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An extension of the Piatetski-Shapiro prime number theorem
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Stephan Baier
Published/Copyright:
July 29, 2016
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.
Published Online: 2016-7-29
Published in Print: 2005-3-1
© 2016 Oldenbourg Wissenschaftsverlag GmbH, Rosenheimer Str. 145, 81671 München
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Articles in the same Issue
- Masthead
- On the universality for L-functions attached to Maass forms
- On an Extension Problem for Contractive Block Hankel Operator Matrices
- Boundary value problems for the inhomogeneous polyanalytic equation I
- On a discrete variant of Bernstein’s polynomial inequality
- Strong Cesàro summability and statistical limit of fourier integrals
- An extension of the Piatetski-Shapiro prime number theorem
Articles in the same Issue
- Masthead
- On the universality for L-functions attached to Maass forms
- On an Extension Problem for Contractive Block Hankel Operator Matrices
- Boundary value problems for the inhomogeneous polyanalytic equation I
- On a discrete variant of Bernstein’s polynomial inequality
- Strong Cesàro summability and statistical limit of fourier integrals
- An extension of the Piatetski-Shapiro prime number theorem
