

the distribution of primes, Riemann zeta function and Riemann hypothesis 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."

"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."

"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."

"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."

"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."

"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."

"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

"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!"

"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."

"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."

"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."

"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."

"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."

"...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."

"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."

"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."

"[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."

"...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."

"Mother Nature has such beautiful harmonies, so you couldn't say that something like [the Riemann Hypothesis] is false."

"...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."

"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."

"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."

"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."

"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."

"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"

"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".

"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?"

"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 2p – 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.''

"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.

