The least prime with a given primitive root and a variant of the larger sieve

For an integer $g$ not equal to $-1$ or a square, we study the least prime $p_g$ such that $g$ is a primitive root modulo $p_g$. We show that, assuming the Generalised Riemann Hypothesis, all such $g$ with $\mathrm{\mid }g\mathrm{\mid }\in [N,2N]$ have $p_g\le (\log \mathrm{\mid }g\mathrm{\mid }{)}^{3.44}$ with at most $O((\log N{)}^{2.44})$ exceptions. This can be directly compared to the uniform bound $p_g\le (\log \mathrm{\mid }g\mathrm{\mid }{)}^{19}$ proven recently by Fan and Pollack. To obtain our result, we prove a new variant of Gallagher’s larger sieve, which may be of independent interest. In addition to our conditional result, we discuss other ``almost all" bounds for $p_g$ that hold unconditionally.