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Computing $\pi$ (x): The Meissel-Lehmer method. (English) Zbl 0564.10006
An earlier paper by J. C. Lagarias and 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 $1\le M\le {x}^{1/3}$, at most $O\left({M}^{-1}{x}^{2/3+ϵ}\right)$ arithmetic operations are needed and at most $O\left({x}^{1/3+ϵ}\right)$ storage locations. Tables of $\pi$ (x), for various values of x from ${10}^{12}$ to $4×{10}^{16}$, are given, showing the discrepancies between $\pi$ (x), Li(x), and Riemann’s approximation $R\left(x\right)={\sum }_{n=1}^{\infty }\mu \left(n\right){n}^{-1}Li\left({x}^{1/n}\right)$.
Reviewer: H.J.Godwin

##### MSC:
 11A41 Elementary prime number theory 68Q25 Analysis of algorithms and problem complexity 11A25 Arithmetic functions, etc. 11N05 Distribution of primes