Inductive construction of the p-adic zeta functions for noncommutative p-extensions of exponent p of totally real fields

We construct the p-adic zeta function for a one-dimensional (as a p-adic Lie extension) noncommutative p-extension F∞ of a totally real number field F such that the finite part of its Galois groupGis a p-group of exponent p. We first calculate theWhitehead groups of the Iwasawa algebra Λ(G) and its canonical Ore localization Λ(G)_S by using Oliver and Taylor's theory of integral logarithms. This calculation reduces the existence of the noncommutative p-adic zeta function to certain congruences between abelian p-adic zeta pseudomeasures. Then we finally verify these congruences by using Deligne and Ribet's theory and a certain inductive technique. As an application we prove a special case of (the p-part of) the noncommutative equivariant Tamagawa number conjecture for critical Tate motives.