I'm hoping that this is an easy question for someone.

How many elements can we expect to have multiplicative order at most $n^{1/c}$ in the set of all finite fields $\mathbb{F}_p$ with $p$ prime with $n \le p \le n^2$, for $c=\frac{9\log{n}}{\log{\log{\log{n}}}}$?

Note that I'm trying to bound the expectation from below.

I'm hoping that this is an easy question for someone.

How many elements can we expect to have multiplicative order at most $n^{1/c}$ in one of the finite fields $\mathbb{F}_p$ with $p$ prime with $n \le p \le n^2$, for $c=\frac{9\log{n}}{\log{\log{\log{n}}}}$?

Note that I'm trying to bound the expectation from below.

CLARIFICATION

We're taking the average order in one field, out of the set of finite fields with $n \le p \le n^2$.