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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 function Li2 (x), and the corresponding higher p-adic (p) polylogarithms Lim (x), have good properties of analytic continuation and are related to p-adic L-functions [11].

The Bloch-Wigner dilogarithm is the imaginary part of Li2 (z) (corrected by a multiple of log |z| Li1 (z) to make its analytic properties better) and hence vanishes on P1 (R), while the Rogers dilogarithm is the restriction of Li2 (z) (corrected by a multiple of log |z| Li1 (z) to make its analytic properties better) to P1 (R) and takes its values most naturally in the circle group R/(π 2 /2)Z. It is reasonable to ask whether there is then a function D(z) with values in C/(π 2 /2)Z or at least C/π 2 Q whose imaginary part is D(z) and whose restriction to P1 (R) is L(z).

E Now much more information about the actual order of K2 (OF ) is available, thanks to the work of Browkin, Gangl, Belabas and others. Cf. [7], [3] of the bibliography to Chapter II. F The statement “the general picture is not yet clear” no longer holds, since after writing it I found hundreds of further numerical examples of identities between special values of polylogarithms and of Dedekind zeta functions and was able to formulate a fairly precise conjecture describing when such identities occur.

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