The Moebius function $\mu(n)$ is defined for squarefree integers $n$ as $(-1)^k$ if $k$ is the number of prime factors of $n$ (and $\mu(n)$ is defined as $0$ if $n$ is not squarefree). The Riemann hypothesis is equivalent to the statement $\sum_{n < x} \mu(n) = O(\sqrt x)$$\forall \epsilon > 0, \sum_{n < x} \mu(n) = o(x^{1/2 + \epsilon})$. If, on the other hand, you choose $\nu(n) = \pm 1$ randomly for squarefree $n$ and zero otherwise, then by the law of the iterated logarithm, with probability one $\sum_{n < x} \nu(n) = O(\sqrt x)$$\sum_{n < x} \nu(n) = O(\sqrt {x \log \log x})$.
Other randomness issues connected with the Riemann hypothesis for L-functions are estimates for sums of the type $\sum e^{2\pi i \alpha n}$ or $\sum \chi(n)$ for a Dirichlet character $\chi$, where $n$ may vary over the integers in an interval or just over primes in an interval (and I should have written $p$ in that case). The mean of these sums is usually easy to figure out and sharp estimates for the deviation from the mean are consequences of (or sometimes equivalent to) the Riemann hypothesis. You can phrase some of these things in terms of primes in arithmetic progressions but then it's not so easy to see the connection with randomness.
