JoshuaZ

While some of my work is in analytic number theory, much is in other subareas, so it is possible I should defer to you on this.

It seems to me less than PNT in terms of what can we actually do with this. Many different areas of math use PNT, and from my standpoint, PNT is helpful not just for what it implies directly but because it lets us make really good heuristics about whether some sets are infinite or not, and what their rough size is. (Granted, one can do that also mostly via Chebyshev). For those purposes, this doesn't really enter in. Similarly, PNT feels like a statement at least I can say explain to my mother without any technical details. This isn't that. But that may also be my own biases of wanting things to cash out to very concrete statements about the integers.

I agree that one striking element is how no one saw this coming. This isn't building on an existing research program, which itself is remarkable. And last night, before I went to bed, I saw a conversation between a bunch of analytic number theorists who seemed to think there was potentially some slack in the quasi-RH argument, which if that's the case means this is going to go even further.

Generalized Riemann hypothesis.

Yeah, completely reasonable to argue that.

Number theorist here. This is a massive big deal, and would likely be a Fields Medal for a human if a human had done it. But it is an exaggeration to say it is the biggest result in 200 years. At a minimum, it is hard to argue that it is a bigger result than the proof of the prime number theorem in 1896 (which this is a strengthening of), or Riemann's original 1859 paper where he laid out the zeta function and its analytic importance, or Dirichlet's proof of infinitely many primes in arithmetic progressions which is the late 1830s.

But yeah, this is still a very big deal. Among other things, it will drastically improve all sorts of Rosser-Schoenfeld type results for the PNT and that's just a start. For comparison, I have a paper form 2018 where this result would cut 3 pages out and make the full result cleaner and much tighter, and there are likely hundreds of papers like this.

I really like the product you've made, and use it regularly. It might be helpful for your user base, and the project as a whole if one responded with a little more than a shrug. Facebook may be more willing to listen to you if you contact them then random users in getting this resolved.

To expand on what phil is saying: It is by itself logically consistent for someone to decide that there's enough fun left in their life and enough stuff that they need to do that they don't want to die now even if they don't necessarily mind dying in the indefinite future.

Uncomputable problems are not necessarily NP-hard. You can for example encode a version of the halting problem such that the instances are simply very long. For example, consider the language L defined by k is in L iff k=2^2^2^n for some n and nth Turing machine halts on the blank tape (some reasonable listing of Turing machines). In order to feed an NP-hard problem to an L-oracle, one needs to ridiculously expand the length of the problem far more than polynomial length so this certainly can't be done in polynomial time.

I'm not sure what could be done that would be dangerous in the name of "ushering in the singularity." The worst plausible scenario is that we put money into research avenues that don't pan out. But that happens all the time anyways. The real worst case scenario is that someone gets too afraid of AI and goes all Sarah Connor. But that's not a likely scenario especially because most of the people who take the Singularity idea seriously are people who are generally optimistic about the long-term trends.