Question

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.

Answer 1

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 and standard that experts use in practice: "As I start to read this paper, am I learning from it?" You would figure that a proof of a big conjecture would have a lot of interesting lemmas, and otherwise that it would teach the reader new things along the way. This is not always obvious either; there have been a few grievous misunderstandings in which initial readers just didn't understand 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 the paper. Never mind what was in 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 some standards) and even some inessential mistakes (rumor has it).

Although there isn't a ton of public discussion about this, I have seen first-hand that experts have glanced at many weird-looking claims of big conjectures, in the arXiv and elsewhere. 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 a diamond in the swamp, or even a diamond in the garbage.)