On Primes in Arithmetic Progression Having a Prescribed Primitive Root Original Research Article

Let gset membership, variantimage\{−1, 0, 1} and let h be the largest integer such that g is an hth power. Let p be a prime. Put wg(p)=2phi(p−1)/(p−1) if (g/p)=−1 (Legendre symbol) and (p−1, h)=1 and wg(p)=0 otherwise, with phi Eulerʹs totient. Let a (mod f) be a primitive residue class. Let πg(x; f, a) denote the number of odd primes pless-than-or-equals, slantx such that p≡a (mod f) and g is a primitive root mod p. It is shown, under the GRH, that -- FORMULA OMITTED -- Thus the function wg(p) behaves as if it were some kind of “probability” that g is a primitive root mod p.