20150906, 19:58  #1 
Sep 2015
3×7 Posts 
New confirmed pi(10^27),... pi(10^29) prime counting function records
I am happy to announce that David Baugh and I have computed the number
of primes below 10^27, the result is: pi(10^27) = 16,352,460,426,841,680,446,427,399 The computation was performed using an unpublished version of my primecount program with backup functionality. primecount counts primes using a highly optimized parallel implementation of the DelegliseRivat algorithm (combinatorial method). The computation took 23.03 CPU core years and the peak memory usage was 235 gigabytes. Half of the computations were run on multiple EC2 spot instances of type r3.8xlarge (16 CPU cores, Intel Xeon E52670 v2). The other half (the computation of the hard special leaves) was run on David's dual socket server (36 CPU cores, Intel Xeon E52699 v3). Our result passes the parity check and the result is also very close to Riemann R(10^27) i.e.:  R(10^27)  16,352,460,426,841,680,446,427,399  < sqrt(10^27) / log(10^27) We will start a full verification shortly by recalculating pi(10^27) a second time using different configuration parameters. We expect this verification to take about 5 months. We'd like to thank J&N Computer Services for giving a discount on David's BigRig server and Shabir Ali for granting free access on his backup server. Regards, Kim Walisch Last fiddled with by kwalisch on 20150906 at 20:11 Reason: Add link to primecount website 
20150906, 20:36  #2 
"Dana Jacobsen"
Feb 2011
Bangkok, TH
3^{2}·101 Posts 

20150906, 21:04  #3 
"Ben"
Feb 2007
E56_{16} Posts 
Nice work!

20150907, 12:51  #4 
Jan 2008
France
569 Posts 
Great! Congratulations

20150908, 12:27  #5 
Nov 2007
Halifax, Nova Scotia
111000_{2} Posts 
Nice work, Kim!

20150908, 13:47  #6 
"Mark"
Apr 2003
Between here and the
2^{4}·421 Posts 

20160601, 00:22  #7 
Aug 2005
11^{2} Posts 
result verified
Last December Kim Walisch and I started a verification run of pi(1e27) using different initial conditions. Kim completed the last piece of the verification calculation of pi(1e27) on May 31, 2016 at 23:00 Luxembourg time. Along with the pieces I finished computing on May 9 , we can now say that our original value for pi(1e27) has been verified. The latest version of primecount is even faster than when we first calculated pi(1e27). I am sure Kim will be providing more details.
Best regards, David Baugh 
20160601, 07:24  #8 
Sep 2015
3×7 Posts 
pi(10^27) verified!
After roughly 5 months of computation David Baugh and myself have verified pi(10^27):
pi(10^27) = 16,352,460,426,841,680,446,427,399 This time the computation took 20.35 CPU core years, this is 11.6% faster than our first computation. The speed up comes from primecount improvements, particularly I have added presieving and wheel factorization to primecount's sieving algorithms. Below are the details of the verification: Code:
x = 1000000000000000000000000000 y = 231112254739 pi(y) = 9199337709 P2 = 4743234949871865833944278 S1 = 45739379279637813150 S2_trivial = 42247262851521121201 S2_easy = 4453498620247012088893172 S2_hard = 16642108769824393833206446 S2 = S2_trivial + S2_easy + S2_hard pi(x) = S1 + S2 + pi(y)  1  P2 pi(x) = 16352460426841680446427399 
20160601, 20:02  #9 
Aug 2006
5,987 Posts 
Fantastic! I've updated the OEIS entry https://oeis.org/A006880.

20160601, 21:07  #10 
Sep 2015
21_{10} Posts 
Thanks, I actually created an account on OEIS earlier today in order to submit it. But you have been faster. So I hope I can use my OEIS account in near future for submitting new prime sum records ;)

20160601, 21:40  #11 
Aug 2006
5,987 Posts 

Thread Tools  
Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Prime counting function records  D. B. Staple  Computer Science & Computational Number Theory  50  20201216 07:24 
The prime counting function  R2357  Math  2  20201009 22:20 
The prime counting function conjecture  jrsousa2  Miscellaneous Math  3  20200913 04:04 
Prime counting function  Steve One  Miscellaneous Math  20  20180303 22:44 
Legendre's prime counting function  pbewig  Information & Answers  0  20110714 00:47 