5. Interpolation at Dirichlet characters
The Kubota–Leopoldt pseudo-measure \zetap was built purely from values of the
Riemann zeta function, with Dirichlet characters never entering the construction.
Remarkably, \zetap nonetheless interpolates Dirichlet L-values as well. This
chapter establishes that interpolation, first for characters of p-power
conductor, then for general conductors (where one even obtains a genuine measure
\zeta_\eta rather than a pseudo-measure), and finally repackages everything as
analytic functions on \Zp via the Mellin transform.
Throughout, p is an odd prime, (\eps_{p^n})_{n} is a compatible system of
primitive p^n-th roots of unity in \Qpbar (so \eps_{p^{n+1}}^p =
\eps_{p^n}), and L/\Qp is a finite extension large enough to contain the values
of the characters under consideration, with ring of integers \cO_L. A Dirichlet
character \chi of conductor p^n is viewed, via \Zpx \twoheadrightarrow
(\Z/p^n\Z)^\times, as a locally constant character of \Zpx. We write \Am_\mu
for the Mahler transform of a measure \mu and recall the standard substitution
e^t = T+1 relating power series to functions of a real variable.