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This is a question about pedagogy.

I want to sketch the proof of the prime number theorem or any other application of complex analysis to number theory in a single lecture, in a complex analysis course for physics students.

Is this possible? How would you go about it?

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    $\begingroup$ As far as I remember, the proof of the prime number theorem is quite involved. I would probably choose a topic more easily physically motivated, such as Riemann surfaces or M\"obius transformations. $\endgroup$ – Mark Grant Jul 23 '18 at 7:36
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    $\begingroup$ You might be interested to know that there is a Stack Exchange site for Mathematics Education and pedagogical questions: matheducators.stackexchange.com $\endgroup$ – J W Jul 23 '18 at 15:13
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"Newman's short proof of the prime number theorem" by Don Zagier might work, in particular since there is an extensive discussion of the steps in that proof in this MSE posting. "The proof has a beautifully simple structure and uses hardly anything beyond Cauchy's theorem."

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    $\begingroup$ I would recommend Newman's own exposition over Zagier's. Specifically, I would recommend the account in Newman's book Analytic Number Theory. $\endgroup$ – Timothy Chow Jul 23 '18 at 21:13
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A bit easier than the Prime Number Theorem is Dirichlet's Theorem on primes in arithmetic progression. There are lots of proofs around, using complex variables: here is one.

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  • $\begingroup$ 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. $\endgroup$ – Vesselin Dimitrov Jul 23 '18 at 19:57
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    $\begingroup$ @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. $\endgroup$ – Timothy Chow Jul 23 '18 at 21:34
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    $\begingroup$ @TimothyChow: This uses a real variable $s \to 1+$. $\endgroup$ – Vesselin Dimitrov Jul 23 '18 at 22:27
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I agree with Mark Grant, since I also remember that the first proofs of the Prime number theorem given by J. Hadamard and C. J. de la Vallée Poussin, were quite long and involved: however, many mathematicians worked to simplify their proofs. Currently you can find proofs it in (graduate) textbooks as the one of Veech (1967) (ch. 6, pp. 200-238): you can read that chapter and figure out if your students will be able to attend fruitfully a lecture dealing with an abridged version of it. On my side, I remark that there are many interesting tools developed/introduced for the proof, for example the Tauberian theory wich is an interesting topic per se.

[1] Veech, W. A. (1967), A second course in complex analysis, New York-Amsterdam: W.A. Benjamin, Inc., pp. IX+246, MR0220903, Zbl 0145.29901

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