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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 $compute the limit distribution of$\lim_{n\to\infty}{|a_{\lfloor |a_{\lfloor n/2 \rfloor}|}=\infty $$and estimate rfloor}| under the rate of growthappropriate normalizations? Thanks! 3 added 652 characters in body Let z_1,z_2,\ldots,z_n be i.i.d. random variables in the unit circle. Consider the polynomial$$ 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 $$and estimate the rate of growth? Thanks! 2 added 24 characters in body Let z_1,z_2,\ldots,z_n be i.i.d. random variables in the unit circle. Consider the polynomial$$ 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 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}}\quad,\quad\ldots\quad,a_{n}=(-1)^{n}z_{1}z_{2}\ldots 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?

Thanks!

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