Terence Tao

So I think we will, in 10 years, we will have it.

many more, much closer results.

It may not have the whole thing.

Yeah, so twin primes is somewhat close.

Riemann hypothesis, I have no, I mean, it has happened by accident, I think.

Right, yeah.

It's states that are sort of viewed multiplicatively.

Like for questions only involving multiplication, no addition, the primes really do behave as randomly as you could hope.

So there's a phenomenon in probability code, square root cancellation, that if you want to poll, say, America on some issue, and you ask one or two voters, and you may have sampled a bad sample, and then you get a really imprecise measurement of the

full average, but if you sample more and more people, the accuracy gets better and better, and the accuracy improves like the square root of the number of people you sample.

So if you sample a thousand people, you can get like a 2-3% margin of error.

So in the same sense, if you measure the primes in a certain applicative sense, there's a certain type of statistic you can measure, and it's called the Riemann's data function, and it fluctuates up and down.

But in some sense, as you keep averaging more and more, if you sample more and more, the fluctuations should go down as if they were random.

And there's a very precise way to quantify that, and the Riemann hypothesis is a very elegant way that captures this.

But

As with many other ways in mathematics, we have very few tools to show that something really genuinely behaves really random.

And this is actually not just a little bit random, but it's asking that it behaves as random as an actually random set, this square root cancellation.

And we know, because of things related to the parity problem, actually, that most of us' usual techniques cannot hope to settle this question.

The proof has to come out of left field.

Yeah, but what that is, no one has any serious proposal.