On the least prime primitive root

We study the uniform upper bound for the least prime that is a primitive root. Let g⁎(q) be the least prime primitive root (mod q) where q is a prime power or twice a prime power of a prime p. The upper bound for g⁎(q) is studied by many authors who succeeded in establishing various conditional upper bounds. However, no uniform bounds were known other than Linnikʼs bound on the least prime in an arithmetic progression. In this paper, we prove that g⁎(q)≪p3.1. The exponent 3.1 is improved from the known exponent 4.5 from Linnikʼs bound for the prime modulus.