|
| |
|
|
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
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
|
|
|
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
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
