Strong Invariance Principle for Random Dirichlet Series

Source abstract

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 $\mathcal{P}$ 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 $Y(s)=\sum_{k=2}^{\infty} \frac{(\log k)^α}{k^{s+1/2}}$ with $α=-1/2$ admits the same approximation as $X(s)$, and a direct relationship between $X(s)$ and $Y(s)$ with $α=-1/2$ is established. The strong invariance principle for $Y(s)$ with $α>-1/2$ is also obtained.