  E. Artin conjectured that any integer $a >1$ which is not a perfect square is
a primitive root modulo $p$ for infinitely many primes $p.$ Let $f_a(p)$ be the
multiplicative order of the non-square integer $a$ modulo the prime $p.$ M. R.
Murty and S. Srinivasan [10] showed that if $\sum_{p<x} \frac 1 {f_a(p)}=
O(x^{1/4})$ then Artin's conjecture is true for $a.$ We relate the
Murty-Srinivasan condition to sums involving the cyclotomic periods from the
subfields of $\mathbb Q(e^{2{\pi}i/p})$ corresponding to the subgroups $<a>
\subseteq \mathbb F*_p.$
