rfurmani

> But why is it important that these zeros are on the vertical line Im z = 0.5

Great question. The formula works whether or not the zeros have real part 0.5, but its implications are different. If you're counting primes up to x you will get the main term (roughly x/log x) and each zero p will give an oscillating error term ~ x^p/p. If the real part is always 0.5 then these error terms are all on sqrt(x) scale, which is what you would expect from random variation. If on the other hand you have a single zero right of the line, say at real part 3/4 (and mirror image one at real part 1/4) then now you get a secondary error term of size x^(3/4) that dominates all of the other error terms, giving some extra structure to the primes: they're no longer random, you can predict where they are more or less likely.

We can't even solve Sonic the Hedgehog without getting stuck in dead-ends in the game :-) https://openai.com/blog/retro-contest/

Adding an extra point, one thing that makes this difficult is that a lot of this kind of research is not about applying existing ideas but coming up with something that has never been done before, and neutral networks are a lot worse with creativity than with pattern matching. It's also hard to get enough training examples, especially since you'd either want these to be in strict logical language or train a system to recognize whether a hundred page proof in natural language is rigorous or not. So it's like a very long maze where noody knows if you're getting closer to the exit or not, and in most rooms you have to come up with something original or discover a new dimension. And there's the aspect whereby a lot of significant research is: create a new abstraction, realize and convince others that this abstraction is useful and gives you new intuition that lets you solve new problems.

Slashdot reference?

It sounds like this has been largely debunked, based on the research cited in The Honest Truth about Dishonesty https://www.amazon.com/Honest-Truth-About-Dishonesty-Everyon...

Did you win when you did that? The strategy varies quite a lot between instances of the game, but often I find that those kinds of “engines” happen too late in the game to really be valuable, and by then someone might have bought gold and provinces.

3) Russia. Look it up.

I’d that doable? Is there a proof-of-concept? It would be an incredible interface, even if it just recognized some basic hand gestures using a bracelet.

They had my company name there, even in incognito mode

Very nice, can I link to it? I'm building a resource for [approaches to the Riemann Hypothesis](https://pnphard.pro/blems/riemann/)

There is also Problems in Analytic Number Theory, which teaches the field as a series of problems and exercises. It's a great way to learn, and one that I haven't seen implemented well in other areas (even the corresponding book for Algebraic Number Theory isn't as good) https://www.amazon.com/Problems-Analytic-Number-Graduate-Mat...

But this isn't just summing up a few positive integers, in which case there's no ambiguity in what the answer is. Once you start summing up infinitely many things you have to bring in some theory and some techniques to justify what the answer is. These techniques generally have a limited scope, but there's a big theory of "divergent series" that shows that if you extend these techniques to more contexts you still get a definition that is compatible with most of what one would want from a limit. For example, taking running averages, or taking it as the coefficients of a series and taking a limit. So classically 1-1+1-1+1... doesn't converge, but if you "squint" and take the running averages of the partial sums (1,0,1,0,...) you would get 1/2. Or if you use the fact that 1+x+x^2+x^3+... = 1/(1-x) and take x=-1 you would get 1/2. Or a myriad other approaches that are perfectly valid with standard convergence, and just all happen to get you that 1-1+1-1+... is 1/2. And yes if you squint very hard you would get that 1+2+4+8+16+... = 1/(1-2)=-1

I just skimmed through it so maybe I'm missing something, but it also appears to be incomplete. It even says "the completion of the rigorous derivation of the above results will be presented in a companion paper". He brings up equation 1.6 but I don't see a rigorous handling of its asymptotics, nor any precise statement of what the growth rate of zeta on or near the critical line is expected to be. The conclusions section also says about as much. I hope it works out and we will see the conclusion, but I'd really like to see some precise growth rates so we can at least compare it to known bounds and example.

For SF people it's playing at the Alamo Drafthouse this weekend

Why not? C does not have a book type and it is standard to use either char or int, depending on if you're concerned about space or about time.

