Pillai's lower bound on the least primitive root

History

This result was established by S. Pilai in 1944.

Statement

There are infinitely many prime numbers $p$ for which the least primitive root modulo $p$ is greater than the double logarithm $\log \log p$.

Another way of putting this is in terms of a limit superior:

$\lim \sup _{p}{\frac {g(p)}{\log \log p}}\geq 1$

where $g(p)$ is the least primitive root modulo $p$.