As anyone who follows the arxiv, I notice every now and then "proofs" and "disproofs" of Riemann Hypothesis. I looked on several such articles, and it seemed to me quite nonsense, but I didn't make the effort to find a mistake. My question is whether someone reads these "proofs"?

BTW, I wanted to refer to some of these papers in the arxiv, but it turned out that there are too many of them.

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    $\begingroup$ I assume the authors themselves read them, so the answer is YES ;) Perhaps your question should be more specific, e.g. "what are the common failed attempts?" or something along those lines. If you do edit the question into something like that, it probably should be community wiki. $\endgroup$ – GMRA Nov 26 '09 at 20:32
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    $\begingroup$ I think the question is not so much "what are the common failed attempts" as "where can one find refutations of specific attempts". But I might have misunderstood Lior. $\endgroup$ – Yemon Choi Nov 26 '09 at 20:43
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    $\begingroup$ It could very well be me who misunderstood him. But either way that is asking for a list, and should be community wiki. $\endgroup$ – GMRA Nov 26 '09 at 22:02
  • $\begingroup$ A short list of ArXiv "proofs" of RH is available in my contribution to mathoverflow.net/questions/59770/… $\endgroup$ – Gerry Myerson Jan 31 '12 at 22:53

Whenever someone claims a proof (or disproof) of a big conjecture, many people leap to the question of whether the proof is correct. The problem then is it that it takes an enormous amount of work to confirm that a proof is correct. Even a clear mistake in a proof could be reparable. Moreover, attempted proofs have inferences that amount to gaps of different sizes. Even in a naive attempt, it can take a lot of work to decide which gaps are so big that the proof has to be called incomplete.

There is a much simpler standard that experts use in practice: "As I start to read this paper, am I learning from it?" You would expect a proof of a big conjecture to have very interesting lemmas, and otherwise to teach you new things along the way. This is not always obvious either; there have been a few grievous misunderstandings in which initial readers dismissed a great paper. Even so, it's a somewhat reliable standard, and it's the most that authors can expect.

When Perelman posted the first of his three papers on geometrization, experts in differential geometry quickly embraced it as exciting and teachable, before they had even checked half of that paper or seen the other two papers. From the beginning, this was very different from most claimed proofs of the Poincare conjecture, even most of the noble failures. The great ideas in these papers were more important than the fact that they had a lot of gaps (by common standards) and even some inessential mistakes (or so I was told).

I know for a fact that experts sometimes do study weird-looking claims of big results, in the arXiv and elsewhere. They have little incentive to broadcast their attention to it if they think that it's shoddy work, but sometimes they try to be fair. For starters, the math arXiv has moderators, and they often take a look. I think that usually (not quite always), several people have looked long enough to decide that they aren't getting anything out of the paper. But hey, there could always be a diamond in the rough, or even a diamond in the garbage.

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    $\begingroup$ "many people leap to the question..." But I think Lior is not talking about Perelman-type proposed solutions. The typical RH proof in Arxiv is by a non-mathematician, or by a mathematician in a remote branch of mathematics who has published no research for many years. I think there is no "leaping" in most of these cases. $\endgroup$ – Gerald Edgar Nov 26 '09 at 23:37
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    $\begingroup$ A counterpoint, Hales proof on sphere packings. The checking team "gave up" after 18 months(?), largely as they were learning nothing new but just verifying inequalities. Whether the proof was "correct" became not so interesting in the end. When reducing a proof to finite computation, there are interesting ways and ones that shed no light. Another paper I am told, is Friedlander and Iwaniec on $X^2+Y^4$ being prime (Annals). There is "nothing new" in that paper, for 2 related papers (Duke+FI, Fouvry+I) first break a parity problem, and experts say FrIw is tour de force in technicalities then. $\endgroup$ – Junkie Apr 12 '11 at 7:10
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    $\begingroup$ @Junkie Actually Hales is probably right. It is true that his proof is very complicated and challenges the traditional routine of checking someone else's result. Hales is well aware of this issue and his new project is to code his result as a formal proof checked by computer. I don't know about Friedlander-Iwaniec, but certainly in the case of Hales it's simplistic to say "they were learning nothing new". People learned a lot from Hales' paper, even if the end stage of traditional refereeing became too complicated. $\endgroup$ – Greg Kuperberg Apr 12 '11 at 11:38

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