An extension of the Piatetski-Shapiro prime number theorem

Balog and Harman proved that for any λ in the interval 1/2 ≤ λ < 1 and any real θ there are infinitely many primes p satisfying (with an asymptotic result). In the present paper we prove that for 59/85 = 0-694... < λ < 1 the above expo­nent -(1-λ)/2+ε may be replaced by - min{max{(35-22λ)/129, 1/7}, 5/18-λ/6}+ε. This result in particular contains the Piatetski-Shapiro prime number theorem in the ver­sion given by Liu and Rivat: We have |{n ≤ N : [nc] prime}) ~ N/(c log N) as N → 8734 if 1 < c < 15/13. For the proof of our result we use exponential sum techniques.