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
$$
a_{1}=(-1)\sum_{i=1}^{n}{z_{i}}\hspace{0.3cm},\quad a_{2}=(-1)^2\sum_{1\leq i< j\leq n}{z_{i}z_{j}}\hspace{0.3cm} \quad\ldots\quad a_{n}=(-1)^{n}z_{1}z_{2}\ldots z_{n}.
$$
    
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? 

> **Update:** If we look at the term at the central symmetric random variable
> $a_{\lfloor n/2 \rfloor}$ $$
> a_{\lfloor n/2 \rfloor}=\text{sum of
> the products of $\lfloor n/2 \rfloor$
> of different $z_{i}$'s} $$ 
> it is not hard to see that it
> has uniform distributed phase in
> $(-\pi,\pi]$. However, its magnitude
> is blowing up extremely fast!
> 
> Simulations showed that for $n=100$
> the mean of $\log |a_{\lfloor n/2
> \rfloor}|\approx 15$ which was
> surprisingly big.
> 
> Does anyone knows how to prove that 
> $$ \lim_{n\to\infty}{|a_{\lfloor n/2
> \rfloor}|}=\infty $$ and estimate the
> rate of growth?




Thanks!