formula

$P(H=n)&=&P\bigg(E_n\cap F_n\cap \bigcap_{p:{\rm prime}}(E_{np}\cap F_{np})^c \bigg)\\ &=&E\bigg[1_{E_n\cap F_n}P\bigg(\bigcap_{p:{\rm prime}}(E_{np}\cap F_{np})^c|E_n,F_n\bigg)\bigg]\\ &=&E[1_{E_n}1_{F_n}P(H=1)]\\ &=&n^{-2s}/\zeta(2s)$