Strong Invariance Principle for Random Dirichlet Series

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.