  Let $F(\sigma)=\sum_{n=1}^\infty \frac{X_n}{n^\sigma}$ be a random Dirichlet
series where $(X_n)_{n\in\mathbb{N}}$ are independent standard Gaussian random
variables. We compute in a quantitative form the expected number of zeros of
$F(\sigma)$ in the interval $[T,\infty)$, say $\mathbb{E} N(T,\infty)$, as
$T\to1/2^+$. We also estimate higher moments and with this we derive
exponential tails for the probability that the number of zeros in the interval
$[T,1]$, say $N(T,1)$, is large. We also consider almost sure lower and upper
bounds for $N(T,\infty)$. And finally, we also prove results for another class
of random Dirichlet series, e.g., when the summation is restricted to prime
numbers.