Oh wow, that's a blast to the path. The author also wrote the bucktooth gopher server, and I implemented a new (semi-OO Perl) version of it that supported Gopher+.

There was always something very satisfying about Gopher, how it made explicit the structure and categories of the data, leaving the front-end to present it how it pleased.

There's a lot of companies out there, and dismissing them outright is the same brand of arrogance that lets startups win over incumbents, but reversed.

> better brand

GM/F/Honda/Toyota/etc are reliable, BMW/Audi/Lexus/etc are luxurious and have a package that's been very carefully tuned

> faster

Doesn't matter to 90%

> longer range

Debatable, also price constrained

> novelty

Not what you look for in a massive purchase that you want to last 10+ years

Here's another one. You'll notice that typically there's slightly more numbers that factor into a product of an odd number of primes (i.e. 2,3,5,7,8,11,12,13,...) than into an even number of primes (4,6,9,10,14,15,...). If this is the case then RH is true. You can make this into a more precise question about the partial sums of the Liouville function and make it an equivalence.

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.

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 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.

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.

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.

Actually, no: ascii 10B>100. The bug above is that you can't do chained inequalities like that in Javascript. Try: should_not_exclude_10B_10C("200")

The video is now on Youtube https://www.youtube.com/watch?v=6eIoYMB_0Xc

I agree completely about the system being broken and that innovation would be fantastic (I wrote a few paragraphs below, but in a very different direction). I'll take a close look at Thinklab, and it'd be great to see a dialogue forming here.

The Riemann Hypothesis! Hear me out, because I think it could serve as a great model for modern math research and collaboration.

One thing that really bothered me during my PhD is that even at the Stanford math department it felt like academia is not set up to encourage work on big questions, nor is it optimal for any objective function I can think of. Is it any wonder that many of the recent breakthroughs (Perelman with Poincare, Zhang with prime gaps) came not from establishment mathematicians, but ones who couldn't or wouldn't play the academia game and ended up almost destitute on the fringes, working away at these problems that they knew inside could be resolved.

Now, why specifically is the Riemann Hypothesis a great place to start? Math problems are like a maze. There are many paths to take, and while you can take satisfaction in exploring one part, you won't know your progress until you see the exit in front of you. In the case of RH the branch factor is absolutely massive, but once you pick a direction, it doesn't take long (relatively) to get to the forefront of what's known. But as it stands it is very hard to get perspective on the problem, or know which ways are good or bad. A mathematician who is not an expert in the field may experiment with various ways of bounding Zeta away from zero, but a domain expert would right away point out positive vertical density results for all a-sets, and bounds on moments of 1/zeta(s). Worse: if you have an idea for a new approach, there's no good way to share it because it will likely be too small for the quantum of academia: the academic paper. Building a collaborative atlas for the Riemann Hypothesis would help solve both of these problems, and could provide a more structured way of working on hard problem than even the polymath projects. It'll be a challenging use-case but at least with 150 years' of work there will be no cold start problem. And I've seen many analytic number theorists who are incredibly smart, hard-working, insightful, great with programming, happy to get their hands dirty, eager to teach others, but neglected by the system and lacking an outlet for their skills. PS: There's still no clear reason to believe RH to be true rather than false.

Sorry to comment on only the subject and not the content, but "#define CTO" just makes sure that CTO is recognized as something but makes it completely empty. All it's good for (standard C pattern) is asking "#ifdef CTO" and then it will happily reply: yes there is a CTO, just don't try to use it! Actually this does fit in nicely with the article, but to be pedantic maybe a typedef would have been better!

This is a comment that often comes up, but rest assured the top competitors really are that fast from scratch. There are some screencasts floating around of Petr competing in Toocoder SRMs, and you can also look at the timings on I site Topcoder and ACM competitions, where there really is no copy and paste. That said, one does end up memorizing some things with practice. For example, after having to code it up a few times it is easy to get a nice short implementation of network flow that you can type in in 2 minutes or so.

As mentioned, it's still closed off while I clean up the codebase, but I was just curious first if there'd be any interest, because if not then there's only downside.