Fast method for computing the number of primes less than a given limit
David C. Mapes
Math. Comp. 17 (1963), 179-185
Primary 10.03; Secondary 10.42
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Abstract: "Fast Method for Computing the Number of Primes Less Than a Given Limit'' describes three processes used during the course of calculation. In the first part of the paper the author proves:
represents the number of numbers less than or equal to x
and not divisible by the first ``a
'' primes. This identity is used to evaluate the formula
where resulting terms of the form
are broken down still further by the previously described method, or numerically evaluated using one or both of two other identities, the choice being dependent on
Following the paper is a table of calculations made using this process which gives the values of for x at intervals of 10 million up to 1000 million, along with the Riemann and the Chebyshev approximations for and the amount they deviate from the true count.
H. Lehmer, On the exact number of primes less than a given
limit, Illinois J. Math. 3 (1959), 381–388. MR
D. N. Lehmer, List of Prime Numbers from 1 to 10,006,721, New York, Hafner Pub. Co., 1956.
- D. H. Lehmer, Illinois J. Math., v. 3, no. 3, p. 381-388, 1959. MR 0106883 (21:5613)
- D. N. Lehmer, List of Prime Numbers from 1 to 10,006,721, New York, Hafner Pub. Co., 1956.
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