4 ms·

Man HR is harsh as hell. While yes, if a proof of RH is found it will probably use advanced tools, ever finding a proof with elementary tools is not totally out

Man HR is harsh as hell. While yes, if a proof of RH is found it will probably use advanced tools, ever finding a proof with elementary tools is not totally out of the question. Regarding the Prime Number Theorem, which can be seen as a baby version of RH, experts in the field such as Hardy have been quoted as saying elementary proofs would not exist. Then 50 years after Hadamard and Poussin found the original proofs, Selberg and Erdos published proofs based on elementary methods. No one can say that it's worthless for someone to try the same for RH.

pfortuny 7y ago
Notice also the decidability of primality in polynomial time, which was proved by some Computer Science graduates (Neeraj Kayal, and Nitin Saxena) in 2011, and led Bombieri (one of the leading experts on Number Theory) to say that Number Theory may have gone astray in theoretical complexity in some fields.
- throwawaymath 7y ago
  That's very interesting, do you know offhand where Bombieri said that?
math-fan 7y ago
I've taken math up to calculus and linear algebra and enjoy watching 3B1B proofs. Is that enough to follow the elementary Prime Number Theorem proof? I've been interested in understanding that for a while. Any good links?
- dilippkumar 7y ago
  Check out the book “Prime Obsession” It was easy for me to follow along.
- pmiller2 7y ago
  You can probably slog through it with that level of knowledge, if you’v got a good grasp of summation notation. Here’s one version of the proof: https://www.math.lsu.edu/~mahlburg/teaching/handouts/2014-7230/Selberg-ElemPNT1949.pdf https://www.math.lsu.edu/~mahlburg/teaching/handouts/2014-72... See also: https://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf https://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.... https://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf https://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.... And, a related, very simple and elementary proof of Chebyschev’s theorem that there is always a prime number between n and 2n: https://en.wikipedia.org/wiki/Proof_of_Bertrand%27s_postulate https://en.wikipedia.org/wiki/Proof_of_Bertrand%27s_postulat...
- fspeech 7y ago
  You will need to learn some complex analysis which is very beautiful. Ash has a very good chapter on the Prime Number Theorem as a reward for learning complex variables: https://faculty.math.illinois.edu/~r-ash/CV.html https://faculty.math.illinois.edu/~r-ash/CV.html You don't need to learn the whole book just some knowledge of contour integral and analytic continuation is sufficient to understand the proof. So you could just start with the last chapter and fill the gaps as you go. (If you go linearly through the book you will encounter things like the normal family not required for your stated goal. Even though the normal family stuff are even more beautifully amazing IMHO the presentation here may be too abstract to grok and could discourage one to soldier on.)
  - User23 7y ago
    Visual Complex Analysis by Tristan Needham is an incredibly approachable textbook on the subject.
oneepic 7y ago
I agree, I'm not saying a simple proof is ever out of the question either. But my failed attempt makes me wonder how much of a creative burst a typical person would have to have in order to come up with a simple solution in this case.
- pmiller2 7y ago
  Don’t conflate “elementary” in the GP post with “simple.” Elementary in this context just means that the proof does not use tools from complex analysis (i.e. it only uses theorems from real analysis, which is basically freshman calculus on steroids). The elementary proof of the prime number theorem is not easy by any means.
- Ragib_Zaman 7y ago
  I didn't mean you were the one implying an elementary proof is unobtainable, that was directed at some of the other replies to your comment like "oh really?". I agree that it's enlightening to give these things a try.
- deleted 7y ago
  [deleted]

pfortuny 7y ago

Notice also the decidability of primality in polynomial time, which was proved by some Computer Science graduates (Neeraj Kayal, and Nitin Saxena) in 2011, and led Bombieri (one of the leading experts on Number Theory) to say that Number Theory may have gone astray in theoretical complexity in some fields.

throwawaymath 7y ago
That's very interesting, do you know offhand where Bombieri said that?

math-fan 7y ago

I've taken math up to calculus and linear algebra and enjoy watching 3B1B proofs. Is that enough to follow the elementary Prime Number Theorem proof? I've been interested in understanding that for a while. Any good links?

dilippkumar 7y ago
Check out the book “Prime Obsession” It was easy for me to follow along.
pmiller2 7y ago
You can probably slog through it with that level of knowledge, if you’v got a good grasp of summation notation. Here’s one version of the proof: https://www.math.lsu.edu/~mahlburg/teaching/handouts/2014-7230/Selberg-ElemPNT1949.pdf https://www.math.lsu.edu/~mahlburg/teaching/handouts/2014-72... See also: https://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf https://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.... https://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf https://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.... And, a related, very simple and elementary proof of Chebyschev’s theorem that there is always a prime number between n and 2n: https://en.wikipedia.org/wiki/Proof_of_Bertrand%27s_postulate https://en.wikipedia.org/wiki/Proof_of_Bertrand%27s_postulat...
fspeech 7y ago
You will need to learn some complex analysis which is very beautiful. Ash has a very good chapter on the Prime Number Theorem as a reward for learning complex variables: https://faculty.math.illinois.edu/~r-ash/CV.html https://faculty.math.illinois.edu/~r-ash/CV.html You don't need to learn the whole book just some knowledge of contour integral and analytic continuation is sufficient to understand the proof. So you could just start with the last chapter and fill the gaps as you go. (If you go linearly through the book you will encounter things like the normal family not required for your stated goal. Even though the normal family stuff are even more beautifully amazing IMHO the presentation here may be too abstract to grok and could discourage one to soldier on.)
- User23 7y ago
  Visual Complex Analysis by Tristan Needham is an incredibly approachable textbook on the subject.

oneepic 7y ago

I agree, I'm not saying a simple proof is ever out of the question either. But my failed attempt makes me wonder how much of a creative burst a typical person would have to have in order to come up with a simple solution in this case.

pmiller2 7y ago
Don’t conflate “elementary” in the GP post with “simple.” Elementary in this context just means that the proof does not use tools from complex analysis (i.e. it only uses theorems from real analysis, which is basically freshman calculus on steroids). The elementary proof of the prime number theorem is not easy by any means.
Ragib_Zaman 7y ago
I didn't mean you were the one implying an elementary proof is unobtainable, that was directed at some of the other replies to your comment like "oh really?". I agree that it's enlightening to give these things a try.
deleted 7y ago
[deleted]