Book contents
- Frontmatter
- Contents
- Preface
- 1 Foundations
- 2 Some important Dirichlet series and arithmetic functions
- 3 The basic theorems
- 4 Prime numbers in residue classes: Dirichlet's theorem
- 5 Error estimates and the Riemann hypothesis
- 6 An “elementary” proof of the prime number theorem
- Appendices
- A Complex functions of a real variable
- B Double series and multiplication of series
- C Infinite products
- D Differentiation under the integral sign
- E The O, o notation
- F Computing values of π(x)
- G Table of primes
- H Biographical notes
- Bibliography
- Index
F - Computing values of π(x)
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- 1 Foundations
- 2 Some important Dirichlet series and arithmetic functions
- 3 The basic theorems
- 4 Prime numbers in residue classes: Dirichlet's theorem
- 5 Error estimates and the Riemann hypothesis
- 6 An “elementary” proof of the prime number theorem
- Appendices
- A Complex functions of a real variable
- B Double series and multiplication of series
- C Infinite products
- D Differentiation under the integral sign
- E The O, o notation
- F Computing values of π(x)
- G Table of primes
- H Biographical notes
- Bibliography
- Index
Summary
We shall describe the principles of various methods for computing π(N) for a given N. We leave it to readers to translate these principles into their chosen programming language or package.
The simplest method is the well-known “sieve of Eratosthenes”. It operates as follows to create a list of primes up to a chosen N, from which one can read off values of π(n). Start with the full list of integers from 2 to N. Delete all multiples of 2, apart from 2 itself. Then delete all multiples of 3 (apart from 3 itself). Since 4 has already been deleted, move on to 5 and delete its multiples. Continue the process, deleting multiples of each m < N½ unless m has already been deleted. Note that when deleting multiples of m, it is enough to start at m2, since the numbers km (for 2 ≤ k < m) have already been deleted. When we have finished, the remaining numbers are the primes in [2, N], since every composite number in this interval is a multiple of some m ≤ N½, and consequently has been deleted.
To implement this process on a computer, what we really do is construct the function
(in the notation of this book, f is up). We start by setting f(n) = 1 for 2 ≤ n ≤ N. The act of “deleting” an integer n equates to changing f(n) from 1 to 0.
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- Information
- The Prime Number Theorem , pp. 234 - 237Publisher: Cambridge University PressPrint publication year: 2003