What is the Euler Totient of Zero? Wolfram MathWorld defines the Euler Totient function as follows:
The totient function phi(n), also called Euler's totient function, is defined    
as the number of positive integers <=n that are relatively prime to (i.e., do 
not contain any factor in common with) n, where 1 is counted as being 
relatively prime to all numbers

By this definition would $\phi(0)=0$?

If you read a little further down in the MathWorld entry, you find this:

By convention, $\phi(0)=1$, although Mathematica defines
  EulerPhi[$0$] equal to $0$ for consistency with its FactorInteger[$0$]
  command.

However, no reference is given for the convention.  It's a rare formula that calls out for $\phi(0)$ to be included.  The Book of Numbers, by Conway and Guy, offers one possibility.  On page 156, they write:

...the $n$th Farey series has length
$$1+\phi(1)+\phi(2)+\phi(3)+\cdots+\phi(n-1)+\phi(n)$$
  (the initial $1$ comes from the fact that we count both $0/1$ and $1/1$).

Note, however, that Conway and Guy do not write $\phi(0)$ for the initial $1$.
