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A147533
Decimal expansion of 2*gamma-1, where gamma is the Euler-Mascheroni constant.
5
1, 5, 4, 4, 3, 1, 3, 2, 9, 8, 0, 3, 0, 6, 5, 7, 2, 1, 2, 1, 3, 0, 2, 4, 1, 8, 0, 1, 6, 4, 8, 0, 4, 8, 6, 2, 0, 8, 4, 3, 1, 8, 6, 7, 1, 8, 7, 9, 8, 4, 7, 1, 9, 7, 6, 1, 1, 5, 3, 4, 4, 6, 9, 7, 6, 9, 7, 3, 5, 4, 5, 3, 5, 5, 5, 3, 2, 9, 3, 4, 1, 8, 7, 3, 8, 9, 4, 1, 2, 6, 5, 8, 3, 4, 9, 3, 4, 9, 9, 0, 2, 9, 2, 6
OFFSET
0,2
COMMENTS
This constant arises in the dynamical system: z(n+1) = F(z(n)) where F(z) = conjugate(zeta(z))/zeta(z) and zeta is the Riemann zeta function. For instance, starting from z(0) with Re(z(0)) > 0 and Im(z(0)) > 0 we get lim_{n->oo} Im(z(2n))/Im(z(2n-1)) = 2*gamma - 1. See Finch's book p. 29 for another appearance.
REFERENCES
Steven R. Finch, "Euler-Mascheroni constant, gamma", Section 1.5 in Mathematical Constants, Cambridge University Press, 2003, pp. 28-32.
Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See Problem 2.10, pages 101 and 115-116.
LINKS
Ovidiu Furdui, Problem H-790, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 54, No. 2 (2016), p. 186; A series involving harmonic numbers and the zeta function at positive integers, Solution to Problem H-790 by Ramya Dutta, ibid., Vol. 56, No. 2 (2018), pp. 190-191.
FORMULA
Equals limit_{n->oo} ((1/n)*Sum_{k=1..n} tau(k) - log(n)), where tau(k) is the number of divisors of k. - Jean-François Alcover, Mar 28 2013, after S. R. Finch's book, p. 29.
Equals Sum_{k>=2} (k-2)*(zeta(k)-1)/k. - Amiram Eldar, Jun 26 2021
Equals Sum_{k>=2} (H(k) - gamma - Sum_{j=2..k} zeta(j)/j), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number (Furdui, 2016). - Amiram Eldar, Dec 02 2021
Equals Integral_{x=0..1} frac(1/x)*frac(1/(1-x)) dx (Furdui, 2013). - Amiram Eldar, Mar 26 2022
EXAMPLE
2*gamma - 1 = 0.15443132980306572121302418016480486208431867187984...
MAPLE
evalf(2*gamma-1); # R. J. Mathar, Jan 26 2009
MATHEMATICA
RealDigits[2*EulerGamma - 1, 10, 100][[1]] (* G. C. Greubel, Aug 31 2018 *)
PROG
(PARI) 2*Euler-1
(Magma) R:= RealField(100); 2*EulerGamma(R) -1; // G. C. Greubel, Aug 31 2018
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Benoit Cloitre, Nov 06 2008
STATUS
approved