E. D. F. Meissel, Uber die Bestimmung der Primzahlmenge innerhalb gegebener Grenzen, Math. Ann. 2 (1870), 636-642.  Home/Search   Document Not in Database   Summary   Related Articles

....about 6p 2 (1 log 2) x nonzero terms. Actual calculations of p(x) by Gauss and others were based on factor tables made using sieve methods. The first efficient algorithm for computing p(x) which does not involve locating all the primes below x is due to the astronomer E. D. F. Meissel ([11] [14] His method is economical of space, and can be viewed as reducing the number of terms in Legendre s sum (1.1) Using his methods 2 he calculated p(10 8 ) 5 , 761 , 455 in 1871, and then after an enormous calculation announced in 1885 that p(10 9 ) was 50,847,478. His value of ....

....it appears that the existing methods for computing p(x) which are based on variants of the Meissel Lehmer method, have asymptotic running times at least c e x 1 e for any e 0 and some c e 0. It is hard to estimate the asymptotic computational complexity of Meissel s original method [11], since it is presented as a collection of rules to be applied according to the human calculator s judgment. We analyze asymptotic running times using as a model of computation a Random Access Machine (RAM) which is a relatively realistic model of the addressible core storage area of a ....

E. D. F. Meissel, Uber die Bestimmung der Primzahlmenge innerhalb gegebener Grenzen, Math. Ann. 2 (1870), 636-642.

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