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A073004 Decimal expansion of exp(gamma). 49
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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
See references and additional links in A094644.
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
LINKS
Paul Erdős and S. K. Zaremba, The arithmetic function Sum_{d|n} log d/d, Demonstratio Mathematica, Vol. 6 (1973), pp. 575-579.
T. H. Gronwall, Some Asymptotic Expressions in the Theory of Numbers, Trans. Amer. Math. Soc., Vol. 14, No. 1 (1913), pp. 113-122.
Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, arXiv:1303.1856 [math.NT], 2013.
Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, Bull. Amer. Math. Soc., 50 (2013), 527-628.
Simon Plouffe, The exp(gamma).
G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187-213.
Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant.
Eric Weisstein's World of Mathematics, Gronwall's Theorem.
Eric Weisstein's World of Mathematics, Mertens Theorem, Equations 2-3.
Eric Weisstein's World of Mathematics, Robin's Theorem.
FORMULA
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
Equals limsup_{n->oo} sigma(n)/(n*log(log(n))) (Gronwall, 1913). - Amiram Eldar, Nov 07 2020
Equals limsup_{n->oo} (Sum_{d|n} log(d)/d)/(log(log(n)))^2 (Erdős and Zaremba, 1973). - Amiram Eldar, Mar 03 2021
Equals Product_{k>=1} (1-1/(k+1))*exp(1/k). - Amiram Eldar, Mar 20 2022
Equals lim_{n->oo} n * Product_{prime p<=n} p^(1/(1-p)). - Thomas Ordowski, Jan 30 2023
EXAMPLE
Exp(gamma) = 1.7810724179901979852365041031071795491696452143034302053...
MATHEMATICA
RealDigits[ E^(EulerGamma), 10, 110] [[1]]
PROG
(PARI) exp(Euler)
(Magma) R:=RealField(100); Exp(EulerGamma(R)); // G. C. Greubel, Aug 27 2018
CROSSREFS
Cf. A001620 (Euler-Mascheroni constant, gamma).
Cf. A001113, A067698, A080130, A091901, A094644 (continued fraction for exp(gamma)), A246499.
Sequence in context: A188485 A093828 A010514 * A256670 A021132 A019936
KEYWORD
cons,nonn,easy
AUTHOR
Robert G. Wilson v, Aug 03 2002
STATUS
approved

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Last modified March 19 06:32 EDT 2024. Contains 370953 sequences. (Running on oeis4.)