  We obtain asymptotic formulas for the $2k$th moments of partially smoothed
divisor sums of the M\"obius function. When $2k$ is small compared with $A$,
the level of smoothing, then the main contribution to the moments come from
integers with only large prime factors, as one would hope for in sieve weights.
However if $2k$ is any larger, compared with $A$, then the main contribution to
the moments come from integers with quite a few prime factors, which is not the
intention when designing sieve weights. The threshold for "small" occurs when
$A=\frac 1{2k} \binom{2k}{k}-1$.
  One can ask analogous questions for polynomials over finite fields and for
permutations, and in these cases the moments behave rather differently, with
even less cancellation in the divisor sums. We give, we hope, a plausible
explanation for this phenomenon, by studying the analogous sums for Dirichlet
characters, and obtaining each type of behaviour depending on whether or not
the character is "exceptional".
