# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a073004 Showing 1-1 of 1 %I A073004 #72 Oct 27 2024 09:06:31 %S A073004 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 A073004 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 A073004 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 A073004 Decimal expansion of exp(gamma). %C A073004 See references and additional links in A094644. %C A073004 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 A073004 Stanislav Sykora, Table of n, a(n) for n = 1..2000 %H A073004 Paul Erdős and S. K. Zaremba, The arithmetic function Sum_{d|n} log d/d, Demonstratio Mathematica, Vol. 6 (1973), pp. 575-579. %H A073004 T. H. Gronwall, Some Asymptotic Expressions in the Theory of Numbers, Trans. Amer. Math. Soc., Vol. 14, No. 1 (1913), pp. 113-122. %H A073004 Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, arXiv:1303.1856 [math.NT], 2013. %H A073004 Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, Bull. Amer. Math. Soc., 50 (2013), 527-628. %H A073004 Simon Plouffe, The exp(gamma). %H A073004 G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187-213. %H A073004 Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant. %H A073004 Eric Weisstein's World of Mathematics, Gronwall's Theorem. %H A073004 Eric Weisstein's World of Mathematics, Mertens Theorem, Equations 2-3. %H A073004 Eric Weisstein's World of Mathematics, Robin's Theorem. %F A073004 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 A073004 Equals limsup_{n->oo} sigma(n)/(n*log(log(n))) (Gronwall, 1913). - _Amiram Eldar_, Nov 07 2020 %F A073004 Equals limsup_{n->oo} (Sum_{d|n} log(d)/d)/(log(log(n)))^2 (Erdős and Zaremba, 1973). - _Amiram Eldar_, Mar 03 2021 %F A073004 Equals Product_{k>=1} (1-1/(k+1))*exp(1/k). - _Amiram Eldar_, Mar 20 2022 %F A073004 Equals lim_{n->oo} n * Product_{prime p<=n} p^(1/(1-p)). - _Thomas Ordowski_, Jan 30 2023 %F A073004 Equals Product_{k>=1} (k/sqrt(2))^((-1)^k/(k*log(2))). - _Antonio Graciá Llorente_, Oct 11 2024 %F A073004 Equals lim_{n->oo} (1/log(n))*Product_{prime p<=n} p/(p - 1) [Mertens] (see Finch at p. 31). - _Stefano Spezia_, Oct 27 2024 %e A073004 Exp(gamma) = 1.7810724179901979852365041031071795491696452143034302053... %t A073004 RealDigits[ E^(EulerGamma), 10, 110] [[1]] %o A073004 (PARI) exp(Euler) %o A073004 (Magma) R:=RealField(100); Exp(EulerGamma(R)); // _G. C. Greubel_, Aug 27 2018 %Y A073004 Cf. A001620 (Euler-Mascheroni constant, gamma). %Y A073004 Cf. A001113, A067698, A080130, A091901, A094644 (continued fraction for exp(gamma)), A246499. %K A073004 cons,nonn,easy %O A073004 1,2 %A A073004 _Robert G. Wilson v_, Aug 03 2002 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE