Strong Invariance Principle for Random Dirichlet Series

It is shown that the random prime Dirichlet series $ X(s)=\sum_{p\in \mathcal{P}}\frac{ξ_p}{p^{s+1/2}} $ can be almost surely approximated by a Brownian motion, where $ \{ξ_p\}_{p\ge 1} $ are independent and identically distributed random variables with zero mean and finite variance, and $ \mathcal{