Fast method for computing the number of primes less than a given limit

Author:
David C. Mapes

Journal:
Math. Comp. **17** (1963), 179-185

MSC:
Primary 10.03; Secondary 10.42

DOI:
https://doi.org/10.1090/S0025-5718-1963-0158508-8

MathSciNet review:
0158508

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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:

*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 and .

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.

**[1]**D. H. Lehmer,*On the exact number of primes less than a given limit*, Illinois J. Math.**3**(1959), 381–388. MR**0106883****[2]**D. N. Lehmer,*List of Prime Numbers from 1 to 10,006,721*, New York, Hafner Pub. Co., 1956.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1963-0158508-8

Article copyright:
© Copyright 1963
American Mathematical Society