# Catalan’s Constant

## The Catalan’s Constant

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

This world record computation of 600,000,000,100 digits by Seungmin Kim 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 if it is transcendental, or even irrational. Take a look at this Wolfram Mathworld entry for the mathematical stuff.

Same as 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 fellow mathematical computing program Prime95.
It was also very hard for me to maintain this server stable, as this takes all components of a computer to the extreme.

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 operations 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

The multi-core efficiency was slightly improved compared to the Lemniscate Constant, but still did not reach the 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)

Disk operation was decreased in half compared to the Lemniscate Constant because of an efficient algorithm, and this contributed to a faster computation.
One caveat is that HDD I/O speeds are great bottlenecks to virtually any other component, and perhaps having Optane DIMMs or more normal RAM can help the speed of the computation speed greatly.

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 lead to a lot of disk I/O, resulting to be similar R/W to the Lemniscate Constant.

Overall, as the Catalan constant is a comparably simple series compared to the AGM-Pi algorithm of the Lemniscate Constant, due to the way the algorithm works, it had lead to a faster computation with less I/O bottleneck.

If you want to take a look at the digits for the Catalan’s constant, you can download it from This Link (Over 1 TB total but don’t worry, it will just redirect to a registry with a link to download).

Note that digits are released as an Attribution-NonCommercial-NoDerivatives 4.0 International License, meaning no commercial purposes and you cannot distribute a remixed, transformed, or built upon version without my consent. You must also give appropriate credit, provide a link to the license, and indicate if changes were made 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.

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