Number theory

"Mathematics is the queen of the sciences and number theory is the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank."

"Most texts on number theory contain inserted historical notes but in this course I have attempted to obtain a presentation of the results of the theory integrated more fully in the historical and cultural framework. Number theory seems particularly suited to this form of exposition, and in my experience it has contributed much to making the subject more informative as well as more palatable to the students. ...for the understanding of a greater part of the subject matter a knowledge of the simplest algebraic rules should be sufficient."

"I believe that the vast majority of statements about the integers are totally and permanently beyond proof in any reasonable system. Here I am using proof in the sense that mathematicians use that word. Can statistical evidence be regarded as proof ? I would like to have an open mind, and say ‘Why not?’. If the first ten billion zeros of the zeta function lie on the line whose real part is 1/2, what conclusion shall we draw? I feel incompetent even to speculate on how future generations will regard numerical evidence of this kind."

"The Riemann hypothesis is, and will hopefully remain for a long time, a great motivation to uncover and explore new parts of the mathematical world. After reviewing its impact on the development of we discuss three strategies, working concretely at the level of the explicit formulas. The first strategy is “analytic” and is based on Riemannian spaces and and its comparison with the explicit formulas. The second is based on algebraic geometry and the . We establish a framework in which one can transpose many of the ingredients of the Weil proof as reformulated by , Tate and Grothendieck. This framework is elaborate and involves noncommutative geometry, es and . We point out the remaining difficulties and show that RH gives a strong motivation to develop algebraic geometry in the emerging world of characteristic one. Finally we briefly discuss a third strategy based on the development of a suitable “”, the role of Segal’s Γ-rings and of topological cyclic homology as a model for “absolute algebra” and as a cohomological tool."

"Hilbert, in his 1900 address to the Paris , listed the Riemann Hypothesis as one of his 23 problems for mathematicians of the twentieth century to work on. Now we find it is up to twenty-first century mathematicians! The Riemann Hypothesis (RH) has been around for more than 140 years, and yet now is arguably the most exciting time in its history to be working on RH. Recent years have seen an explosion of research stemming from the confluence of several areas of mathematics and physics."

"At the beginning of the new millennium the most famous unsolved problem in complex analysis, if not in all of mathematics, is to determine whether the Riemann hypothesis holds."

"A substantial portion of Weil’s research was motivated by an effort to prove the Riemann hypothesis concerning the zeroes of the Riemann zeta function. He was continually looking for new ideas from other fields that he could bring to bear on a proof. He commented on this matter in a 1979 interview:... Asked what theorem he most wished he had proved, he responded, “In the past it sometimes occurred to me that if I could prove the Riemann hypothesis, which was formulated in 1859, I would keep it secret in order to be able to reveal it only on the occasion of its centenary in 1959. Since 1959, I have felt that I am quite far from it; I have gradually given up, not without regret.”"

"It tells us that they are very nicely distributed, about as evenly and as good as altogether possible. One cannot expect a completely even distribution, of course. But it tells us that at least in mathematics, certainly in number theory, we live in Leibniz’ “best possible of all worlds”, just as the good Candide in Voltaire’s ' is told by his teacher that he lives in the best of all possible worlds. Well, in number theory at least, one has the best relation possible among primes, even though we cannot prove it yet. It would give me great satisfaction to see a proof, because it would demonstrate that there are some things that are right in this world. There are so many other things that do not work as they should, but at least for the prime numbers, and of course also for the zeros of the zeta function, they are distributed as well as they could be."

"The dependence of so many results on Riemann's challenge is why mathematicians refer to it as a hypothesis rather than a conjecture. The word 'hypothesis' has the much stronger connotation of a necessary assumption that a mathematician makes in order to build a theory. 'Conjecture', in contrast, represents simply a prediction of how mathematicians believe their world behaves. Many have had to accept their inability to solve Riemann's riddle and have simply adopted his prediction as a working hypothesis. If someone can turn the hypothesis into a theorem, all those unproven results would be validated."

