Polylogarithms, Dedekind Zeta Functions, and the Algebraic K-Theory of Fields

Abstract

The Dedekind zeta function ζF(s) of an algebraic number field F is the most important invariant of F. Its Euler product tells how the unramified primes of Q split in F. Information about the ramified primes and about the behavior at infinity is contained in three integers Δ, n₊ and n₋: the first is the absolute value of the discriminant of F and is a positive integer whose prime divisors are precisely the primes ramifying in F, and the other two (= r ₁ + r ₂ and r ₂ in the standard notation) give the dimensions of the (+1)- and (− l)-eigenspaces of complex conjugation on $F{ \otimes_{\mathbb{Q}}}\mathbb{R}\left( {{\mathbb{R}^{{{r_1}}}} \times {\mathbb{C}^{{{r_2}}}}} \right)$ . These invariants are in turn determined by ζF(s) via its functional equation

$$ \varsigma_F^{*}(s) : = {\Delta^{{ \frac{s}{2} }}}{\left( {{\pi^{{ - \frac{s}{2} }}}\Gamma \left( {\frac{s}{2}} \right)} \right)^{{{n_{ + }}}}}{\left( {{\pi^{{ - \frac{s}{2} }}}\Gamma \left( {\frac{{s + 1}}{2}} \right)} \right)^{{{n_{ - }}}}}{\varsigma_F}(s) = \varsigma_F^{*}\left( {1 - s} \right) $$

((1))

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