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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
    Commented Jul 23, 2018 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
    Commented Jul 23, 2018 at 15:13
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    $\begingroup$ Haha one single lecture, when I took a course on analytic number theory the proof of the PNT was spread over several lectures. $\endgroup$
    – Hollis Williams
    Commented Oct 13, 2020 at 22:34

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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
    Commented Jul 23, 2018 at 21:13
  • $\begingroup$ @TimothyChow Beautiful book, but the corrected edition. The original one is full of annoying errata. $\endgroup$
    – efs
    Commented Dec 31, 2021 at 17:37
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    $\begingroup$ If anyone’s interested, I have a detailed writeup of Newman’s proof (adapted from Zagier’s paper) in sections 15-16 of my complex analysis lecture notes. $\endgroup$
    – Dan Romik
    Commented Jan 1, 2022 at 4:55
  • $\begingroup$ Update to my earlier comment: my lecture notes have been superseded by my book Topics in Complex Analysis, which is available for download under Open Access. See here for more information. The proof of the prime number theorem is the subject of chapter 2. $\endgroup$
    – Dan Romik
    Commented Sep 7 at 4:29
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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
    Commented Jul 23, 2018 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
    Commented Jul 23, 2018 at 21:34
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    $\begingroup$ @TimothyChow: This uses a real variable $s \to 1+$. $\endgroup$
    – Vesselin Dimitrov
    Commented Jul 23, 2018 at 22:27
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    $\begingroup$ @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. $\endgroup$
    – KConrad
    Commented Jan 1, 2022 at 2:44
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    $\begingroup$ @LSpice The author is Pete L. Clark, and the material is essentially Chapter 17 of alpha.math.uga.edu/~pete/4400FULL.pdf $\endgroup$
    – duje
    Commented Jan 2, 2022 at 20:51
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For physics students with familiarity with the Dirac delta function, the pdf "Dirac’s Delta Function and Riemann’s Jump Function J(x) for the Primes" available in my blog post Riemann’s Jump Function J(x) for the Primes should be easy to digest in one class.

With my background in mathematical physics, my math class in complex analysis would have been substantially more interesting if the following topics had been worked in to some degree (at least mentioned): the relationship between complex analysis and electrostatics; the integral transforms--Fourier, Laplace, Mellin--with physical applications; the Cauchy problem and relation to the heat equation and Brownian motion leading into Green/Green's functions; Young's double slit problem and quantum mechanics; the Heaviside operational calculus, its relation to the Laplace transform, and applications to transient EM signals in cables--slipping in the fractional calculus, the Euler beta function integral (related to string theory) and the Pochhammer integration curve; local and global conformal transformations and their relation to string theory. (Cartier's "Mathemagics" would have thrilled me as extracurricula reading.)

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I agree with Mark Grant's comment above, 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 reasonably short proofs in (graduate) textbooks as [1], chapter 6, pp. 200-238: you may 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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    $\begingroup$ @MarkGrant's comment referenced here. $\endgroup$
    – LSpice
    Commented Dec 31, 2021 at 18:34
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    $\begingroup$ @LSpice again thank you. Added link to the comment in the body of the question. $\endgroup$
    – Daniele Tampieri
    Commented Dec 31, 2021 at 18:42
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You can start by defining the Riemann Zeta function as $\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$ for $s$ real and then prove the Euler product formula, $\zeta(s) = \prod_{p} \frac{1}{1-p^{-s}}$ for $s>1$ where the product runs over all prime numbers. Then you can view the zeta function as a function of a complex variable. This will illustrate a nontrivial connection between a complex function and the prime numbers. After this, I suggest that you go through Riemann's epoch 8 page paper with the class as outlined in Edwards book on the zeta function. Here, Riemann uses tools from complex analysis such as contour integration to study zeta. He proves analytic continuation, the functional equation for zeta, and goes on to examine the connection between the zeros of the zeta function and the distribution of the prime numbers.

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  • $\begingroup$ @KConrad yes corrected $\endgroup$
    – Mustafa Said
    Commented Jan 2, 2022 at 3:23

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