What is...?

In the 1850s, Kummer discovered some striking congruences mod powers of a prime number p between values of the Riemann zeta function at negative odd integers.  This was part of his attempt to understand structural aspects of certain algebraic extensions of the rational numbers.  A century later, these congruences were packaged together into a “p-adic zeta function,” which turns out to encode significant arithmetic data.  I will introduce Kummer’s congruences, explain what p-adic zeta functions (and more generally p-adic L-functions) are, and describe how they turn out to be important in number theory and beyond.  This talk is aimed at nonexperts.