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

On the Gross order of vanishing conjecture for large vanishing orders