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 $\phi(0)=0$?

Yes, because we must conclude that there are zero positive integers less than or equal to zero. We do not even have to consider the remainder of the criteria or the fact that $1$ is relatively prime to all numbers.