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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

The $k^{th}$ prime is greater than
$k(\ln k +\ln\ln k -1)$ for $k\geq 2$


Author: Pierre Dusart
Journal: Math. Comp. 68 (1999), 411-415
MSC (1991): Primary 11N05; Secondary 11A41
MathSciNet review: 1620223
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Abstract: ROSSER and SCHOENFELD have used the fact that the first 3,500,000 zeros of the RIEMANN zeta function lie on the critical line to give estimates on $\psi(x)$ and $\theta(x)$. With an improvement of the above result by BRENTet al., we are able to improve these estimates and to show that the $k^{th}$ prime is greater than $k(\ln k +\ln\ln k -1)$ for $k\geq 2$. We give further results without proof.


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Additional Information

Pierre Dusart
Affiliation: LACO, ESA 6090, Faculté des Sciences, 123 avenue Albert Thomas, 87060 Limoges Cedex, FRANCE
Email: dusart@unilim.fr

DOI: http://dx.doi.org/10.1090/S0025-5718-99-01037-6
PII: S 0025-5718(99)01037-6
Keywords: Distribution of primes, arithmetic functions
Received by editor(s): June 17, 1996
Article copyright: © Copyright 1999 American Mathematical Society