"In the spring of 1997, Connes went to Princeton to explain his new ideas to the big guns: Bombieri, Selberg and Sarnak. Princeton was still the undisputed Mecca of the Riemann Hypothesis despite Paris's push to reclaim its dominance. Selberg had become godfather to the problem - nothing could pass muster before being vetted by a man who had spent half a century doing battle with the primes. Sarnak was the young gun whose rapier-like intellect would cut through anything that was found slightly wanting. He'd recently joined forces with Nick Katz, also at Princeton, one of the undisputed masters of the mathematics developed by Weil and Grothendieck. Together they had proved that the strange statistics of random drums that we believe describe the zeros in Riemann's landscape are definitely present in the landscapes considered by Weil and Grothendieck. Katz's eyes were particularly sharp, and little escaped his penetrating stare. It was Katz who, some years before, had found the mistake in Wiles's first erroneous proof of Fermat's Last Theorem. And finally there was Bombieri, sitting in state as the undisputed master of the Riemann Hypothesis. He had earned his Fields Medal for the most significant result to date about the error between the true number of primes and Gauss's guess - a proof of something mathematicians call the 'Riemann Hypothesis on average'. In the quiet of his office overlooking the woods that surround the Institute, Bombieri has been marshalling all his insights of previous years for a final push for the complete solution. Bombieri, like Katz, has a fine eye for detail. A keen philatelist, he once had the chance to purchase a very rare stamp to add to his collection. After scrutinising it carefully he discovered three flaws. He returned the stamp to the dealer, pointing out two of them. The third subtle flaw he kept to himself - in case he is offered an improved forgery at a future date. Any aspiring proof of the Riemann Hypothesis is subjected to an equally painstaking examination."

"The conjecture discovered jointly by Peter Swinnerton-Dyer and Bryan Birch in the early 1960s both surprised the mathematical world, and also forcefully reminded mathematicians that computations remained as important as ever in uncovering new mysteries in the ancient discipline of number theory. Although there has been some progress on their conjecture, it remains today largely unproven, and is unquestionably one of the central open problems of number theory. It also has a different flavour from most other number-theoretic conjectures in that it involves exact formulae, rather than inequalities or asymptotic questions."

"Just as Weil's conjectures were about counting solutions to equations in a situation where the number of solutions is known to be finite, the BSD conjecture concerns the simplest class of polynomial equations—elliptic curves—for which there is no simple way to decide whether the number of solutions is finite or infinite."

"Thanks to their conjecture, Birch and Swinnerton-Dyer are two names that (to mathematicians) are as inextricably linked as the names of Laurel and Hardy, although many have been tricked into believing that there are in fact three mathematicians behind the conjecture - Birch, Swinnerton and Dyer. Birch, with his rather bumbling manner, plays Stan Laurel to Swinnerton-Dyer's rather dour Oliver Hardy."

"This remarkable conjecture relates the behaviour of a function L at a point where it is not at present known to be defined to the order of a group Ш which is not known to be finite."

"The twin conjectures of Hodge and Tate have a status in algebraic and arithmetic geometry similar to that of the Riemann hypothesis in analytic number theory."

"Admittedly, Fermat's Last Theorem was always called a theorem and never a conjecture. But that is unusual, and probably came about because Fermat claimed in notes that he scribbled in his copy of Diophantus's Arithmetica that he had a marvellous proof that was unfortunately too large to write in the margin of the page. Fermat never recorded his supposed proof anywhere, and his marginal comments became the biggest mathematical tease in the history of the subject. Until Andrew Wiles provided an argument, a proof of why Fermat's equations really had no interesting solutions, it actually remained a hypothesis - merely wishful thinking."

"The Wiles proof is a master symphony of the major mathematical themes that have evolved in this century: Hecke's theory of modular forms, Artin L-functions, Grothendieck's theory of schemes, the theory of ℓ-adic representations and the Tate module, the Langlands program, Serre's p-adic modular forms, the Eichler-Shimura theory, Iwasawa theory and its generalizations by Coates to elliptic curves, Kolyvagin's Euler systems. The list is by no means exhaustive, to say the least. As Barry Mazur has so aptly put it, "Not to mention the proof, the names alone of the major contributors will not fit into any margin." It is indeed very satisfying to see the evolution of such mathematical ideas."

"FLT deserves a special place in the history of civilization. Because of its simplicity, it has tantalized amateurs and professionals alike, and its remarkable fecundity has led to the development of large areas of mathematics such as, in the last century, algebraic number theory, ring theory, algebraic geometry, and in this century, the theory of elliptic curves, representation theory, Iwasawa theory, formal groups, finite flat group schemes and deformation theory of Galois representations, to mention a few. It is as if some supermind planned it all and over the centuries had been developing diverse streams of thought only to have them fuse in a spectacular synthesis to resolve FLT. No single brain can claim expertise in all of the ideas that have gone into this "marvelous proof". In this age of specialization, where "each one of us knows more and more about less and less", it is vital for us to have an overview of the masterpiece such as the one provided by this book."