A147533 Decimal expansion of 2*gamma-1, where gamma is the Euler-Mascheroni constant.

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

Author

Benoit Cloitre, Nov 06 2008

Keywords

• cons
• nonn

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.

Examples

			2*gamma - 1 = 0.15443132980306572121302418016480486208431867187984...
		

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

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• 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.

Crossrefs

Cf. A000005, A001008, A001620, A002805.

Programs

Magma

R:= RealField(100); 2*EulerGamma(R) -1; // G. C. Greubel, Aug 31 2018

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 *)

PARI

2*Euler-1

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