Hacker News Reader

> 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.

One million thanks for this comment. This is the most concise and clear explanation of why the critical line is important that I have ever read.

> but how did the author learned to love, and more importantly to "fear" the RH? What's there to fear?

It's made abundantly clear: they learned to love it through the course they were fortunate enough to take as an undergraduate, and they learned to fear it later, as a researcher, when they realised that what there was to fear was wasting their career attacking something that was probably just too big and too hard to be sensible for them to attack.

I understand that the real proof of the RH is the friendships we make along the way, but what would happen if we just acted with the assumption that it is true?

This is the difference between a scientist and a mathematician. A scientist would consider the empirical evidence more than sufficient to consider RH to be true.

Of course there is hope that a proof of RH would reveal some deeper understanding -- not merely a conformation.

I'm not so sure a scientist would consider RH true at this point. Math has a rich history of conjectures appearing true for the first n dimensions, the first k inputs, the first z partial/tangential proofs, etc. The evidence for RH isn't any stronger than that for a plethora of eventually-proven-false conjectures.

Do you know if anyone has tabulated stats on conjecture proofs? That could be a fun dataset to examine.

As Richard K. Guy's strong law of small numbers states: There aren't enough small numbers to meet the many demands made of them.

There's been a lot of research that has assumed it to be true and built on top of it to find deeper implications.

Still, a number of people who've worked most closely with the Riemann zeta function, even the ones who led the computations and numerical verifications, have expressed doubts on whether the Riemann Hypothesis is actually true: https://arxiv.org/abs/math/0311162

The title relates to the article and the video is a supporting piece of media.

The article talks about the personal story of how the author first met the Riemann Hypothesis.

If one could predict with pinpoint accuracy all the prime numbers approaching infinity with certainty and with trivial resources (without having to test each one for prime-ness), would it have any impact on modern day cryptography?

On RSA-style prime-based crypto potentially but barring a much stronger result than merely being able to cheaply find primes things like ECC would probably still be safe.

Being able to find primes faster would make modern public key crypto better I guess, since you could find large keys faster and thus use larger keys.

Unless the theory also leads to faster factorization methods

Not an expert, but the key thing is usually factoring, e.g. finding which prime numbers multiplied together make a certain composite number. The hard part is finding which are the right prime numbers (there are lots of prime numbers). Finding numbers that are prime,just generally,is a very quick operation already.

The fear was alluded to.

If so much technology is built around a hypothesis, it is essentially building a large house of cards if the hypothesis turns out to be incorrect.

Imagine if the world set up a monetary system and what would happen after 100 years if suddenly everyone could print money when a fatal flaw in a system was discovered.

The part why he fears it is covered in the text below the video.

> title claims.

Discussions based on the title are useless. Good articles frequently have titles that are bad or not even relevant. Title of an article:

1. May not be chosen by the author. Its often editor decision.

2. Article can have multiple changing titles. For clickbait reasons.

Discussions based on title are generally worthless. In the HN you can ask someone to change the title to be more relevant.