the Mobius function

[Marc LeBrun]

 > Is there a natural ... assignment for mu(0)?

Without disagreeing with the many good comments, it's worth mulling the 
consequences of which properties we give primacy to in defining "natural" 
extensions to sequences.

Mostly the thread here has emphasized mu's multiplicative character, which 
is fine.  However that might equally well be seen as a derived trait rather 
than a defining attribute.

Recall that the mu sequence arises in many guises: for example it's "also" 
precisely the weights you need to invert a sum over divisors, as well as 
the coefficients you get for the Lambert series for x and the Dirichlet 
series for 1/zeta(x).

These all "happen to" coincide exactly when operating within the normal 
scope of the definitions.  However "natural" extensions to any of the 
definitions might very well diverge with respect to the others, and, while 
individually reasonable, they may not be mutually consistent (eg as with 
0^0 arguably being either 1 or 0).

And generalizations have ramifications.  For example, deriving mu from the 
inverse Mobius transform, mu(0) arises as the weight of the term a(n/0), 
thereby implying the existence of sequence elements of non-finite 
index.  Until we're ready to tangle with that hair ball, we had better hope 
mu(0) is (as Don Knuth put it) "not just zero, but strongly zero"!