Abstract
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.
Original language | English |
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Pages (from-to) | 87-98 |
Number of pages | 12 |
Journal | Analysis |
Volume | 25 |
Issue number | 1 |
DOIs | |
Publication status | Published - 1 Mar 2005 |