Distribution of largest entry in a random vector

This problem was solved in Extreme statistics of complex random and quantum chaotic states (2007). The first two moments of the maximum absolute value of the elements of the random unit vector, $t=\max(|z_1|^2,|z_2|^2,\ldots |z_n|^2)$, are $$\langle t\rangle=\frac{H(n,1)}{n},\;\;\langle t^2\rangle=\frac{H^2(n,1)+H(n,2)}{n(n+1)}$$ where $H(n,k)$ is the $k$-th harmonic number. The cumulative density $F(t)$ in the large-$N$ limit is $$F(t)\rightarrow (1-e^{-nt})^n\approx\exp\left(-e^{-nt+\ln n}\right),\;\;\text{for}\;\;n\rightarrow\infty,$$ which is a Gumbel distribution.

$\newcommand{\C}{\mathbb C}$$\newcommand{\R}{\mathbb R}$Let $U=(U_1,\dots,U_n)$ be a random vector uniformly distributed on the unit sphere in $\C^n$. Then the distribution of the random vector $V:=(X_1,Y_1,\dots,X_n,Y_n)$ is uniform on the unit sphere in $\R^{2n}$, where $X_j:=\Re U_j$ and $Y_j:=\Im U_j$. So, $V$ equals $W:=(W_1,\dots,W_{2n})$ in distribution, where $$W_j:=\frac{Z_j}{\sqrt{Z_1^2+\dots+Z_{2n}^2}} $$ and $Z_1,\dots,Z_{2n}$ are iid standard normal random variables (r.v.'s). So, the random vector $(|U_1|^2 ,\dots,|U_n|^2)$ equals $R:=(R_1,\dots,R_n)$ in distribution, where $$R_j:=\frac{T_j}{T_1+\dots+T_n} $$ and $T_k:=Z_{2k-1}^2+Z_{2k}^2$, so that $T_1,\dots,T_n$ are iid standard exponential r.v.'s. So, $\max_j|U_j|^2$ equals $\max_j R_j$ in distribution. The distribution of $\max_j R_j$ was found a long time ago; see e.g. Irwin (1955) and historical references therein going back to as far as 1897. In particular, according to formula (1) in Irwin's paper,
\begin{equation}\label{eq:fisher} P(\max_j |U_j|^2>x)=P(\max_j R_j>x)=\sum_{j=1}^n(-1)^{j-1}\binom nj(1-jx)_+^{n-1} \end{equation} for $x\in[0,1]$.