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Claimed Proof of Riemann Hypothesis
An anonymous reader writes "Xian-Jin Li claims to have proven the Riemann hypothesis in this preprint on the arXiv." We've mentioned recent advances in the search for a proof but if true, I'm told this is important stuff. Me, I use math to write dirty words on my calculator.
there are "proofs" of the Riemann hypothesis on the arXiv every few weeks. Don't believe it 'til it's vetted.
Good explanation here too:
http://www.irregularwebcomic.net/1960.html
"The weirdest thing about a mind, is that every answer that you find, is the basis of a brand new cliche" -
I think you misunderstand the scope and purpose of arXiv. arXiv is a repository for *preprints*.
By uploading the file to arXiv before submitting it, not only do you ensure that those that can't afford $10,000+ subscription fees can access the article, but you open up your findings to a much wider international audience.
The lack of peer review is not necessarily a liability in this situation
The Riemann zeta function is \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^{s}} [written for LaTeX], or "the sum of 1/(n^s) as n goes from 0 to infinity (increasing by 1 repeatedly)" [in more human-readable form].
Riemann was interested in the zeros to this function, where s is a complex number. He conjectured that all zeros (aside from those of the form s = -2c, where c is a positive integer) would have to be of the form (1/2) + ki, where k is a constant and i is the square root of -1.
This paper is saying that they've found a way to verify this intuition by patching a hole in a previous attempt.
Assuming that everything is correct (a big assumption), this would finally solve a long-standing problem (dating back to 1859).
Details of the actual solution are a bit heavy. Those actually interested in this sort of number theory might want to start here.
Misery loves company. Online misery loves unsuspecting random strangers.
No. Every number field has its own zeta function. The standard Riemann hypothesis concerns that of the rationals.
It's important because the zeros of the zeta function tell you how the prime numbers are distributed and prime numbers are to number theory as elements are to chemistry, everything you could care about is built out of them. The RH is also related to host of other more esoteric, but no less important conjectures; the truth of a large part of modern mathematics relies on knowing if the RH is true or false.
Although it's unlikely to impact the storage capacity of a flash drive any time soon the zeta function shows up in high energy physics and thus does have real world consequences.
The Riemann Hypothesis and RSA encryption both have to do with prime numbers, but the relationship between the two pretty much ends there. To break RSA you need to know how to factor large numbers quickly. RH, on the other hand, pretains to the distribution of prime numbers. It's pretty unlikely that a proof would make computers any faster at factorizing.
So this begs the question that a lot of people have been asking on this thread: why should you care? There tongue-in-cheek answer is that a solution is worth $1,000,000. While that response may suffice for non-mathematicians, mathematicians would have another, more important reason to celebrate. RH and its generalization, the Grand Riemann Hypothesis, have an absolutely enormous number of profound impliations in number theory, and it is difficult to overstate how critical a proof of either would be. (The implications are too technical to write about here, but you can read about them in most good survey books on analytic number theory; for example, see section 5.8 of Iwaniec & Kowalski). A successful proof probably won't affect your life in any meaningful way (unless you work with analytic number theory for a living), but it would be monumental in the world of math - indeed, this is precisely why there's a reward for solving it. If that's not enough for you, just remember that many mathematicians are motivated not by fame or money but by the beauty and elegance of mathematics, and any proof of RH would establish a truly beautiful and amazing result.
Of course, there's also the question: is Li's proof correct? I certainily don't know, and I doubt anyone will for quite some time, but there's an interesting story. Li's Ph.D. adviser was Louis de Branges who, as noted on this very website, claimed to prove RH in 2004. His proof has not been accepted by the mathematical community and is widely considered to be incorrect, in large part because the method he wclaims to use was shown, in a 2000 paper co-authored by none other than Xian-Jin Li, to have holes in it.