Have any of your students approached you and said something to the effect of, "I saw this guy on youtube say 1+2+3+4+...=-1/12..."? Perhaps this talk will prepare you for the next time (no guarantees). The plan is to present one or two proofs of the functional equation relating \zeta(s) and \zeta(1-s). As a byproduct, we obtain special values of the zeta function, including \zeta(-1)=-1/12. We'll start with the definition of the zeta function (and a word or two about why it's interesting), the concept of analytic continuation, and (time permitting) we'll touch upon the gamma function, Poisson summation, theta functions, crazy contour integrals, Bernoulli numbers, and so on. Some knowledge of complex analysis is presupposed.
The functional equation of the Riemann zeta function
The functional equation of the Riemann zeta function