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In mathematics, the Riemann hypothesis is an open problem in the field of number theory. It is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2.
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- The result has caught the imagination of most mathematicians because it is so unexpected, connecting two seemingly unrelated areas in mathematics; namely, number theory, which is the study of the discrete, and complex analysis, which deals with continuous processes.
- 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.
- 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.
The Music of the Primes
- 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.
- p. 8
- 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.
- p. 184
- Hodge conjecture
- P versus NP problem
- Poincaré conjecture (solved)
- Birch and Swinnerton-Dyer conjecture