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Computing $\pi(x)$: The Meissel-Lehmer Method

J. C. Lagarias, V. S. Miller and A. M. Odlyzko
Mathematics of Computation
Vol. 44, No. 170 (Apr., 1985), pp. 537-560
DOI: 10.2307/2007973
Stable URL: http://www.jstor.org/stable/2007973
Page Count: 24
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Computing $\pi(x)$: The Meissel-Lehmer Method
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Abstract

E. D. F. Meissel, a German astronomer, found in the 1870's a method for computing individual values of $\pi(x)$, the counting function for the number of primes $\leqslant x$. His method was based on recurrences for partial sieving functions, and he used it to compute $\pi(10^9)$. D. H. Lehmer simplified and extended Meissel's method. We present further refinements of the Meissel-Lehmer method which incorporate some new sieving techniques. We give an asymptotic running time analysis of the resulting algorithm, showing that for every $\varepsilon > 0$ it computes $\pi(x)$ using at most $O(x^{2/3 + \varepsilon})$ arithmetic operations and using at most $O(x^{1/3 + \varepsilon})$ storage locations on a Random Access Machine (RAM) using words of length $\lbrack \log_2 x \rbrack + 1$ bits. The algorithm can be further speeded up using parallel processors. We show that there is an algorithm which, when given $M$ RAM parallel processors, computes $\pi(x)$ in time at most $O(M^{-1}x^{2/3 + \varepsilon})$ using at most $O(x^{1/3 + \varepsilon})$ storage locations on each parallel processor, provided $M \leqslant x^{1/3}$. A variant of the algorithm was implemented and used to compute $\pi(4 \times 10^{16})$.

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