Not offered in Spring 2017
(Simon Marshall) This course will be an introduction to the theory of L-functions. We will begin with the Riemann zeta function and Dirichlet L-functions, and later introduce an abstract framework for dealing with more general L-functions. The topics covered will include analytic continuation and functional equation, convex and subconvex bounds, counting zeros, consequences of the Riemann hypothesis, zero-free regions, the Siegel zero, and the prime number theorem.
We will use Fourier and Mellin transforms, and contour integration, frequently. We will make occasional use of the Phragmen-Lindelof convexity principle and the Hadamard factorisation theorem. Familiarity with algebraic number theory will not be required in the first half of the course, but may be needed towards the end.