On the exact number of primes less than a given limit

Abstract

The problem of counting the exact number of primes x, wthout actually listing them all, dates from Legendre [l] who observed that the number of primes p for which x II px is one less than where [z] denotes, as usual, the greatest integer z, and the pi range over all the primes less than or equal to xII. Since then, a large number of writers [2] have suggested variants and improvements of this result. Foremost among these was the astronomer Meisse| [3] whose formula (5) s derived