characteristic function of some random variables

Question

I feel curious about which random variable's characteristic function is 1 ? And which random variable's characteristic function is 0 ? By definition, it seems that it is impossible to get 0 characteristic function.

Answer 1

If $X$ is a random variable, then its characteristic function is defined by $$\phi(\xi) = \mathbb{E}\exp(i \xi X).$$ In particular, any characteristic function $\phi$ satisfies $$\phi(0)= 1.$$ This means that there cannot exist a characteristic function which is identically zero.

The function $\phi=1$ is the characteristic function of the random variable $X=0$.