Suppose and are two random variables. If E( ) = E( )E( ), are

Suppose ξ and ζ are two random variables. If E(ξ ζ ) = E(ξ )E(ζ ), are ξ and ζ independent? What about the converse: if ξ and ζ are independent, is E(ξ ζ ) = E(ξ )E(ζ )?