Let $z_1,z_2,\ldots,z_n$ be i.i.d. random variables in the unit circle. Consider the polynomial $$ p(z)=\prod_{i=1}^{n}{(t-z_i)}=t^n+a_{1}t^{n-1}+\cdots+a_{n-2}t^2+a_{n-1}t+a_n $$ where the $a_i$ are the symmetric functions $$ a_{1}=(-1)\sum_{i=1}^{n}{z_{i}}\quad,\quad a_{2}=\sum_{1\leq i< j\leq n}{z_{i}z_{j}}\quad,\quad\ldots\quad,a_{n}=(-1)^{n}z_{1}z_{2}\ldots z_{n}. $$

How can we estimate the random variable $Z$ defined as $$ Z=\sum_{j=1}^{n}{|a_{j}|} $$ asymptotically as $n\to\infty$?

It is not very difficult to estimate $|\sum_{j=1}^{n}{a_{j}}|$ by estimating $\log p(1)$ via the CLT. However, $Z$ seems to be much more difficult. Any idea of what can work here?

Thanks!