  We give a method for producing explicit bounds on $g(p)$, the least primitive
root modulo $p$. Using our method we show that
$g(p)<2r\,2^{r\omega(p-1)}\,p^{\frac{1}{4}+\frac{1}{4r}}$ for $p>10^{56}$ where
$r\geq 2$ is an integer parameter. This result beats existing bounds that rely
on explicit versions of the Burgess inequality. Our main result allows one to
derive bounds of differing shapes for various ranges of $p$. For example, our
method also allows us to show that $g(p)<p^{5/8}$ for all $p\geq 10^{22}$ and
$g(p)<p^{1/2}$ for $p\geq 10^{56}$.
