Computing $\pi (x)$: the Meissel, Lehmer, Lagarias, Miller, Odlyzko method
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- by M. Deleglise and J. Rivat PDF
- Math. Comp. 65 (1996), 235-245 Request permission
Abstract:
Let $\pi (x)$ denote the number of primes $\le x$. Our aim in this paper is to present some refinements of a combinatorial method for computing single values of $\pi (x)$, initiated by the German astronomer Meissel in 1870, extended and simplified by Lehmer in 1959, and improved in 1985 by Lagarias, Miller and Odlyzko. We show that it is possible to compute $\pi (x)$ in $O(\frac {x^{2/3}} {\log ^2x})$ time and $O(x^{1/3}\log ^3x\log \log x)$ space. The algorithm has been implemented and used to compute $\pi (10^{18})$.References
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Additional Information
- M. Deleglise
- Affiliation: Département de Mathématiques, Université Lyon 1, 43 Blvd. du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
- Email: deleglis@lmdi.univ-lyon1.fr
- J. Rivat
- Affiliation: Département de Mathématiques, Université Lyon 1, 43 Blvd. du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
- Email: rivat@caissa.univ-lyon1.fr
- Received by editor(s): January 12, 1994
- Received by editor(s) in revised form: December 1, 1994
- © Copyright 1996 American Mathematical Society
- Journal: Math. Comp. 65 (1996), 235-245
- MSC (1991): Primary 11N05, 11Y70
- DOI: https://doi.org/10.1090/S0025-5718-96-00674-6
- MathSciNet review: 1322888