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## Riemanns hypothesis

 Quote by Graviton can anyone explain to me thoroughly the riemanns hypothesis...i dont understand it.
This was posted as an off topic reply to another thread. I think it was meant to be a new thread so here it is.

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 Recognitions: Homework Help Science Advisor Günter Reimann's hypothesis was that the Nazi control of the economy leading up to and through the war hurt the German economy by excessive regulation and control of intellectual property and activity.
 Recognitions: Homework Help Science Advisor gauss gave an estimate of the number of primes less than a given integer. riemann observed that gauss' estimate was vastly too large, as it included also all squares and cubes and 4th powers etc...of primes, so he tried to correct the estimate. his assertion gave an estimate based on the assumption, which he tried to prove, that all zeroes of the zeta function have imaginaary part = 1/2. aparently this remains unproved. please read riemann's own paper on the topic, instead of inquiring of relative imbeciles (compared to riemann) here.

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## Riemanns hypothesis

Ah, Bernhard Riemann's hypothesis. In that case see the original paper (plus translation) mathwonk mentions here:
http://www.maths.tcd.ie/pub/HistMath.../Riemann/Zeta/

 Recognitions: Science Advisor Just thought I'd mention that the work of Euler, Riemann and de la Vallee Poussin on the distribution of prime numbers are discussed in Jameson, The Prime Number Theorem, Cambridge University Press, 2003. Those with access to a university library can try to find Donald Zagier, "The first 50 million prime numbers", The Mathematical Intelligencer 0 (1977): 7-19. There are many other memorable expositions such as Warren D. Smith, "Cruel and unusual behavior of the Riemann zeta function", available as a postscript file here. Those of you who value on-line exposition might consider making a donation to keep Murray Watkins writing about math, incidentally. (I don't know him personally, only from his entertaining writings.)

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 Quote by Xevarion An important point I should mention is that everyone believes Riemann is true. The reason people still care about it is not to find out whether it's true. It's to find out what methods would be required for the proof, and what other deep insights into number theory we would obtain as a consequence of the techniques developed for the proof. More reading can be found on Wikipedia.
Then you get stuff like

http://en.wikipedia.org/wiki/Euler%2...ers_conjecture

where it takes 200 years to find a counterexample. An actual proof would still be important, because just because everyone thinks 'surely it must be true' doesn't mean it actually is

 Quote by Office_Shredder Then you get stuff like http://en.wikipedia.org/wiki/Euler%2...ers_conjecture where it takes 200 years to find a counterexample. An actual proof would still be important, because just because everyone thinks 'surely it must be true' doesn't mean it actually is
I don't find that very surprising, though. I mean, problems about sums of powers were not really well-understood until relatively recently with the development of the Hardy-Littlewood circle method and Vinogradov's method, see e.g. Waring's Problem. Even now I don't think most people would say that we really understand that kind of Diophantine equation.

I think if you asked a mathematician if that conjecture was true in 1900, he would probably say yes, but he would be able to give you no reason why beyond "numerical evidence". We have significantly more than numerical evidence for Riemann. (I'm not an expert in the field, so I couldn't give a good description of what, but if you really want to know more about it I can go ask Sarnak.)

By the way, regarding Euler's conjecture -- I think it is actually not too hard to find small counterexamples if k is big (using the notation from the Wikipedia article). I recall in high school my friend and I were interested in Waring's problem and wrote a C++ program to search for this kind of thing, and I believe we found some pretty reasonable ones for higher powers. It would be interesting to ask what is the smallest number $b = b(k)$ for which the conjecture is violated.

 Recognitions: Homework Help Science Advisor If you have not yourself read Riemann's paper, then I suggest you are doing a disservice to perpetuate the notion that one should not read the original work to get the best possible idea of what it says. i have also perused several popular treatments of the topic, and even scholarly ones, but they are too lengthy to give a concise idea of what he was doing. riemann himself made it more clear in my opinion than later expositors, and riemanns work has fewer prerecquisites than do the later books. It is a mistake to shy away from original sources until one has tried them at least. i ask you to read my review of riemanns collected works in translation at math reviews.