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Euler-Mascheroni Constant Digits

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The Euler-Mascheroni constant

gamma=0.577215664901532860606512090082402431042...

(OEIS A001620) 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. It was subsequently computed to 40 correct decimal placed by Soldner in 1809 and verified by Gauss and Nicolai in 1812 (Havil 2003, pp. 89-90). No quadratically converging algorithm for computing gamma is known (Bailey 1988).

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

The Earls sequence (starting position of n copies of the digit 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, ... (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 n-digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 8, 18, 346, 2778, 84514, ... (OEIS A000000), which end at digits 16, 658, 6600, 91101, 1384372, ... (OEIS A000000).

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
0A0000000111111004100651001509998531000176899998397
1A00000016951006997410014310006019996653100002318
2A00000011097967982199796998927999811299986624
3A0000000910897699731001941000766999946099984204
4A000000110901014987099783100144410007542100011681
5A00000029999801020010011010021041000198599996372
6A00000021490988101031001709995309996871100014127
7A0000002131161014987799682998692999748799988819
8A00000007811033101141001359985349998182100006202
9A0000001111131018100039983799954910001940100011256

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.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."Yee, A. J. "Euler's Constant-116 Million Digits on a Laptop: New World Record." 2006. http://www.numberworld.org/euler116m.html.Yee, A. J. "y-cruncher - A Multi-Threaded Pi-Program." http://www.numberworld.org/y-cruncher/.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/.

Cite this as:

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

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