  We prove the Gross order of vanishing conjecture in special cases where the
vanishing order of the character in question can be arbitrarily large. In
almost all previously known cases the vanishing order is zero or one. One major
ingredient of our proofs is the equivalence of this conjecture to the
Gross-Kuz'min conjecture. We present here a direct proof of this equivalence,
using only the known validity of the Iwasawa Main Conjecture over totally real
fields.
