Catalan’s Constant

Catalan’s Constant

Update: This computation has been superseded by my own newer computation of 1,200,000,000,100 digits, which is twice the number of digits below. It was computed using the Pilehrood (2010-short) algorithm from Fri Aug 20 06:45:21 2021 to Fri Oct 8 05:10:07 2021 and verified using the Guillera (2019) algorithm from Fri Jan 21 04:14:46 2022 to Wed Mar 9 12:38:14 2022, both with y-cruncher v0.7.8 Build 9507. The digits and both validation files are available at This Link. The rest of this post describes the earlier 600,000,000,100-digit computation.

Please cite:
Kim, S. Normality Analysis of Current World Record Computations for Catalan’s Constant and Arc Length of a Lemniscate with a=1. arXiv Preprint arXiv:1908.08925
if this article or the calculated digits were useful.

This world record computation of 600,000,000,100 digits was done from Sat May 25 22:37:01 2019 to Tue Jun 18 18:59:44 2019 using the Pilehrood (2010-short) algorithm. This time, I have also verified the calculation using the Guillera (2008) algorithm from Fri Jun 7 11:13:58 2019 to Tue Jul 16 10:29:12 2019. Validation file generated by y-cruncher v0.7.7 Build 9501 for computation, and y-cruncher v0.7.7 Build 9499 for the verification run:
Computation: https://web.archive.org/web/20190724102605/http://www.numberworld.org/y-cruncher/records/2019_6_18_catalan.txt
Verification: https://web.archive.org/web/20190724102625/http://www.numberworld.org/y-cruncher/records/2019_7_16_catalan.txt

\[G = \beta(2) = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)^2} = \frac{1}{1^2} - \frac{1}{3^2} + \frac{1}{5^2} - \frac{1}{7^2} + \frac{1}{9^2} - \cdots\]

The definition of Catalan’s constant (Wikipedia)

Catalan’s constant is defined by the above equation, where β is the Dirichlet beta function, which is closely related to the Riemann zeta function (I think both of them are learned by undergrad students in mathematics). Even though it looks irrational in a numerical scope, it is unproven whether it is transcendental or even irrational. For more information on Catalan’s constant, see this Wolfram Mathworld entry.

As with the Lemniscate Constant, I have used y-cruncher by Mr. Alexander J. Yee for this computation. This program is commonly used for stress testing and benchmarking overclocked PC builds (obviously this program performs a very rigorous computation), along with the fellow mathematical computing program Prime95.
Maintaining server stability was challenging, as the computation pushed all components of the computer to their limits.

Computation:
System information:
Operating System: Linux 3.10.0-693.21.1.el7.x86_64 x86_64 (CentOS 7)
Processor(s):
Name: Intel(R) Xeon(R) Gold 6140 CPU @ 2.30GHz
Logical Cores: 72
Physical Cores: 36
Sockets: 2
NUMA Nodes: 2
Base Frequency: 2,294,522,799 Hz

I have used two CPUs that are compatible with AVX-512 operations that are used crucially in vector-heavy programs like y-cruncher. However, y-cruncher has severe I/O bottlenecks (although I managed to select a better algorithm than the Lemniscate Constant and optimized some important aspects of the computation), so the number of cores did not assist my operation well. I have used RAID scratch storage for my operations, but the speed of the disk was very slow compared to the required R/W operations.

Start Date: Sat May 25 22:37:01 2019
End Date: Tue Jun 18 18:59:44 2019
Total Computation Time: 2028121.582 seconds
Start-to-End Wall Time: 2060562.370 seconds
CPU Utilization: 564.31 % + 151.19 % kernel overhead
Multi-core Efficiency: 7.84 % + 2.10 % kernel overhead

Although the multi-core efficiency improved slightly compared to the Lemniscate Constant, it still did not reach optimal efficiency (Dr. Ian Cutress’s Lemniscate Constant calculation had an efficiency of 94.04 % and CPU utilization of 9027.61 %, which means that the CPU was not bottlenecked by other factors).

Memory:
Usable Memory: 201,159,380,992 ( 187 GiB)
Logical Peak Disk Usage: 3,962,541,437,992 (3.60 TiB)
Logical Disk Bytes Read: 543,482,162,425,752 ( 494 TiB)
Logical Disk Bytes Written: 474,688,611,298,000 ( 432 TiB)

The use of a more efficient algorithm reduced disk operations by half compared to the Lemniscate Constant, resulting in a faster computation.
It is worth noting that HDD I/O speeds can be a bottleneck for virtually any other component, and using Optane DIMMs or more RAM may greatly improve computation speed.

Verification:
System information:
Operating System: Linux 3.10.0-693.21.1.el7.x86_64 x86_64 (CentOS 7)
Processor(s):
Name: Intel(R) Xeon(R) Gold 6140 CPU @ 2.30GHz
Logical Cores: 72
Physical Cores: 36
Sockets: 2
NUMA Nodes: 2
Base Frequency: 2,294,527,776 Hz

Start Date: Fri Jun 7 11:13:58 2019
End Date: Tue Jul 16 10:29:12 2019
Total Computation Time: 3218837.554 seconds
Start-to-End Wall Time: 3366914.430 seconds
CPU Utilization: 639.34 % + 150.50 % kernel overhead
Multi-core Efficiency: 8.88 % + 2.09 % kernel overhead

Memory:
Usable Memory: 201,226,489,856 ( 187 GiB)
Logical Peak Disk Usage: 3,986,470,844,768 (3.63 TiB)
Logical Disk Bytes Read: 947,069,197,181,784 ( 861 TiB)
Logical Disk Bytes Written: 829,531,225,016,496 ( 754 TiB)

A less efficient algorithm led to substantially more disk I/O, resulting in R/W similar to that of the Lemniscate Constant.

Overall, the simpler series of Catalan’s constant, compared to the AGM-Pi algorithm of the Lemniscate Constant, allowed for a faster computation with less of an I/O bottleneck.

To view the digits for Catalan’s constant, you can download them from This Link (note that the total size is over 1 TB, but the link will simply redirect to a registry with a download link). The digits of the newer 1,200,000,000,100-digit computation are at This Link instead.

Note that the digits are released under an Attribution-NonCommercial-NoDerivatives 4.0 International License, which prohibits commercial use and distribution of remixed, transformed, or built-upon versions without consent. Proper attribution and indication of changes are required even if it is not a prohibited use case.

Archive for computation results in the y-cruncher website:
https://web.archive.org/web/20190722034426/http://www.numberworld.org/y-cruncher/
Special thanks to Mr. Alexander J. Yee for developing and releasing y-cruncher and providing advice, and the Internet Archive for hosting the computed digits.