Teaching prime number theorem in a complex analysis class for physicists 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?
 A: 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.)
A: "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."
A: 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. 
A: 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.
A: 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.
