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The calculation of π(N)

  • H. Lindgren (a1)

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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.

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References

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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.
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).
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The calculation of π(N)

  • H. Lindgren (a1)

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