On the least prime primitive root modulo a prime

We derive a conditional formula for the natural density E(q) of prime numbers p having its least prime primitive root equal to q, and compare theoretical results with the numerical evidence. 1. Theoretical result concerning the density of primes with a given least prime primitive root Let us denote, following Elliott and Murata [4], by g(p) and G(p) the least primitive and the least prime primitive root mod p, respectively. The first aim of this paper is to derive from the work of Matthews [5] a conditional (under the generalized Riemann hypothesis) formula for the density of primes p such that G(p) = q, where q is a given prime, and to compare this formula with the numerical evidence. Next we give for each prime q ≤ 349 the least prime p such that G(p) = q, if such p exists below 2, and we compare G(p) with (log p)(log log p), which, according to a conjecture of E. Bach [2], is the maximal order of G(p) (i.e., 0 < lim sup G(p) (log p)(log log p)2 < ∞). We also numerically investigate the average value of the least prime primitive root. In order to formulate the theorem, we denote by pn the nth prime and, for a given set M , by |M | its cardinality. Now we can state Theorem. Assume that the Riemann hypothesis holds for each of the fields Q( k √ 1, l1 √ p1, . . . , ln √ pn), where k = l.c.m. li is squarefree. Then the set of primes p such that G(p) = pn has a natural density equal to