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We construct the p-adic zeta function for a one-dimensional (as a p-adic Lieextension) non-commutative p-extension of a totally real number field such thatthe finite part of its Galois group is a pgroup with exponent p. We firstcalculate the Whitehead groups of the Iwasawa algebra and its canonical Orelocalisation by using Oliver-Taylor's theory upon integral logarithms. Thiscalculation reduces the existence of the non-commutative p-adic zeta functionto certain congruence conditions among abelian p-adic zeta pseudomeasures. Thenwe finally verify these congruences by using Deligne-Ribet's theory and certaininductive technique. As an application we shall prove a special case of (thep-part of) the non-commutative equivariant Tamagawa number conjecture forcritical Tate motives. The main results of this paper give generalisation ofthose of the preceding paper of the author.

Inductive construction of the p-adic zeta functions for non-commutative p-extensions of totally real fields with exponent p