  We construct the p-adic zeta function for a one-dimensional (as a p-adic Lie
extension) non-commutative p-extension of a totally real number field such that
the finite part of its Galois group is a pgroup with exponent p. We first
calculate the Whitehead groups of the Iwasawa algebra and its canonical Ore
localisation by using Oliver-Taylor's theory upon integral logarithms. This
calculation reduces the existence of the non-commutative p-adic zeta function
to certain congruence conditions among abelian p-adic zeta pseudomeasures. Then
we finally verify these congruences by using Deligne-Ribet's theory and certain
inductive technique. As an application we shall prove a special case of (the
p-part of) the non-commutative equivariant Tamagawa number conjecture for
critical Tate motives. The main results of this paper give generalisation of
those of the preceding paper of the author.
