7. The residue at s = 1
The Riemann zeta function \zeta(s) has a simple pole at s=1 with residue
1. On the p-adic side we constructed \zp not as a measure but as a
pseudo-measure, precisely so as to accommodate a
potential pole at the trivial character. In this chapter we show that this pole
is real and simple, and we compute its residue — the p-adic analogue of the
analytic class number formula. Throughout, p is a fixed odd prime, a is a
fixed topological generator of \Zpx, and \logp denotes the Iwasawa
p-adic logarithm.
As in the chapter on the value at s=1, it is convenient to phrase everything
through the analytic branches \zpi of the p-adic zeta function: for
i \in \set{1, 2, \ldots, p-1} one sets
\zpi(s) = \int_{\Zpx} \Teich(x)^{i}\,\ang{x}^{1-s} \cdot \zp,
where \Teich is the Teichmüller character and
\ang{x} = \Teich(x)^{-1}x is the projection of x \in \Zpx to its
1-units. The behaviour of \zp at the trivial character is encoded by the
behaviour of \zeta_{p,p-1} at s=1.