Small prime k-th power residues modulo p

Let $p$ be a prime number. For each positive integer $k\geq 2$, it is widely believed that the smallest prime that is a k-th power residue modulo p should be $O(p^{\epsilon})$, for any $\epsilon>0$. Elliott proved that such a prime is at most $p^{\frac{k-1}{4}+\epsilon}$, for each $\epsilon > 0$. In this talk, we discuss the number of prime k-th power residues modulo p in the interval $[1,p^{\frac{k-1}{4}+\epsilon}]$ for $\epsilon > 0$.