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Prominent Mathematicians Rebuke Recent Riemann Hypothesis Proof
Bryan writes "Xian-Jin Li's purported proof of the Riemann Hypothesis (reported on recently) has been rebuked by Fields Medalist Terence Tao. Fortunately, Dr. Li's proof fails alongside a respectable graveyard of previous attempts." Relatedly, jim.shilliday writes "The proof cites and appears to be based in part on the work of the leading French theorist Alain Connes. A few hours ago, Connes posted a comment on his blog stating that the purported proof is so badly flawed that he stopped reading it."
I guess they mean that there's no shame in having failed, since many other respectable attempts also failed.
There are a lot of results based on assuming the conjecture is true, including a variety of factoring and root finding algorithms that are computationally very useful.
Until it is proven you really don't know if these algorithms are giving correct answers.
This is why it is so important and has a big prize associated to it.
I believe you're mixing this up with another hard problem that hasn't been proven yet. You're thinking about the NP = P problem. The difference is that here we don't know what will be the outcome, whereas for the RH most people assume it's true. Having a proof for this wouldn't really change anything (apart from validating large parts of mathematics that assume it is true)
Just wanted to point out that Professor Connes is also a Fields medalist (1982).
I guess it is a testament to Xian-Jin Li excellent reputation and the importance of the topic that these two mathematical superstars took the time to look at his proof.
One possible explanation for your understanding (which in my understanding, is wrong), is the Miller-Rabin primality test algorithm.
The primality problem (telling whether a number is prime), although hard, was never proved to be NP-complete.
The Miller-Rabin primality test is a (actually, the 1st and possibly the only) polynomial deterministic algorithm that is based on the Riemann hypothesis (polinomial deterministic meaning "fast and accurate"). Proving RH would prove that Miller-Rabin is exact and therefore shown that primality testing is in P.
http://en.wikipedia.org/wiki/Miller-Rabin_primality_test
Unfortunately, algorithm freaks were faster than math freaks (well, the algorithm freaks involved were math freaks too) and a new algorithm called AKS was developed that did everything Miller-Rabin did without relying on the Riemann Hypothesis.
http://en.wikipedia.org/wiki/AKS_primality_test
So, to this day, we know primality testing is polynomial. The _real_ problem in cryptography is prime *factoring* (if it's not prime, then find 2 numbers that when multiplied produce the original number). Although it is not know whether that problem is P or NP-complete or both, it is believed to be outside NP because it is much harder than plain primality testing.
http://en.wikipedia.org/wiki/Integer_factorization
My english is sow-sow. Sowhat?