Euler-Mascheroni Constant

5 minute read

The Euler-Mascheroni Constant

Update: To people who came here from Dr. Ian Cutress’s Twitter or TechTechPotato Youtube: Welcome. I am the very person that asked Dr. Ian to compute this dreadful mathematical constant for over 3 months to verify my initial computation. But I guess we did it. I give big credit to Dr. Ian Cutress because without him, my computation would have never become official. This website is meant to provide additional information about computing world records in addition to the information in the y-cruncher website, so have some more look. Thank you.

Another update:

And also, ECC is pretty important yes, although I never got an ECC corrected error for any of my computations.

Please cite this webpage if my world record of the Euler-Mascheroni Constant were useful and also the below citation if you used computations from my other posts or the digit analysis methodologies:
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

Out of all the mathematical constants that I have set a world record computation for until now, the Euler-Mascheroni Constant is so far the most mathematically significant constant (the only ones more significant are probably π and e).

This world record computation of 600,000,000,100 digits by Seungmin Kim was done from Mon Aug 19 17:21:44 2019 to Sat Jan 11 18:06:11 2020 using the Brent-McMillan with Refinement ( n = 2^38 ) algorithm. Verification calculation has been done by Dr. Ian Cutress using the Brent-McMillan ( n = 2^39 ) algorithm from Wed Feb 12 09:34:27 2020 to Tue May 26 12:55:30 2020.

Validation file generated by y-cruncher v0.7.7 Build 9501 for computation, and y-cruncher v0.7.8 Build 9503 for the verification run:
Verification by Dr. Ian Cutress:

\[\gamma = \lim_{n\to\infty}\left(-\ln n + \sum_{k=1}^n \frac1{k}\right) = \int_1^\infty\left(-\frac1x+\frac1{\lfloor x\rfloor}\right) dx\]

The definition of Euler–Mascheroni constant (Wikipedia)

Euler–Mascheroni constant is defined by the above equation and denoted as the symbol γ. The definition itself will be familiar to many freshmen studying calculus, as it is the infinite limit difference of the harmonic series and the logarithm, and can be converted to the area of the blue region in the figure below. I am pretty sure this figure is in a calculus textbook related to the integral test for convergence. 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.

The area of the blue region converges to the Euler–Mascheroni constant. (Wikipedia)

Same as all the other constants, I have used y-cruncher by Mr. Alexander J. Yee, basically the only program that can do this task 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. Compared to the earlier constants, this constant is very intensive to compute since it is not just one series expansion. The time to compute and disk writes went basically out of bounds compared to the earlier constants.

System information:
Operating System: Linux 3.10.0-693.21.1.el7.x86_64 x86_64 (CentOS 7)
Name: Intel(R) Xeon(R) Gold 6140 CPU @ 2.30GHz
Logical Cores: 72
Physical Cores: 36
Sockets: 2
NUMA Nodes: 2
Base Frequency: 2,294,552,832 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 had an even more severe I/O bottleneck in this computation as the required read/writes were more intensive than any of my computations, so the number of cores did not assist my operation well. It was better than the Lemniscate Constant but worse than Catalan’s Constant. I have used RAID scratch storage for my operations, but the speed of the disk was very slow compared to the required read/write operations.

Start Date: Mon Aug 19 17:21:44 2019
End Date: Sat Jan 11 18:06:11 2020
Total Computation Time: 11899422.659 seconds
Start-to-End Wall Time: 12534266.770 seconds
CPU Utilization: 451.31 % + 191.22 % kernel overhead
Multi-core Efficiency: 6.27 % + 2.66 % kernel overhead

The computation took roughly 5 times more than Catalan’s Constant with about 80-90% efficiency to it, so the disk writes were the fundamental issue. I definitely cannot do this computation one more time.

Usable Memory: 201,226,489,856 ( 187 GiB)
Logical Peak Disk Usage: 4,380,033,959,120 (3.98 TiB)
Logical Disk Bytes Read: 3,146,383,900,360,116 (2.79 PiB)
Logical Disk Bytes Written: 2,755,641,530,520,684 (2.45 PiB)

Looks like we have a new unit called PiB. This is the first computation that I have done that exceeds 1 Pebibyte. HDD I/O speeds are still great bottlenecks to virtually any other component, and thus having Optane DIMMs, SSDs with modules that tolerate a huge amount of read/writes, or more normal RAM can help the speed of the computation speed greatly.

For verification results, check the link at the start of the post.

If you want to take a look at the digits for the Euler–Mascheroni 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:
Special thanks to Mr. Alexander J. Yee for developing and releasing y-cruncher and providing advice, Senior Editor Dr. Ian Cutress for verifying my computation, and the Internet Archive for hosting the computed digits.