A080130 Decimal expansion of exp(-gamma).

5, 6, 1, 4, 5, 9, 4, 8, 3, 5, 6, 6, 8, 8, 5, 1, 6, 9, 8, 2, 4, 1, 4, 3, 2, 1, 4, 7, 9, 0, 8, 8, 0, 7, 8, 6, 7, 6, 5, 7, 1, 0, 3, 8, 6, 9, 2, 5, 1, 5, 3, 1, 6, 8, 1, 5, 4, 1, 5, 9, 0, 7, 6, 0, 4, 5, 0, 8, 7, 9, 6, 7, 0, 7, 4, 2, 8, 5, 6, 3, 7, 1, 3, 2, 8, 7, 1, 1, 5, 8, 9, 3, 4, 2, 1, 4, 3, 5, 8, 7, 6, 7, 3, 1

Offset: 0

Author

Benoit Cloitre, Jan 26 2003

Keywords

• cons
• nonn

Comments

By Mertens's third theorem, lim_{k->oo} (H_{k-1}*Product_{prime p<=k} (1-1/p)) = exp(-gamma), where H_n is the n-th harmonic number. Let F(x) = lim_{n->oo} ((Sum_{k<=n} 1/k^x)*(Product_{prime p<=n} (1-1/p^x))) for real x in the interval 0 < x < 1. Consider the function F(s) of the complex variable s, but without the analytic continuation of the zeta function, in the critical strip 0 < Re(s) < 1. - Thomas Ordowski, Jan 26 2023

Examples

			0.56145948356688516982414321479088078676571...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.5 p. 29, 2.7 p. 117 and 5.4 p. 285.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 202.
• Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, Bull. Amer. Math. Soc., 50 (2013), 527-628; arXiv:1303.1856 [math.NT], 2013.
• Doron Zeilberger and Noam Zeilberger, Fractional Counting of Integer Partitions, 2018; Local copy [Pdf file only, no active links]

Crossrefs

Cf. A000010, A000142, A001113, A001620, A002852, A007838, A073004, A079650, A322364, A322365, A322380, A322381.

Programs

Magma

R:= RealField(100); Exp(-EulerGamma(R)); // G. C. Greubel, Aug 28 2018

Maple

evalf(exp(-gamma), 120);  # Alois P. Heinz, Feb 24 2022

Mathematica

RealDigits[N[Exp[-EulerGamma], 200]][[1]] (* Arkadiusz Wesolowski, Aug 26 2012 *)

PARI

default(realprecision, 100); exp(-Euler) \\ G. C. Greubel, Aug 28 2018

Formula

Equals lim inf_{n->oo} phi(n)*log(log(n))/n. - Arkadiusz Wesolowski, Aug 26 2012

From Alois P. Heinz, Dec 05 2018: (Start)

Equals lim_{n->oo} A322364(n)/(n*A322365(n)).

Equals lim_{n->oo} A322380(n)/A322381(n). (End)

Equals lim_{k->oo} log(k)*Product_{prime p<=k} (1-1/p). - Amiram Eldar, Jul 09 2020

Equals lim_{n->oo} A007838(n)/A000142(n). - Alois P. Heinz, Feb 24 2022

Equals Product_{k>=1} (1+1/k)*exp(-1/k). - Amiram Eldar, Mar 20 2022

Equals A001113^(-A001620). - Omar E. Pol, Dec 14 2022

Equals lim_{n->oo} (A001008(p_n-1)/A002805(p_n-1))*(A038110(n+1)/A060753(n+1)), where p_n = A000040(n). - Thomas Ordowski, Jan 26 2023