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%I #87 Jan 28 2023 12:02:44
%S 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,
%T 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,
%U 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
%N Decimal expansion of exp(-gamma).
%C 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
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.5 p. 29, 2.7 p. 117 and 5.4 p. 285.
%H G. C. Greubel, <a href="/A080130/b080130.txt">Table of n, a(n) for n = 0..10000</a>
%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 202.
%H Jeffrey C. Lagarias, <a href="http://arxiv.org/abs/1303.1856">Euler's constant: Euler's work and modern developments</a>, Bull. Amer. Math. Soc., 50 (2013), 527-628; arXiv:1303.1856 [math.NT], 2013.
%H Doron Zeilberger and Noam Zeilberger, <a href="http://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/fcp.html">Fractional Counting of Integer Partitions</a>, 2018; <a href="/A080130/a080130.pdf">Local copy</a> [Pdf file only, no active links]
%F Equals lim inf_{n->oo} phi(n)*log(log(n))/n. - _Arkadiusz Wesolowski_, Aug 26 2012
%F From _Alois P. Heinz_, Dec 05 2018: (Start)
%F Equals lim_{n->oo} A322364(n)/(n*A322365(n)).
%F Equals lim_{n->oo} A322380(n)/A322381(n). (End)
%F Equals lim_{k->oo} log(k)*Product_{prime p<=k} (1-1/p). - _Amiram Eldar_, Jul 09 2020
%F Equals lim_{n->oo} A007838(n)/A000142(n). - _Alois P. Heinz_, Feb 24 2022
%F Equals Product_{k>=1} (1+1/k)*exp(-1/k). - _Amiram Eldar_, Mar 20 2022
%F Equals A001113^(-A001620). - _Omar E. Pol_, Dec 14 2022
%F 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
%e 0.56145948356688516982414321479088078676571...
%p evalf(exp(-gamma), 120); # _Alois P. Heinz_, Feb 24 2022
%t RealDigits[N[Exp[-EulerGamma], 200]][[1]] (* _Arkadiusz Wesolowski_, Aug 26 2012 *)
%o (PARI) default(realprecision, 100); exp(-Euler) \\ _G. C. Greubel_, Aug 28 2018
%o (Magma) R:= RealField(100); Exp(-EulerGamma(R)); // _G. C. Greubel_, Aug 28 2018
%Y Cf. A000010, A000142, A001113, A001620, A002852, A007838, A073004, A079650, A322364, A322365, A322380, A322381.
%K cons,nonn
%O 0,1
%A _Benoit Cloitre_, Jan 26 2003
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