Timeline for Teaching prime number theorem in a complex analysis class for physicists
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 2, 2022 at 20:51 | comment | added | duje | @LSpice The author is Pete L. Clark, and the material is essentially Chapter 17 of alpha.math.uga.edu/~pete/4400FULL.pdf | |
| Jan 1, 2022 at 2:44 | comment | added | KConrad | @VesselinDimitrov the method of proving $L(1,\chi) \not= 0$ for all nontrivial Dirichlet characters $\chi$ might use complex analysis. Of course Dirichlet himself could do no such thing, since complex analysis was in a rather primitive state at the time of his proof (1837). Nowadays a common method of proving the nonvanishing of $L(1,\chi)$ uses Landau's theorem about Dirichlet series with nonnegative coefficients, and that theorem is about analytic continuation, while the application of it to a proof of $L(1,\chi) \not= 0$ is a proof by contradiction using a cancellation of zeros and poles. | |
| Dec 31, 2021 at 23:13 | comment | added | LSpice | @GerryMyerson, thanks. I tried that, with too many results. Do you happen to remember the author? | |
| Dec 31, 2021 at 22:05 | comment | added | Gerry Myerson | @LSp, sorry, I'm sure it required no password when I posted it, and I didn't download a copy of whatever was there at the time. All I can suggest is typimg something like Dirichlet's Theorem on primes in arithmetic progression into Google, to see what comes up. | |
| Dec 31, 2021 at 18:38 | comment | added | LSpice | Your link seems to require a password to access. | |
| Jul 23, 2018 at 22:27 | comment | added | Vesselin Dimitrov | @TimothyChow: This uses a real variable $s \to 1+$. | |
| Jul 23, 2018 at 21:34 | comment | added | Timothy Chow | @VesselinDimitrov : The proof in Serre's Course in Arithmetic certainly uses complex variables in what looks to me like an essential way. It derives it by studying the logarithm of the relevant $L$-functions. | |
| Jul 23, 2018 at 19:57 | comment | added | Vesselin Dimitrov | Does any of those proofs of Dirichlet, though, use complex variables in an essential way? I remember saying this to myself when I saw Dirichlet's theorem on primes in arithmetic progressions listed in a syllabus for a course in complex analysis. PNT on the other hand gives a true showcase of one complex variable, and indeed Hadamard's solution came simultaneous with the historical development of that subject. | |
| Jul 23, 2018 at 13:10 | history | answered | Gerry Myerson | CC BY-SA 4.0 |