On Primes in Arithmetic Progression Having a Prescribed Primitive Root

Abstract

Let g∈Z\{-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)=2φ(p-1)/(p-1) if (g/p)=-1 (Legendre symbol) and (p-1, h)=1 and wg(p)=0 otherwise, with φ Euler's totient. Let a (modf) be a primitive residue class. Let πg(x; f, a) denote the number of odd primes p≤x such that p≡a (modf) and g is a primitive root modp. 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 modp. © 1999 Academic Press.