Small prime th power residues

Let k ≥ 2 k\geq 2 be an integer. Let ϵ > 0 \epsilon >0 be given. It is widely believed that the smallest prime that is a k k th power residue modulo a prime q q should be O ( q ϵ ) O(q^{\epsilon }) for any ϵ > 0 \epsilon >0 . Elliott has proved that for large primes q ≡ 1 mod k q\equiv 1\bmod k such a prime is at most q k − 1 4 + ϵ q^{\frac {k-1}{4}+\epsilon } for each ϵ > 0 \epsilon >0 . We show that for large primes q ≡ 1 mod k q\equiv 1\bmod k , the number of primes p ≤ q k − 1 4 + ϵ p\leq q^{\frac {k-1}{4}+\epsilon } such that p p is a k k th power residue mod q \bmod \, q is at least q 9 ϵ 20 k q^{\frac {9\epsilon }{20k}} .