New values of pi(x) (18 Apr 1996)

New values of pi(x)

Date: Thu, 18 Apr 1996 09:30:37 +0200
From: Marc Deleglise
To: Chris Caldwell

[See the later note of 19 Jun 1996]

I am pleased to send you some new values of pi(x): ( 2e18 means 2.10^(18) and so on...)

pi(2e18) 		=  48 645 161 281 738 535
pi(3e18) 		=  72 254 704 797 687 083
pi(4e18) 		=  95 676 260 903 887 607
pi(4185296581467695669) = 100 000 000 000 000 000 
pi(5e18) 		= 118 959 989 688 273 472
pi(6e18) 		= 142 135 049 412 622 144
pi(7e18) 		= 165 220 513 980 969 424
pi(8e18) 		= 188 229 829 247 429 504
pi(9e18) 		= 211 172 979 243 258 278
pi(1e19) 		= 234 057 667 276 344 607
These values have been checked
  1. by computing pi(x) and pi(x + 1e7) and checking that the number of primes in the short interval agrees with the two values of pi.
  2. or by computing them two times with different values of 2 parameters y and z used during the computation.
(Thanks to Paul Zimmermann from INRIA Nancy who lend me some hours of computation on his Silicon Graphics and between other values got the pi(418....) = 10^17).

The method is presented in Math of Comp 1996 by Deleglise & Rivat: Computing Pi(x), the Meissel, Lehmer, Lagarias, Miller, Odlyzko method [DR96]. The program is an improved implementation of the precedent version. The asymptotic time and space complexity are unchanged (O(x^(2/3)/logx^2) for time and O(x^(1/3)logx^3) for space);

It is running about 2 times faster and needs less memory. The last value took 40 hours of computation on a DEC-Alpha 5/250 and needed about 80Mo memory. I got some other intermediate values that are waiting to be checked. I hope to send you some new bigger values after a while.

Marc Deleglise.