Let $U$ be a $d\times d$ unitary matrix, and $U\_{i,j}$ be its matrix elements. I am interested in the following quantity
$$\int dU \max\_j |U\_{1,j}|^2 \ , $$
where $dU$ is the uniform Haar measure over ${\rm SU}(d)$. 


Please let me know if you have any idea for calculating this integral for general $d$.



Average of the maximum matrix element over the Haar measure
The answer to the question as stated (maximum of row elements) has been solved in [Extreme statistics of complex random and quantum chaotic states](https://arxiv.org/abs/0708.0176), see also this [MO posting](https://mathoverflow.net/a/298776/11260): 


$$\int dU \max\_j |U\_{1,j}|^2 =\frac{1}{d}\sum\_{j=1}^d \frac{1}{j}.$$


For large $d$ this tends to $(1/d)\log d$. The complete probability distribution of the row-maximum is known.




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The maximum of *all* matrix elements is more difficult and only large-$d$ asymptotics has a closed-form expression, see [Maxima of entries of Haar distributed matrices](https://link.springer.com/article/10.1007/s00440-004-0376-5) (alternative
[link](http://users.stat.umn.edu/~jiang040/papers/haar1.pdf)).


If $W\_d$ is the maximum matrix element in absolute value of a Haar-distributed $d\times d$ unitary matrix, then $W\_d^2\rightarrow (2/d)\log d$ in probability for $d\rightarrow\infty$, so twice as large as for the row-maximum.


