Apéry’s Constant

5 minute read

The Apéry’s Constant

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 1,200,000,000,100 digits by Seungmin Kim was done from Thu May 21 15:11:49 2020 to Mon Jun 22 08:38:33 2020 using the Wedeniwski (1998) algorithm. This time again after Catalan’s Constant, I have verified the calculation using the Amdeberhan-Zeilberger (1997) algorithm from Wed Jun 24 09:02:14 2020 to Sun Jul 26 22:22:36 2020.
Validation file generated by y-cruncher v0.7.8 Build 9506 for computation, and y-cruncher v0.7.8 Build 9506 for the verification run:
Computation: https://web.archive.org/web/20200810061235/http://www.numberworld.org/y-cruncher/records/2020_6_22_zeta3.txt
Verification: https://web.archive.org/web/20200810062529/http://www.numberworld.org/y-cruncher/records/2020_7_26_zeta3.txt

\[\zeta(3) = \sum_{n=1}^\infty\frac{1}{n^3} = \lim_{n \to \infty}\left(\frac{1}{1^3} + \frac{1}{2^3} + \cdots + \frac{1}{n^3}\right)\]

The definition of Apéry’s Constant (Wikipedia)

Apéry’s Constant is defined by the above equation, ζ is the Riemann zeta function. The zeta function is well regarded as the key to many unsolved problems in mathematics, physics, and chemistry, and I think it is learned by undergrad junior to senior students in mathematics. It has been named after Roger Apéry, the mathematician that proved that \(\zeta(3)\) was irrational and generated great insight to the zeta function itself. Because Apéry’s Constant was proved as irrational, we know the digits will continue indefinitely. Take a look at this Wolfram Mathworld entry for the mathematical stuff.

Interesting fact: Dr. Sebastian Wedeniwski, the discoverer of the Wedeniwski (1998) algorithm, was the person behind ZetaGrid, which was one of the largest distributed computing projects of the early 2000s and had the purpose of finding roots of the zeta function to test if there are any counterexamples of the Riemann hypothesis. He is now the Chief Information Officer (executive position) at the Standard Chartered Bank at Singapore after 18 years at IBM, currently in charge of all informational management of the multinational banking group. Mr. Alexander Yee (the person who created y-cruncher) also works at Citadel Securities, a huge hedge fund located in Chicago, after his time at Google. I guess the people from mathematical computing academic diciplines meet in the financial industry.

Same as all other mathematical constants, 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 and linear algebra program Linpack.
It was a complicated constant to compute based on that I have doubled the number of digits from all my computations before, but it was pretty easy compared to that messy I/O bottleneck of the Euler-Mascheroni Constant.

System information:
Operating System: Linux 3.10.0-327.36.1.el7.x86_64 x86_64
Name: Intel(R) Xeon(R) CPU E5-2670 v3 @ 2.30GHz
Logical Cores: 48
Physical Cores: 24
Sockets: 2
NUMA Nodes: 2
Base Frequency: 2,299,961,280 Hz

My first computation used two Xeon CPU sockets from the Haswell era (and thus supports AVX2 SIMD vector operations that are used crucially in vector computations like y-cruncher) as I decided the Xeon Scalable Skylake Purley processors that support AVX-512 was overkill since I was gonna confront I/O bottlenecks anyways. The time would not have had much difference from using just one CPU socket. I have optimized I/O throughput further by changing the Bytes/Seek parameter and allocating more I/O Buffer to reach RAID-level performance in my file systems.

Start Date: Thu May 21 15:11:49 2020
End Date: Mon Jun 22 08:38:33 2020
Total Computation Time: 2702135.464 seconds
Start-to-End Wall Time: 2741203.782 seconds
CPU Utilization: 997.73 % + 33.91 % kernel overhead
Multi-core Efficiency: 20.79 % + 0.71 % kernel overhead

The multi-core efficiency did improve compared to the constants before this (to a level of a high-end desktop CPU maybe, this is because there was more RAM than before), but still did not reach the utilization possible by the CPUs (Dr. Ian Cutress’s Lemniscate Constant calculation had a multi-core efficiency of 94.04 % and CPU utilization of 9027.61 %, which means that the CPU was not bottlenecked by other factors). This is also the first time I use Cilk Plus Work-Stealing multiprocessing framework along with the dynamic version of y-cruncher.

Working Memory: 499,483,876,096 ( 465 GiB)
Total Memory: 499,786,337,280 ( 465 GiB)
Logical Largest Checkpoint: 2,295,791,263,000 (2.09 TiB)
Logical Peak Disk Usage: 7,799,479,894,560 (7.09 TiB)
Logical Disk Bytes Read: 795,796,887,749,864 ( 724 TiB)
Logical Disk Bytes Written: 696,627,866,292,616 ( 634 TiB)

Disk operation was decreased by a fourth compared to the Euler-Mascheroni Constant because the algorithm is easier, and this contributed to a faster computation along with more RAM. Disk writes were overall similar to Catalan’s Constant since there were two times the digits and half the algorithm difficulty.
One caveat is that HDD I/O speeds are again great bottlenecks to virtually any other component, and perhaps having Optane DIMMs or even more normal RAM can help the speed of the computation speed greatly.

System information:
Operating System: Linux 3.10.0-693.21.1.el7.x86_64 x86_64
Name: Intel(R) Xeon(R) Gold 5220 CPU @ 2.20GHz
Logical Cores: 72
Physical Cores: 36
Sockets: 2
NUMA Nodes: 2
Base Frequency: 2,194,831,008 Hz

Start Date: Wed Jun 24 09:02:14 2020
End Date: Sun Jul 26 22:22:36 2020
Total Computation Time: 2620847.662 seconds
Start-to-End Wall Time: 2812821.324 seconds
CPU Utilization: 934.98 % + 33.72 % kernel overhead
Multi-core Efficiency: 12.99 % + 0.47 % kernel overhead

Working Memory: 536,619,505,280 ( 500 GiB)
Total Memory: 536,870,912,000 ( 500 GiB)
Logical Largest Checkpoint: 2,288,826,179,160 (2.08 TiB)
Logical Peak Disk Usage: 7,793,874,082,416 (7.09 TiB)
Logical Disk Bytes Read: 784,068,383,106,800 ( 713 TiB)
Logical Disk Bytes Written: 687,145,764,017,680 ( 625 TiB)

Changed stuff around for the verification computation. Slightly more RAM and two Cascade Lake Xeon CPU sockets that support AVX-512. CPU utilization was similar even when the number of cores increased, and this is explained by having slightly more RAM. Total Computation Time and Disk R/W were slightly less despite a more inefficient algorithm.

Overall, more RAM and better optimization based on insight of both the hardware and the software led to a better computing experience compared to before. Apéry’s Constant is a very important mathematical constant in expanding the horizon of human knowledge in mathematics. I hope computing and sharing the results can result in more insight that can be used by mathematicians for better insight.

If you want to take a look at the digits for the Apéry’s Constant, you can download it from This Link (Almost 2 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/20200810060943/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.