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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 bee

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.

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.