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Euler-Mascheroni Constant Digits

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The decimal digits of the Euler-Mascheroni constant begin

gamma=0.577215664901532860606512090082402431042...

(OEIS A001620). It was calculated to 16 digits by Euler in 1781 and to 32 decimal places by Mascheroni (1790), although only the first 19 decimal places were correct (Havil 2003, pp. 89-90). Soldner computed 22 decimal places in 1809, and Nicolai subsequently computed 40 places at the request of Gauss (Lagarias 2013). Bailey (1988) reported that no quadratically converging algorithm for computing gamma was then known.

The following table summarizes some record computations.

decimal digitdatereference
108×10^6Oct. 1999X. Gourdon and P. Demichel (Gourdon and Sebah)
116580041Dec. 8, 2006Alexander J. Yee (Yee 2006; United Press International 2007)
2×10^9?S. Kondo
29844489545Mar. 13, 2009A. Yee and R. Chan

The Earls sequence (starting position of the first occurrence of n copies of the decimal representation of n) for gamma is given for n=1, 2, ... by 5, 139, 163, 10359, 86615, 193446, 236542, 6186099, 36151186, ... (OEIS A224826).

gamma-constant primes occur at 1, 3, 40, 185, 1038, 22610, 179849, ... (OEIS A065815) decimal digits.

The starting positions of the first occurrence of n=0, 1, 2, ... in the decimal expansion of gamma (excluding the initial 0 to the left of the decimal point) are 11, 5, 4, 14, 9, 1, 7, 2, 16, 10, ... (OEIS A229192).

Scanning the decimal expansion of gamma until all positive n-digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 8, 18, 346, 2778, 84514, ..., which end at digits 16, 658, 6600, 91101, 1384372, ....

It is not known if gamma is normal, but the following table giving the counts of digits in the first 10^n terms shows that the decimal digits are very uniformly distributed up to at least 10^9.

d\nOEIS1010010^310^410^510^610^710^810^9
00111111004100651001509998531000176899998397
116951006997410014310006019996653100002318
211097967982199796998927999811299986624
30910897699731001941000766999946099984204
4110901014987099783100144410007542100011681
529999801020010011010021041000198599996372
621490988101031001709995309996871100014127
72131161014987799682998692999748799988819
807811033101141001359985349998182100006202
91111131018100039983799954910001940100011256

See also

Constant Digit Scanning, Constant Primes, Earls Sequence, Euler-Mascheroni Constant, Euler-Mascheroni Constant Continued Fraction

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References

Bailey, D. H. "Numerical Results on the Transcendence of Constants Involving pi, e, and Euler's Constant." Math. Comput. 50, 275-281, 1988.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.Lagarias, J. C. "Euler's Constant: Euler's Work and Modern Developments." Bull. Amer. Math. Soc. 50, 527-628, 2013. https://doi.org/10.1090/S0273-0979-2013-01423-X.Mascheroni, L. Adnotationes ad calculum integralem Euleri, Vol. 1 and 2. Ticino, Italy, 1790 and 1792. Reprinted in Euler, L. Leonhardi Euleri Opera Omnia, Ser. 1, Vol. 12. Leipzig, Germany: Teubner, pp. 415-542, 1915.Sloane, N. J. A. Sequences A001620/M3755, A065815, A224826, and A229192 in "The On-Line Encyclopedia of Integer Sequences."United Press International. "Student at Northwestern Breaks Math Record." Apr. 9, 2007. http://www.upi.com/NewsTrack/Quirks/2007/04/09/student_at_northwestern_breaks_math_record/.Yee, A. J. "Euler's Constant-116 Million Digits on a Laptop: New World Record." 2006. https://www.numberworld.org/euler116m.html.Yee, A. J. "y-cruncher - A Multi-Threaded Pi-Program." https://www.numberworld.org/y-cruncher/.

Referenced on Wolfram|Alpha

Euler-Mascheroni Constant Digits

Cite this as:

Weisstein, Eric W. "Euler-Mascheroni Constant Digits." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Euler-MascheroniConstantDigits.html

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