P-adic L-functions and their special values

This talk will begin by recalling classical facts about the relationship between values of the Riemann zeta function at negative integers and the arithmetic of cyclotomic extensions of the rational numbers. We will then consider a generalisation of this theory due to Iwasawa, and along the way we shall define the p-adic Riemann zeta function. Time permitting, I will also say something about what zeta values at positive integers have to do with the fundamental group of the projective line minus three points