeros of the zeta function, which was first made by
Riemann himself...Even a single exception to Riemann's conjecture would
have enormously strange consequences for the distribution of prime
numbers...If the Riemann hypothesis turns out to be false, there will be
huge oscillations in the distribution of primes. In an orchestra, that
would be like one loud instrument that drowns out the others - an
aesthetically distasteful situation."*
E. Bombieri from
"Prime Territory" (*The Sciences*, Sept./Oct. 1992)
*"Right now, when we tackle problems without knowing the truth of the
Riemann hypothesis, it's as if we have a screwdriver. But when we have it,
it'll be more like a bulldozer."*
P. Sarnak, quoted in
"Prime Time" by E. Klarreich (*New Scientist*,
11/11/00)
*"I have a feeling that the *[*Riemann*]* hypothesis will be cracked in the next few
years. I see the strands coming together. Someone will soon get
the million
dollars."*
M. Berry, quoted in
"Prime Time" by E. Klarreich (*New Scientist*, 11/11/00)
*"Proving the Riemann hypothesis won't end the story. It will
prompt a sequence of even harder, more penetrating questions. Why do
the primes achieve such a delicate balance between randomness and
order? And if their patterns do encode the
behaviour of quantum chaotic systems, what other jewels will we
uncover when we dig deeper?
*
*Those who believe mathematics holds the key to the Universe might
do well to ponder a question that goes back to the ancients: What
secrets are locked within the primes?"
*
E. Klarreich,
"Prime Time" (*New Scientist*, 11/11/00)
*"One of the remarkable aspects of the distribution of prime numbers
is their tendency to exhibit global regularity and local irregularity. The
prime numbers behave like the 'ideal gases' which physicists are so fond of.
Considered from an external point of view, the distribution is - in broad
terms - deterministic, but as soon as we try to describe the situation at a
given point, statistical fluctuations occur as in a game of chance where it
is known that on average the heads will match the tail but where, at any one
moment, the next throw cannot be predicted. Prime numbers try to occupy all
the room available (meaning that they behave as randomly as possible), given
that they need to be compatible with the drastic constraint imposed on them,
namely to generate the ultra-regular sequence of integers.
*
*This idea underpins the majority of conjectures concerning prime numbers:
everything which is not trivially forbidden should actually happen..."*
G. Tenenbaum and M. Mendèfs France, from
*The Prime
Numbers and Their Distribution* (AMS, 2000) page 51
*"It is often remarked that prime numbers finally found a legitimate practical
application in the domain of cryptography. The cryptographic relevance is not disputed,
but there are many other applications of the majestic primes...It seems fair to regard the
prime number concept as ubiquitous, since the primes appear in so many disparate domains
of thought."*
R. Crandall and C. Pomerance, from
*Prime
Numbers: A Computational Perspective* (Springer-Verlag, 2001)
*"Euclid may have been the first to define primality in his Elements approximately
300 BC...He realized that the even perfect numbers are all closely related to the primes
of the form 2*^{p} – 1 for some prime *p* (now called Mersennes). So the
quest for these jewels began near 300 BC."
C. Caldwell, from The Prime
Pages
*"I have sometimes thought that the profound
mystery which envelops our conceptions relative to prime numbers depends
upon the limitations
of our faculties in regard to time, which like space may be in essence
poly-dimensional and that this and other such sort sort of truths
would become self-evident to a being whose mode of perception is according
to *superficially* as opposed to our own limitation to *linearly* extended
time."*
J.J.
