Explicit upper bound on the least primitive root modulo p2

In this paper, we give an explicit upper bound on h(p), the least primitive root modulo p(2). Since a primitive root modulo p is not primitive modulo p(2) if and only if it belongs to the set of integers less than p which are pth power residues modulo p(2), we seek the bounds for N-1(H) and N-2(H) to find H which satisfies N-1(H) - N-2(H) > 0, where, Ni(H) denotes the number of primitive roots modulo p not exceeding H, and N-2(H) denotes the number of pth powers modulo p(2) not exceeding H. The method we mainly use is to estimate the character sums contained in the expressions of the N-1(H) and N-2(H) above. Finally, we show that h(p) < P-0.74 for all primes p. This improves the recent result of Kerr et al.