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The calculation of π(N)

Published online by Cambridge University Press:  09 April 2009

H. Lindgren
H. Lindgren
Affiliation:
Patent Office, Canberra
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The aim of this paper is to dervie two formulae for π(N) that need involve only a few of the smallest primes. Here m is a small integer, the b's are integers that will be found later, and Pij…k denotes the number of products figi… ≧ N, in which f, g,?…,h are unequal integers greater than 1 and prime to the first m primes. The suffixes run through all partitions of all integers.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 3 , Issue 3 , August 1963 , pp. 257 - 266
Copyright
Copyright © Australian Mathematical Society 1963

References

The earliest practical formula for π(N) is Meissel's of 1870, described in Uspensky and Heaslet's Elementary Number Theory, pp. 120–2. More recent is Lehmer's, D. H. formula described in his paper “On the exact number of primes less than a given limit”, Illinois J. Math., 3 (1959) 381–8.Google Scholar
The expansion similar to (9) of e −x forms the subject of contributions (in English) to Nordisk Matematisk Tidskrift by wKolberg, O., Carlitz, L., and Herzog, F. (8 (1960) 33–4, 9 (1961) 117–22, and 10 (1962) 78–9 respectively).Google Scholar