Sylvester, from "On certain inequalities relating to prime numbers", *Nature* **38** (1888) 259-262,
and reproduced in *Collected Mathematical Papers*, Volume 4, page
600 (Chelsea, New York, 1973)
*"Although the prime numbers are rigidly determined, they somehow feel like experimental data."*
T. Gowers, *Mathematics: A Very Short Introduction* (Oxford Univ. Press, 2002), p.121
*"It is evident that the primes are randomly distributed but,
unfortunately, we don't know what 'random' means.''*
R. C. Vaughan (February 1990)
*"God may not play dice with the universe, but
something strange is going on with the prime numbers."*
C. Pomerance, suggesting something that P. Erdös *might have* said,
referring to the famous quote of Einstein. From
"Homage to an Itinerant Master" by D. Mackenzie (*Science* 275:759, 1997) (this mistakenly attributes the quote to Erdös
himself)
*"Given the millennia that people have contemplated prime numbers, our continuing ignorance concerning the primes is stultifying."*
R. Crandall and C. Pomerance, from
*Prime
Numbers: A Computational Perspective* (Springer-Verlag, 2001)
Oxford University professor of mathematics Marcus du Sautoy's recent book *The Music of the Primes* (Fourth Estate,
2003) contains a wealth of relevant quotes. He expressed an interest in the quotes collected here when
in the process of writing the book. His book has been widely acclaimed, and his prose has been
praised for its 'poetic' qualities. The choice of language noticeably contrasts with the somewhat
drier styles of K. Sabbagh and J. Derbyshire, both of whom also had popular books on the Riemann
Hypothesis published in early 2003. Some of the words and phrases used in the quotes below are:
* music (repeatedly), sense of wonder, timeless, Nature's gift, ultimate challenge, secret source,
inner harmony, elixir, metamorphosis, miraculous, stunning, yearning, enthralled, frightened, fascinated
and teased, embarrassment, anathema, misty waters, vast ocean, vast expanse, awesome vista,
looming out of the mist, utterly alien, unleash the full force, radically new vistas, hidden harmonies,
cacophony, poetry, alchemist, treasure, jewels, crown, riddle, cryptic, most enigmatic,
mystical ley line, masters of disguise, diabolical malice*
*
*
* "The primes are jewels studded throughout the vast expanse of the infinite universe of numbers
that mathematicians have explored down the centuries. For mathematicians they instil a sense of
wonder: 2, 3, 5, 7, 11, 13, 17, 19, 23,... - timeless numbers that exist in the same world independent of
our physical reality. They are Nature's gift to the mathematician." **(p.5)***
**
** "Yet despite their apparent simplicity and fundamental character, prime numbers remain the
most mysterious objects studied by mathematicians. In a subject dedicated to finding patterns and
order, the primes offer the ultimate challenge." **(p.5)**
**
** "Prime numbers present mathematicians with one of the strangest tensions in their subject.
On the one hand a number is either prime or it isn't. No flip of a coin will suddenly make a number
divisible by some smaller number. Yet there is no denying that the list of primes looks like a
randomly chosen sequence of numbers. Physicists have grown used to the idea that a quantum die
decides the fate of the universe, randomly choosing at each throw where scientists will find matter.
But it is something of an embarrassment to have to admit that these fundamental numbers on which
mathematics is based appear to have been laid out by Nature flipping a coin, deciding at each toss the
fate of each number. Randomness and chaos are anathema to the mathematician.
Despite their randomness, prime numbers - more than any other part of our mathematical heritage -
have a timeless, universal character. Prime numbers would be there regardless of whether we had
evolved sufficiently to recognise them." **(p.6)**
**
** "Riemann's insight followed his discovery of a mathematical looking-glass through which he
could gaze at the primes. Alice's world was turned upside down when she stepped through her
looking-glass. In contrast, in the strange mathematical world beyond Riemann's glass, the chaos of
the primes seemed to be transformed into an ordered pattern as strong as any mathematician could
hope for. He conjectured that this order would be maintained however far one stared into the never-ending
world beyond the glass. His prediction of an inner harmony on the far side of the mirror would explain why
outwardly the primes look so chaotic. The metamorphosis provided by Riemann's mirror, where chaos
turns to order, is one which most mathematicians find almost miraculous. The challenge that Riemann
left the mathematical world was to prove that the order he thought he could discern was really there."
**(p.9)**
**
** "As mathematicians navigate their way across the mathematical terrain, it as though all paths
will necessarily lead at some point to the same awesome vista of the Riemann Hypothesis."
**(p.10)**
**
** "The search for the secret source that fed the primes had been going on for over two millennia.
The yearning for this elixir had made mathematicians all too susceptible to Bombieri's ****[April Fools announcement of a "proof" of the Riemann Hypothesis in 1997]. ***For years, many
had simply been too frightened to go anywhere near this notoriously difficult problem."
**(p.13)***
**
** "A solution to the Riemann Hypothesis offers the prospect of charting the misty waters of the
vast ocean of numbers. It represents just a beginning in our understanding of Nature's numbers. If we
can only find the secret of how to navigate the primes, who knows what else lies out there, waiting for
us to discover?"
**(p.18)**
**
** "...Gauss liked to call ****[number theory]*** 'the Queen of Mathematics'. For Gauss, the jewels in
the crown were the primes, numbers which had fascinated and teased generations of mathematicians."
