Euler-Mascheroni Constant Digits

The Euler-Mascheroni constant


(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.


See also

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

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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., A. J. "y-cruncher - A Multi-Threaded Pi-Program." Press International. "Student at Northwestern Breaks Math Record." Apr. 9, 2007.

Cite this as:

Weisstein, Eric W. "Euler-Mascheroni Constant Digits." From MathWorld--A Wolfram Web Resource.

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