Hello,

Consider the multiplicative group $(\mathbb{Z}/p)^*$ for a given prime $p$. We know that the number of generators in this group is $\phi(p-1)$ --- the Euleur totient function. The question is, for $0 \leq a < a + \log^{c} p < b \leq p-1$ where $c$ is a constant (say $c=10$), how many generators of the group belongs to $[a,b]$? In other words, what is the density of generators in a given interval $[a,b]$ (compared to the density $\phi(p-1)/p-1$)? Is it easier if $b=p-1$?

For a given prime $p$, what is the densest interval in term of generators?

onegenerator have length $O((\log p)^2)$, assuming the Generalized Riemann Hypothesis. Unconditionally, you can’t even get anywhere near that. Thus, I doubt you can get any nontrivial answer. – Emil Jeřábek Mar 21 '12 at 19:48