**(p.22)***
**
** "It seems paradoxical that the fundamental objects on which we build our order-filled world of
mathematics should behave so wildly and unpredictably."
**(p.45)**
**
** "Armed with his prime number tables, Gauss began his quest. As he looked at the proportion
of numbers that were prime, he found that when he counted higher and higher a pattern started to
emerge. Despite the randomness of these numbers, a stunning regularity seemed to be looming out
of the mist."
**(p.47)**
**
** "The revelation that the graph appears to climb so smoothly, even though the primes themselves
are so unpredictable, is one of the most miraculous in mathematics and represents one of the high points
in the story of the primes. On the back page of his book of logarithms, Gauss recorded the discovery
of his formula for the number of primes up to ****N*** in terms of the logarithm function. Yet despite the
importance of the discovery, Gauss told no one what he had found. The most the world heard of his
revelation were the cryptic words, 'You have no idea how much poetry there is in a table of logarithms.'"
**(p.50)***
**
** "As we shall see, Riemann's Hypothesis can be interpreted as an example of a general
philosophy among mathematicians that, given a choice between an ugly world and an aesthetic
one, Nature always chooses the latter." **(p.55)**
**
** "Gauss had heard the first big theme in the music of the primes, but it was one of his students,
Riemann, who would truly unleash the full force hidden of the hidden harmonies that lay behind the
cacophony of the primes." **(p.58)**
**
** "Riemann had found a passageway from the familiar world of numbers into a mathematics
which would have seemed utterly alien to the Greeks who had studied prime numbers two thousand
years before. He had innocently mixed imaginary numbers with his zeta function and discovered,
like some mathematical alchemist, the mathematical treasure emerging from this admixture of elements
that generations had been searching for. He had crammed his ideas into a ten-page paper, but
was fully aware that his ideas would open up radically new vistas on the primes."
**(p.58)**
**
** "For centuries, mathematicians had been listening to the primes and hearing only disorganised
noise. These numbers were like random notes wildly dotted on a mathematical stave with no discernible
tune. Now Riemann had found new ears with which to listen to these mysterious tones. The sine-like
waves that Riemann had created from the zeros in his zeta landscape revealed some hidden harmonic
structure." **(p.93)**
**
** "These zeros did not appear to be scattered at random. Riemann's calculations indicated that
they were lining up as if along some mystical ley line running through the landscape." **(p.99)**
**
** "In an interview, Hilbert explained that he believed the Riemann Hypothesis to be the most
important problem 'not only in mathematics but absolutely the most important.'" **(p.114)**
**
** "Littlewood's proof...revealed that prime numbers are masters of disguise. They hide their true
colours in the deep recesses of the universe of numbers, so deep that witnessing their true nature may
be beyond the computational power of humankind. Their true behaviour can be seen only through the
penetrating eyes of abstract mathematical proof." **(p.130)**
**
** "Littlewood wrote to Hardy about ****[Ramanujan]***: 'it is not surprising that he would have been
*[misled]*, unsuspicious as he presumably is of the diabolical malice inherent in the primes'."
**(p.139)***
**
** "We have all this evidence that the Riemann zeros are vibrations, but we don't know what's
doing the vibrating." **(p.280)**
**
** "Maybe we have become so hung up on looking at the primes from Gauss's and Riemann's
perspective that what we are missing is simply a different way to understand these enigmatic numbers.
Gauss gave an estimate for the number of primes, Riemann predicted that the guess is at worst the
square root of ****N*** off its mark, Littlewood showed that you can't do better than this. Maybe there
is an alternative viewpoint that no one has found because we have become so culturally attached to the
house that Gauss built." **(p.312)***
**
** "Until ****[the RH is proved]***, we shall listen enthralled by this unpredictable mathematical music,
unable to master its twists and turns. The primes have been a constant companion in our exploration
of the mathematical world yet they remain the most enigmatic of all numbers. Despite the best efforts
of the greatest mathematical minds to explain the modulation and transformation of this mystical music,
the primes remain an unanswered riddle. We still await the person whose name will live for ever as the
mathematician who made the primes sing." **(p.312)*
To conclude, a somewhat daunting quote about the prime numbers from
someone who was as familiar with them as anyone has ever been:
*
*
*"It will be millions of years before we'll have any understanding, and even then
it won't be a complete understanding, because we're up against the infinite."
*
*
*
P. Erdös (interview with P. Hoffman, *Atlantic Monthly*, Nov. 1987, p. 74)
If you know of any other quotes which belong on this page, please
get in touch.
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