Real zeros of random Dirichlet series

Let $F(\sigma )$ be the random Dirichlet series $F(\sigma )={\sum }_{p\in \mathcal{P}}\frac{X_p}{p^{\sigma }}$, where $\mathcal{P}$ is an increasing sequence of positive real numbers and $(X_p{)}_{p\in \mathcal{P}}$ is a sequence of i.i.d. random variables with $\mathbb{P}(X_1=1)=\mathbb{P}(X_1=-1)=1\mathrm{/}2$. We prove that, for certain conditions on $\mathcal{P}$, if ${\sum }_{p\in \mathcal{P}}\frac{1}{p}<\mathrm{\infty }$ then with positive probability $F(\sigma )$ has no real zeros while if ${\sum }_{p\in \mathcal{P}}\frac{1}{p}=\mathrm{\infty }$, almost surely $F(\sigma )$ has an infinite number of real zeros.