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"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 Univ. Press, 2002), p.118

"[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."

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 Mathematics", Prospect, January 2002.

"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."

Don Zagier, Bonn University inaugural lecture

"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: Exploring the Infinite Landscape at the Base of the Number System" (The Sciences, Sept/Oct 1992)

"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

"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, 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 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)

http://www.secamlocal.ex.ac.uk/people/staff/mrwatkin/isoc/quotes.htm

"...a variety of evidence suggests that underlying Riemann's zeta function is some unknown classical, mechanical system whose trajectories are chaotic and without [time-reversal] symmetry, with the property that, when quantised, its allowed energies are the Riemann zeros. These connections between the seemingly disparate worlds of quantum mechanics and number theory are tantalising."

M. Berry - "Quantum Physics on the Edge of Chaos" (New Scientist, 19 November 1987)

http://www.secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/zetafn.htm