Explicit upper bounds on the least primitive root

Abstract:

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 $p\geq 10^{56}$.