Following is from http://matwbn.icm.edu.pl/ksiazki/aa/aa90/aa9023.pdf:
Theorem 1. Let $\varepsilon > 0$. For all but $O(t^{1/3+\varepsilon})$ positive integers $n \leq t$, the derivative of the polynomial $f(x)=1+x+x^2+\dots+x^n$ is irreducible.
I'm having trouble understand, what exactly is the $O(t^{1/3+\varepsilon})$ trying to say in this case?
When I look at big O notation definition (one of):
$f=O(g)$ if and only if $\exists c, N>0$ such that $\forall x > N, f(x) < cg(x)$.
Say that we have $h(n)$ number of cases for which the polynomial in the clam is not irreducible, then the statement says that $h(n)=O(t^{1/3+\varepsilon})$ for some $t \geq n$ (or for all?), and so by definition of big O notation we have some $c,N$ such that $h(n)\leq c\cdot t^{1/3+\varepsilon}$ for $n>N$. Now it seems this is not correct interpretation (it does not seem to make sense), I guess what confuses me is the presence of $t$ , and then also the presence of $\varepsilon$. Could anybody explain the precise meaning of this?