OFFSET
0,1
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
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]
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_{k->oo} log(k)*Product_{prime p<=k} (1-1/p). - Amiram Eldar, Jul 09 2020
Equals Product_{k>=1} (1+1/k)*exp(-1/k). - Amiram Eldar, Mar 20 2022
EXAMPLE
0.56145948356688516982414321479088078676571...
MAPLE
evalf(exp(-gamma), 120); # Alois P. Heinz, Feb 24 2022
MATHEMATICA
RealDigits[N[Exp[-EulerGamma], 200]][[1]] (* Arkadiusz Wesolowski, Aug 26 2012 *)
PROG
(PARI) default(realprecision, 100); exp(-Euler) \\ G. C. Greubel, Aug 28 2018
(Magma) R:= RealField(100); Exp(-EulerGamma(R)); // G. C. Greubel, Aug 28 2018
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Benoit Cloitre, Jan 26 2003
STATUS
approved