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Computing $\pi$ (x): The Meissel-Lehmer method. (English) Zbl 0564.10006
An earlier paper by {\it J. C. Lagarias} and {\it A. M. Odlyzko} [Lect. Notes Math. 1052, 176-193 (1984; Zbl 0536.10008)] described two methods for computing $\pi$ (x), the number of primes $p\le x$. In the present paper an extended account of the first method is given, with an analysis of its complexity. It is also shown how the use of parallel processing affects the complexity; with M RAM (random access machine) parallel processors, where $1\le M\le x\sp{1/3}$, at most $O(M\sp{-1} x\sp{2/3+\epsilon})$ arithmetic operations are needed and at most $O(x\sp{1/3+\epsilon})$ storage locations. Tables of $\pi$ (x), for various values of x from $10\sp{12}$ to $4\times 10\sp{16}$, are given, showing the discrepancies between $\pi$ (x), Li(x), and Riemann’s approximation $R(x)=\sum\sp{\infty}\sb{n=1}\mu (n)n\sp{-1} Li(x\sp{1/n})$.
Reviewer: H.J.Godwin

11A41Elementary prime number theory
68Q25Analysis of algorithms and problem complexity
11A25Arithmetic functions, etc.
11N05Distribution of primes
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