formula

$P(E_{i_1}\cap \cdots \cap E_{i_p})&=&\sum_{n=1}^{\infty}P(X=ni_1\cdots i_p)\\ &=&\sum_{n=1}^{\infty}\frac{1}{(ni_1\cdots i_p)^s\zeta(s)}\\ &=&\frac{1}{(i_1\cdots i_p)^s}$