Prime numbers, Riemann, and Langlands

Prime numbers have held a mystery over number theory since before Euclid. To introduce a powerful new tool to the subject, Riemann defined his analytic zeta function; with it, he described the Prime Number Theorem and conjectured Riemann's Hypothesis. More than 100 years after 1859, R.P. Langlands generalized Riemann's function and - among other things - explained how every zeta function in number theory (whether due to Artin, or to an elliptic curve, or whatever) might be one of his generalized functions. In this short talk, we shall summarize briefly the contents of prime numbers, Riemann, and Langlands.