Tempering the Polylogarithm

converges when |z| < 1, defining the classical polylogarithm function, equal to − log(1 − z) when s = 1. In general, the behavior of these functions at z = 1 is complicated: it is known, for example, that Lis has an analytic continuation to the cut plane C − 1,∞). The fact that the ‘modified’ polylogarithm of Bloch, Ramakrishnan, Wigner, Wojtkowiak, Zagier, . . . , defined as the real, or imaginary part of ∑