I am reading the book by E. Kowalski on Probabilistic Number Theory. Link to the book: https://people.math.ethz.ch/~kowalski/probabilistic-number-theory.pdf
I am stuck on Exercise $3.2.4$. It seems that it should follow "elementarily" from Theorem $3.2.3$, but I can't think of any approaches that use the Theorem directly. Any hints and references are welcome. I am learning the basics of Probability theory on the go, so it would be helpful if you state the deep results from Probability theory used in the solution (if any).
Here $Z_D$ is the random Euler product defined by $Z_D(s) = \prod_p \left(1-X_p p^{-s}\right)^{-1}$, with $(X_p)_p$ being a sequence of independent random variables indexed by the primes, which are identically distributed, with distribution uniform on the unit circle $\mathbb{S}^1\subset\mathbb{C}^{\times}$.