6.4. Coleman's generalisation to s = k
The result extends to every positive integer s=k\geq1. For s,z\in\C let
\Li_s(z)=\sum_{n\geq1}z^n n^{-s} be the polylogarithm, with its analytic
continuation to \C\setminus\{z\in\R:z\geq1\}; then \Li_s(1)=\zeta(s) and
\Li_1(z)=-\log(1-z). Coleman constructed p-adic polylogarithms
\Li_{k,p}(z), locally analytic on \Cp\setminus\{1\}, and proved:
Let \theta be a non-trivial Dirichlet character of conductor N, let
k\geq1 be an integer, and let \varepsilon_N be a primitive N-th root of
unity. Then
(i) \displaystyle L(\theta,k)=\frac{1}{G(\theta^{-1})}\sum_{c\in(\Z/N\Z)^\times}\theta^{-1}(c)\,\Li_{k}(\varepsilon_N^{\,c});
(ii) \displaystyle L_p(\theta,k)=\big(1-\theta(p)p^{-k}\big)\frac{1}{G(\theta^{-1})}\sum_{c\in(\Z/N\Z)^\times}\theta^{-1}(c)\,\Li_{k,p}(\varepsilon_N^{\,c}).
The case k=1 recovers Theorem Theorem 6.1.1, using
\Li_1(z)=-\log(1-z). Part (i) rests on the analytic continuation of the
polylogarithm and the special values of Definition 2.1.3; part (ii)
is Coleman's theorem on the p-adic polylogarithms \Li_{k,p}.
Part (i) is the classical evaluation: exactly as in
Theorem 6.2.1 one expands L(\theta,k) as a Gauss-sum
combination of \sum_{n\geq1}\varepsilon_N^{\,nc}n^{-k}=\Li_k(\varepsilon_N^{\,c}),
the value at z=\varepsilon_N^{\,c} of the polylogarithm. Part (ii) is Coleman's
theorem. The mechanism is the s=1 argument run one level up: rather than
antidifferentiating F_\theta once to obtain \widetilde F_\theta, one takes the
k-th Coleman primitive, whose values at roots of unity are the p-adic
polylogarithms \Li_{k,p}(\varepsilon_N^{\,c}). Reading off L_p(\theta,k) by the
same restriction-to-\Zpx computation (1-\varphi\circ\psi) then produces the
Euler factor (1-\theta(p)p^{-k}) (which for k=1 recovers
Theorem 6.1.1) in front of the polylogarithm sum.
Theorem Theorem 6.1.1 (ii) is an instance of Perrin-Riou's p-adic
Beilinson conjectures, which describe non-critical special values of p-adic
L-functions of motives in terms of arithmetic data. Specialised to the
Kubota–Leopoldt p-adic L-function, they express L_p(\theta,k) via p-adic
regulators of cyclotomic units; the right-hand sides of
Theorem 6.1.1 (ii) and Theorem 6.4.1 (ii) can be
read in these terms. This result also yields a p-adic analogue of the analytic
class number formula.