2. What a p-adic L-function should be
This chapter motivates the definition and study of p-adic L-functions. We
begin with the classical complex L-functions, recall how special values encode
arithmetic, and then lean slowly towards the p-adic world. The running example
— and the central object of these notes — is the Riemann zeta function. The
chapter closes by fixing the interpolation property that the Kubota–Leopoldt
p-adic L-function \zeta_p must satisfy; constructing such an object is
the goal of the next several chapters.
Throughout, p is a fixed prime and \ell ranges over rational primes.