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%I #64 Jan 30 2023 04:55:10
%S 1,7,8,1,0,7,2,4,1,7,9,9,0,1,9,7,9,8,5,2,3,6,5,0,4,1,0,3,1,0,7,1,7,9,
%T 5,4,9,1,6,9,6,4,5,2,1,4,3,0,3,4,3,0,2,0,5,3,5,7,6,6,5,8,7,6,5,1,2,8,
%U 4,1,0,7,6,8,1,3,5,8,8,2,9,3,7,0,7,5,7,4,2,1,6,4,8,8,4,1,8,2,8,0,3,3,4,8,2
%N Decimal expansion of exp(gamma).
%C See references and additional links in A094644.
%C The Riemann hypothesis holds if and only if the inequality sigma(n)/(n*log(log(n))) < exp(gamma) is valid for all n >= 5041, (G. Robin, 1984). - _Peter Luschny_, Oct 18 2020
%H Stanislav Sykora, <a href="/A073004/b073004.txt">Table of n, a(n) for n = 1..2000</a>
%H Paul Erdős and S. K. Zaremba, <a href="https://users.renyi.hu/~p_erdos/1974-34.pdf">The arithmetic function Sum_{d|n} log d/d</a>, Demonstratio Mathematica, Vol. 6 (1973), pp. 575-579.
%H T. H. Gronwall, <a href="https://www.jstor.org/stable/1988773">Some Asymptotic Expressions in the Theory of Numbers</a>, Trans. Amer. Math. Soc., Vol. 14, No. 1 (1913), pp. 113-122.
%H Jeffrey C. Lagarias, <a href="http://arxiv.org/abs/1303.1856">Euler's constant: Euler's work and modern developments</a>, arXiv:1303.1856 [math.NT], 2013.
%H Jeffrey C. Lagarias, <a href="https://doi.org/10.1090/S0273-0979-2013-01423-X">Euler's constant: Euler's work and modern developments</a>, Bull. Amer. Math. Soc., 50 (2013), 527-628.
%H Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap24.html">The exp(gamma)</a>.
%H G. Robin, <a href="http://zakuski.utsa.edu/~jagy/Robin_1984.pdf">Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann</a>, J. Math. Pures Appl. 63 (1984), 187-213.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Euler-MascheroniConstant.html">Euler-Mascheroni Constant</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GronwallsTheorem.html">Gronwall's Theorem</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MertensTheorem.html">Mertens Theorem</a>, Equations 2-3.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RobinsTheorem.html">Robin's Theorem</a>.
%F By Mertens theorem, equals lim_{m->infinity}(1/log(prime(m))*Product_{k=1..m} 1/(1-1/prime(k))). - _Stanislav Sykora_, Nov 14 2014
%F Equals limsup_{n->oo} sigma(n)/(n*log(log(n))) (Gronwall, 1913). - _Amiram Eldar_, Nov 07 2020
%F Equals limsup_{n->oo} (Sum_{d|n} log(d)/d)/(log(log(n)))^2 (Erdős and Zaremba, 1973). - _Amiram Eldar_, Mar 03 2021
%F Equals Product_{k>=1} (1-1/(k+1))*exp(1/k). - _Amiram Eldar_, Mar 20 2022
%F Equals lim_{n->oo} n * Product_{prime p<=n} p^(1/(1-p)). - _Thomas Ordowski_, Jan 30 2023
%e Exp(gamma) = 1.7810724179901979852365041031071795491696452143034302053...
%t RealDigits[ E^(EulerGamma), 10, 110] [[1]]
%o (PARI) exp(Euler)
%o (Magma) R:=RealField(100); Exp(EulerGamma(R)); // _G. C. Greubel_, Aug 27 2018
%Y Cf. A001620 (Euler-Mascheroni constant, gamma).
%Y Cf. A001113, A067698, A080130, A091901, A094644 (continued fraction for exp(gamma)), A246499.
%K cons,nonn,easy
%O 1,2
%A _Robert G. Wilson v_, Aug 03 2002
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