Dedekind Zeta Function - Special Values

Special Values

Analogously to the Riemann zeta function, the values of the Dedekind zeta function at integers encode (at least conjecturally) important arithmetic data of the field K. For example, the analytic class number formula relates the residue at s = 1 to the class number h(K) of K, the regulator R(K) of K, the number w(K) of roots of unity in K, the absolute discriminant of K, and the number of real and complex places of K. Another example is at s = 0 where it has a zero whose order r is equal to the rank of the unit group of O_K and the leading term is given by

Combining the functional equation and the fact that Γ(s) is zero at all integers less than or equal to zero yields that ζ_K(s) vanishes at all negative even integers. It even vanishes at all negative odd integers unless K is totally real (i.e. r₂ = 0; e.g. Q or a real quadratic field). In the totally real case, Carl Ludwig Siegel showed that ζ_K(s) is a non-zero rational number at negative odd integers. Stephen Lichtenbaum conjectured specific values for these rational numbers in terms of the algebraic K-theory of K.