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  is known (Bailey 1988).

The following table summarizes some record computations.

decimal digit	date	reference
	Oct. 1999	X. Gourdon and P. Demichel (Gourdon and Sebah)
	Dec. 8, 2006	Alexander J. Yee (Yee 2006; United Press International 2007)
	?	S. Kondo
	Mar. 13, 2009	A. Yee

The Earls sequence (starting position of  copies of the digit ) for  is given for , 2, ... by 5, 139, 163, 10359, 86615, 193446, 236542, 6186099, 36151186, ... (OEIS A224826).

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

The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of  (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  until all -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  is normal, but the following table giving the counts of digits in the first  terms shows that the decimal digits are very uniformly distributed up to at least .

	OEIS	10	100
0	A000000	0	11	111	1004	10065	100150	999853	10001768	99998397
1	A000000	1	6	95	1006	9974	100143	1000601	9996653	100002318
2	A000000	1	10	97	967	9821	99796	998927	9998112	99986624
3	A000000	0	9	108	976	9973	100194	1000766	9999460	99984204
4	A000000	1	10	90	1014	9870	99783	1001444	10007542	100011681
5	A000000	2	9	99	980	10200	100110	1002104	10001985	99996372
6	A000000	2	14	90	988	10103	100170	999530	9996871	100014127
7	A000000	2	13	116	1014	9877	99682	998692	9997487	99988819
8	A000000	0	7	81	1033	10114	100135	998534	9998182	100006202
9	A000000	1	11	113	1018	10003	99837	999549	10001940	100011256

REFERENCES:

Bailey, D. H. "Numerical Results on the Transcendence of Constants Involving , , 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/.