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The next dilogarithm variant we mention is the p-adic dilogarithm, studied by R. Coleman and other authors. We ﬁx a prime number p and deﬁne (p) Li2 (x) = n>0, p∤n xn . n2 (14) This function can be written as Li2 (x)−p−2 Li2 (xp ), so in the complex domain it is simply a combination of ordinary dilogarithms and of no independent interest, but because we have omitted the terms in (14) with p’s in the denominator, the power series converges p-adically for all p-adic numbers x with (p) valuation |x|p < 1.
The deﬁnition and the name, of course, come from the analogy with the Taylor series of the ordinary logarithm around 1, − log(1 − z) = ∞ n=1 zn n for |z| < 1 , which leads similarly to the deﬁnition of the polylogarithm Lim (z) = ∞ n=1 zn nm for |z| < 1, m = 1, 2, . . The relation 1 d Lim (z) = Lim−1 (z) (m ≥ 2) dz z is obvious and leads by induction to the extension of the domain of deﬁnition of Lim to the cut plane C [1, ∞); in particular, the analytic continuation of the dilogarithm is given by z Li2 (z) = − 1 0 log(1 − u) du u for z ∈ C [1, ∞) .
1]), has many other beautiful properties. In particular: (II) D(z), which is a real-valued function on C, can be expressed in terms of a function of a single real variable, namely D(z) = z 1 D +D 2 z¯ 1 − 1/z 1 − 1/¯ z +D 1/(1 − z) 1/(1 − z¯) (2) The Dilogarithm Function 11 which expresses D(z) for arbitrary complex z in terms of the function D(eiθ ) = ℑ[Li2 (eiθ )] = ∞ sin nθ . ) Formula (2) is due to Kummer. 6 4 (III) All of the functional equations satisﬁed by Li2 (z) lose the elementary correction terms (constants and products of logarithms) when expressed in terms of D(z).