A lot of people in this thread have said that I’m way too underconfident about RH, and thinking about that, they are probably right. At this point, there are two “obviously false” statements that we can’t even disprove:
1) There are zeros of the zeta function arbitrarily close to the line Re s =1. 2) A positive fraction of the non-trivial zeros lie off the line (with the zeros ordered in the obvious way by the size of the imaginary part).
We also can’t prove the related Lindelof hypothesis which is an easy consequence of the Riemann hypothesis.
All three of these are things that one would expect to be done before RH is proved, and we don’t seem very close to resolving any of these with the possible exception of statement 2. There’s some reason to believe that 2 might be disproven from further tightening of Hardy and Littlewood type results in the same way that Levinson and Conrey showed that at least two fifths of the zeros lie on the line (I lack the technical knowledge to evaluate the plausibility that Conrey’s type of result can be further tightened, although the fact that his result in the tightest known form has stood for about 20 years suggests that further tightening is very non-trivial.) Note that we can prove the slightly weaker statement that almost all the zeros lie with in epsilon of the 1⁄2 line, which is sort of a hybrid of the negations of 1 and 2, but that result is about a hundred years old (one thing I actually wish I understood more is how if at all that result connects to the Hardy and Littlewood type results. My impression is that they really don’t in any obvious way but I’m not sure.) Analogs of RH have also been proven in many different contexts (including the appropriate analogs for finite fields and for certain p-adic analogs); I don’t know much about most of those, but it seems like those results don’t give obvious hints for how to resolve the original case in any useful fashion.
I guess the real reason for my estimate is that there seem to be so many promising techniques for approaching the problem that I don’t understand much in detail, such as using operator theory, or connecting the location of the zeros to the behavior of some quasicrystals.
I was almost certainly underconfident in my above estimate, and I think everyone was very right to call me out on that, so I won’t be taking any bets on it. I’m not sure how to revise it. Upwards is clearly the correct direction, but thinking about the probability estimate in more details leaves me feeling very confused.
A lot of people in this thread have said that I’m way too underconfident about RH, and thinking about that, they are probably right. At this point, there are two “obviously false” statements that we can’t even disprove:
1) There are zeros of the zeta function arbitrarily close to the line Re s =1.
2) A positive fraction of the non-trivial zeros lie off the line (with the zeros ordered in the obvious way by the size of the imaginary part).
We also can’t prove the related Lindelof hypothesis which is an easy consequence of the Riemann hypothesis.
All three of these are things that one would expect to be done before RH is proved, and we don’t seem very close to resolving any of these with the possible exception of statement 2. There’s some reason to believe that 2 might be disproven from further tightening of Hardy and Littlewood type results in the same way that Levinson and Conrey showed that at least two fifths of the zeros lie on the line (I lack the technical knowledge to evaluate the plausibility that Conrey’s type of result can be further tightened, although the fact that his result in the tightest known form has stood for about 20 years suggests that further tightening is very non-trivial.) Note that we can prove the slightly weaker statement that almost all the zeros lie with in epsilon of the 1⁄2 line, which is sort of a hybrid of the negations of 1 and 2, but that result is about a hundred years old (one thing I actually wish I understood more is how if at all that result connects to the Hardy and Littlewood type results. My impression is that they really don’t in any obvious way but I’m not sure.) Analogs of RH have also been proven in many different contexts (including the appropriate analogs for finite fields and for certain p-adic analogs); I don’t know much about most of those, but it seems like those results don’t give obvious hints for how to resolve the original case in any useful fashion.
I guess the real reason for my estimate is that there seem to be so many promising techniques for approaching the problem that I don’t understand much in detail, such as using operator theory, or connecting the location of the zeros to the behavior of some quasicrystals.
I was almost certainly underconfident in my above estimate, and I think everyone was very right to call me out on that, so I won’t be taking any bets on it. I’m not sure how to revise it. Upwards is clearly the correct direction, but thinking about the probability estimate in more details leaves me feeling very confused.