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Abstract:
The dilogarithm function Li\\sb 2(z)=\\sum\\sp∞\\sbn=1z\\sp n/n\\sp 2 (\\vert z\\vert lt;1) has many beautiful properties and plays a role in connection with problems in many parts of mathematics, most recently K-theory. This function cannot be extended in a one-valued way to the entire complex plane, but the modified function D(z)=\\cal I(Li\\sb 2(z)+\\log (1-z)\\log \\vert z\\vert) introduced by D. Wigner and S. Bloch can be extended as a continuous real-valued function on \\Bbb C which is real-analytic except at 0 and 1. For the higher polylogarithm functions Li\\sb m(z)=\\sum\\sp∞\\sbn=1z\\sp n/n\\sp m a similar one-valued modification D\\sb m(z) was defined in principle, but not written down in closed form or studied in detail, by D. Ramakrishnan. \\par In the present paper the function D\\sb m(z) is given explicitly and some of its properties are studied. It turns out that values of certain Kronecker double series related to special values of Hecke L-series for imaginary quadratic fields can be given in terms of the functions D\\sb m(z) (generalizing a result of S. Bloch involving D\\sb 2=D) and that the function D\\sb m is related to a certain Green's function for the quotient of the upper half-plane by the group of translations τ \\mapsto τ +n, n\\in \\Bbb Z. Most interestingly, a conjecture is presented according to which the value of the Dedekind zeta-function ζ\\sb F(s) as s=m for an arbitrary number field F and integer mgt;1 can be expressed in closed form in terms of finitely many values of D\\sb m(z) at arguments z\\in F.
