  The UMD property of a Banach space is one of the most useful properties when
one thinks about possible applications. This is in particular due to the
boundedness of the vector-valued Hilbert transform for functions with values in
such a space.
  Looking at operators instead of at spaces, it is easy to check that the
summation operator does not have the UMD property. The actual asymptotic
behavior however of the UMD constants computed with martingales of length n is
unknown.
  We explain, why it would be important to know this behavior, rephrase the
problem of finding these UMD constants and give some evidence of how they
behave asymptotically.
