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 \(\epsilon > 0\).