Comment by rfurmani | Hacker News Reader

Comment by rfurmani | Hacker News Reader

Great article!

One of the things about the Riemann Hypothesis is that the search space for a proof is more wide than it is deep. For each given idea or approach it doesn't take relatively long to get to the forefront of what is known, and an expert can often tell you right from the beginning that the whole class of approaches may not work due to some known phenomena, or that they would have to involve certain complications to not pick up on various almost-counterexamples. Furthermore, in these 150 years not only has the right path/approach not been found, but there isn't a truly compelling reason to believe that RH is true, aside from numerical evidence and a belief in beauty.

I think it'd be fascinating to put together an online resource to organize the possible approaches, list the knowledge prerequisites, show the potential counterexamples and stumbling blocks to each approach. This would help anyone interested in the problem, and once it is sufficiently developed it would allow non-specialists to contribute productively. This could be LaTeX on github, it could be more of a traditional wiki, but now I really want to get this going.

There was actually an entire conference in 1996 pretty much dedicated to talks about "How not to prove the Riemann Hypothesis" to prevent people from wasting time on approaches that were known not to work:

http://www.aimath.org/research/conferences/rh-conf.html

Looks like a great conference, regrettably before my time (and unfortunately no videos of course). They look like fairly standard topics about the zeta function, where does "how not to prove RH" come in?

I'm not trying to be rude here, but academics in traditionally textual disciplines that lament lack of video media leave me totally agog.

"there isn't a truly compelling reason to believe that RH is true, aside from numerical evidence and a belief in beauty."

You wouldn't consider the analogue of the Riemann hypothesis for the zeta function of a smooth projective variety over a finite field, proven by Deligne, to be a compelling reason to believe the RH is true?

No because that's a polynomial. The proofs that involve computing moments already have an analog for the Riemann zeta function, but for zeta(s) they just represent zero-density results, while in the Deligne case they give you the full on Riemann Hypothesis.

I am curious which particular proofs you are referring to, could you please give a link or reference or something?

I'm having some difficulty finding it, but Terry Tao had a nice exposition in one of his posts, I believe the crux was computing/bounding traces of powers and then being able to conclude the result. My work is almost purely on the analytic side (hence perhaps the RH skepticism), so I don't know the right reference off-hand.

Was it https://terrytao.wordpress.com/2013/07/19/the-riemann-hypoth...?

I don't get anything when I ctrl-F for "moment" or "density" there (other than in sidebar links).

> there isn't a truly compelling reason to believe that RH is true, aside from numerical evidence and a belief in beauty

Numerical evidence isn't proof, and yes I know about Littlewood and Skewes, but it starts to get suggestive.

It was discovered only really with the work of Siegel that Riemann made his hypothesis after computing the first few zeros and seeing they were on the critical line. In many ways the complexity of the zeta function is controlled by log log log T, and some heuristics say we shouldn't expect to see counterexample until height around e^e^e^3. Here's one example. The function S(T) measures the difference between the number of zeros up to height T and the number asymptotically expected. So in particular it jumps by 1 whenever there is a zero. If there's a zero off the critical line then there will be two zeros symmetric around the critical line, so S(T) would jump by 2. The biggest value of S(T) seen is around 1.6 so in a very real sense there isn't "room" for a counterexample just yet.

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