The Riemann zeta function $\zeta(s)$ is defined by a power series $\sum_{n\geq1} n^{-s}$ which converges for $Re(s)>1$, but which has a meromorphic continuation to the entire complex plane. The Riemann hypothesis is a deep conjecture about the zeros of this function, but its poles are well-understood (there is only one, at $s=1$). This function is really the tip of the iceberg though -- the Riemann zeta function is just one example of an $L$-function. There are some tantalizingly simple examples of $L$-functions discovered almost 100 years ago by Emil Artin which do have meromorphic continuation but whose poles are not well-understood, and other examples coming from systems of polynomial equations where even the meromorphic continuation is still open. I will give an overview of the situation, with no proofs, and attempting to stress the wide and surprising array of techniques (arithmetic, geometric, $p$-adic, analytic) that have been used to solve the innocuous-looking problem of extending an $L$-function $\sum_{n\geq1} a_n.n^{-s}$ to the complex plane.

Speaker's webpage

http://www2.imperial.ac.uk/~buzzard/