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.)