Strong Invariance Principle for Random Dirichlet Series

It is shown that the random prime Dirichlet series $X(s)={\sum }_{p\in \mathcal{P}}\frac{{\xi }_p}{p^{s+1\mathrm{/}2}}$ can be almost surely approximated by a Brownian motion, where $\{{\xi }_p{\}}_{p\ge 1}$ are independent and identically distributed random variables with zero mean and finite variance, and $\mathcal{P}$ is the set of prime numbers. As a consequence, the functional law of the iterated logarithm, limit laws of the iterated logarithm, and Chung’s law of the iterated logarithm for $X(s)$ are obtained. It is also shown that the random Dirichlet series $Y(s)={\sum }_{k=2}^{\mathrm{\infty }}\frac{(\log k{)}^{\alpha }}{k^{s+1\mathrm{/}2}}$ with $\alpha =-1\mathrm{/}2$ admits the same approximation as $X(s)$, and a direct relationship between $X(s)$ and $Y(s)$ with $\alpha =-1\mathrm{/}2$ is established. The strong invariance principle for $Y(s)$ with $\alpha >-1\mathrm{/}2$ is also obtained.