10 ms·

The video is great. But it fails on two counts. One is that it doesn't say what the title claims. We know now what the Rieman Hypothesis is, but how did the au

The video is great. But it fails on two counts. One is that it doesn't say what the title claims. We know now what the Rieman Hypothesis is, but how did the author learned to love, and more importantly to "fear" the RH? What's there to fear? The name of the famous movie (with the bomb) doesn't even have "fear" in it, so the author had something in mind with "fear", but after watching the video, I have no clue what. Second is the promise in the beginning of the video to give us a hint why the RH is important. What we learn is that somehow each zeta zero adds another harmonic to some type of series approximation of the (cousin of the) prime counting function. Which is great; if we add enough harmonics we get to approximate this function as precisely as we want. But why is it important that these zeros are on the vertical line Im z = 0.5 ? I have absolutely no clue. Don't get me wrong, I find that watching this video was a very good investment of 16 minutes of my life. But there is no need to overpromise, especially since the video delivers a lot as is.

Covzire 6y ago
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?
- simias 6y ago
  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.
- gameswithgo 6y ago
  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
- bawolff 6y ago
  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.
red_trumpet 6y ago
The part why he fears it is covered in the text below the video.
nabla9 6y ago
> 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.
wodenokoto 6y ago
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.
gimboland 6y ago
> 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.
- Tade0 6y ago
  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?
  - waynecochran 6y ago
    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.
    - hansvm 6y ago
      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.
      - jjgreen 6y ago
        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.
  - rfurmani 6y ago
rfurmani 6y ago
> 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.
- drbacon 6y ago
  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.
mensetmanusman 6y ago
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.

Covzire 6y ago

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?

simias 6y ago
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.
gameswithgo 6y ago
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
bawolff 6y ago
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.

red_trumpet 6y ago

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

nabla9 6y ago

> 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.

wodenokoto 6y ago

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.

gimboland 6y ago

> 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.

Tade0 6y ago
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?
- waynecochran 6y ago
  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.
  - hansvm 6y ago
    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.
    - jjgreen 6y ago
      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.
- rfurmani 6y ago

rfurmani 6y ago

> 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.

drbacon 6y ago
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.

mensetmanusman 6y ago

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.