The average least quadratic nonresidue modulo m and other variations on a theme of Erdős

For each m⩾3, let n2(m) denote the least quadratic nonresidue modulo m. In 1961, Erdős determined the mean value of n2(p), as p runs over the odd primes. We show that the mean value of n2(m), without the restriction to prime values, is . For each prime p, let G(p) denote the least natural number n so that the subgroup generated by {1,2,…,n} is all of ×(Z/pZ). Assuming the Generalized Riemann Hypothesis, we show that G(p) possesses a finite mean value ≈3.975. For K a quadratic extension of Q, let nK denote the smallest rational prime which is inert in K and rK the least prime which is split in K. We show that with quadratic fields ordered by the absolute value of their discriminant, rK and nK have the same mean value, which is ≈4.981.