Explaining Riemann's Hypothesis: A Comprehensive Guide

[ramsey2879]

can anyone explain to me thoroughly the riemanns hypothesis...i don't 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.

 

[CRGreathouse]

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.

 

[mathwonk]

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.

 

[CRGreathouse]

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

 

[Xevarion]

Somehow I think that posting a link to Riemann's original paper isn't going to explain things very well to the OP.

A good understanding of the Riemann Hypothesis (the statement of it, that is) requires some knowledge of complex analysis and a bit of familiarity with the history of number theory. There are many popular books about the Riemann Hypothesis. I'd recommend reading one of those, such as Prime Obsession by some dude whose name I've forgotten.

The fundamental idea is that somehow the Riemann zeta function is intimately connected with the nature of the primes. The origin of this was with Euler's famous http://mathworld.wolfram.com/EulerProduct.html" , but Riemann was the one who realized the true importance of the zeta function, which is why it's named after him.

Mathwonk's post misleadingly connects the prime number theorem with the Riemann Hypothesis. (I don't think mathwonk meant to be misleading, but the vagueness of that post makes the distinction unclear.) It is also true that the prime number theorem (which states that the number of primes less than n is proportional to n/log(n)) is closely related to the nature of the zeta function. In fact, the prime number theorem is equivalent to the statement that there are no zeros of the zeta function on the line in the complex plane with real part 1. This was proven independently by Hadamard and de la Vallee Poussin (probably I spelled his name wrong) near the end of the 19th century. (In the 1950s or something, Erdos and Selberg independently came up with "elementary" proofs of the prime number theorem. But I don't know much about these proofs or whether they shed further light on the significance of the zeta function.)

The Riemann Hypothesis actually states that all (except for a few trivial) zeros of the zeta function lie on the line in the complex plane with real part 1/2. Of course, this would imply the prime number theorem, by the equivalence I mentioned above. However it is a MUCH stronger statement, and the other things that would follow from a proof of Riemann are countless. (Search on Google for "equivalent to the Riemann Hypothesis"...) So far, I believe results are something like this:
- All nontrivial zeros have real part in the interval (epsilon, 1-epsilon) where epsilon is very small, and possibly depends on certain other things which I don't remember.
- The Hypothesis has been computationally verified up to extraordinarily large numbers.

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 http://en.wikipedia.org/wiki/Riemann_hypothesis" .

 

[Chris Hillman]

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.)

 

[Office_Shredder]

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 http://en.wikipedia.org/wiki/Riemann_hypothesis" .

Then you get stuff like

http://en.wikipedia.org/wiki/Euler's_sum_of_powers_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

 

[Xevarion]

Then you get stuff like

http://en.wikipedia.org/wiki/Euler's_sum_of_powers_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. "[URL Problem[/URL]. 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.

 

[mathwonk]

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.