On the zeros of this polynomial I am studying, for a fixed integer $N\geq 2$, the zeros of the sequence of polynomials $p_{n}(z):=a_{0}+a_{1} z+\ldots + a_{n}z^{n}$ where
$$
a_{0}:=N,  \quad a_{k}:= \frac{n^{k}}{k!}\big[ \log^{k}(2)+\ldots + \log^{k}(N) \big] ,\quad k=1,\ldots, n.
$$
From some numerical experiments with the software Maple, I "suspect" that (for a fixed $n>2$) all the zeros of $p_{n}$ are simples, but I can not prove it formally.
Some idea or suggestions to prove/disprove this fact?
Many thanks in advance for your comments.
 A: (This just got too long for a comment.)
It's by no means a proof, but it would seem miraculous if it had multiple roots, simply because polynomials with real coefficients generically have single roots, e.g. they're dense and for any fixed degree their complement has measure 0. Actually proving it in particular cases can be challenging, much like proving irrationality of any particular number. Why is simplicity your focus here?
Your problem can be equivalently rephrased as: are there any complex solutions $z_0$ to the following:
$$N + \sum_{m=1}^N \sum_{k=1}^n \frac{(n z_0 \log m)^k}{k!} = 0 = \sum_{m=1}^N \log m \sum_{k=0}^{n-1} \frac{(n z_0 \log m)^k}{k!},$$
for fixed integers $n > 2$ and $N \geq 2$? (This is simply setting your polynomial and its derivative simultaneously equal to zero.) This does not seem immediately helpful. I suppose it is doing something like differential Galois theory--trying to solve an algebraic equation where we've also allowed logarithms. Maybe there's an effectively computable theory of intersections of such fields? I'm not aware of it, at any rate.
