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Attempts to prove the Riemann Hypothesis

So I'm compiling a list of all the attacks and current approaches to Riemann Hypothesis. Can anyone provide me sources (or give their thoughts on possible proofs of it) on promising attacks on Riemann Hypothesis?

My current understanding is that the field of one element is the most popular approach to RH.

It would be good if someone started a Polymath project with the aim of proving RH. Surely, if everyone discussing possible ways to prove RH, it would be proven in about a year or so or at least people would have made a bit more progress towards the proof or disproof.

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    "Surely, ..., , it would be proven in about a year". That's a lot of wishful thinking.2011-10-29
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    I suppose that is a bit optimistic. However, I can't believe there hasn't been some sort of polymath project in the aim towards proving this.2011-10-29
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    Consider the whole literature on the RH as a huge polymath project!2011-10-29
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    It ain't gonna happen. I don't think you understand either the difficulties inherent in the RH or the nature of mathematical research. There are only a small number of mathematicians with the skill-set to approach the RH, and they already know each other. They don't need the internet to generate their collaborations. It isn't like the other polymath projects, which focused on problems that are accessible with minimal background and which everyone knew were "just out of reach". There's a reason none of them resulted in bombshell results (or even in papers in top journals).2011-10-29
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    You should read the book "The Riemann Hypothesis" by Karl Sabbagh. The problem has not been solved for lack of trying.2011-10-29
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    Assuming by contradiction. :-)2011-10-29
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    I'm just curious: is there any simple way of proving that RH is not independent on the axioms? There is always a third possibility beside proving or disproving it. Intuitively it seems obvious that the RH cannot be a new axiom, but I think the same can be said about the continuum hypothesis.2011-10-29
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    And sorry for the offtopic, but I had a chuckle when I saw your username, given the question you asked :D2011-10-29
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    As for unconventional approaches: I've found [these](http://dx.doi.org/10.1016/0024-3795(86)90255-7) [three](http://dx.doi.org/10.1016/0024-3795(87)90250-3) [papers](http://dx.doi.org/10.1016/0024-3795(90)90306-W) to be terribly interesting...2011-10-29
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    @user9176 you can't prove RH is independent of the axiom without proving RH is true. Say RH was unprovable, then you can't find a zero of the line to disprove it. Hence, it must be true. Well, the proof uses more detail.2011-10-29
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    @simplicity: Keep in mind that exactly the same argument can be made about the continuum hypothesis: if it is independent of the axioms, then you cannot find an uncountable subset of $\mathbb{R}$ which is not $c$, thus you prove it....The problem of independence is not that simple. If it is independent, it means that we can have two different mathematical models, so that RH is true in one and not true in the other.... The independence doesn't mean that you cannot find a zero, it only means that you don't have enough info to decide if there is a zero off the line...2011-10-29
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    @user9176 http://mathoverflow.net/questions/27755/knuths-intuition-that-goldbach-might-be-unprovable searching for this. I see you are right. Well, there are algorithms to calculate all the zeros RH. But, I'm pretty sure I read a paper from a logician that was saying it is his dream to prove RH independent of ZFC. The mathematician was famous and so I assume he wasn't a crank.2011-10-29
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    @J.M. That was quite a surprise. Thanks!2011-10-29
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    See page 4 of http://www-math.mit.edu/~poonen/papers/h10_notices.pdf for an informed opinion on the independence of RH.2011-10-29
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    You know that joke: a man meets the Devil, who says him: "if you gave me your soul, i grant your dearest wish". The guy answer: "well, ok, explain me how to prove the Riemann Hypothesis". The Devil says he need to think about that a couple of days. One week later, no news from him, so the man takes his phone and call in hell. "So what?", he asks. Then the Devil answers: "You know what? Keep your soul, i don't want it any more" :)2014-02-25

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