A147533 - OEIS

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

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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

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.

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

Cf. A000005, A001008, A001620, A002805.