A228725 Decimal expansion of the generalized Euler constant gamma(1,2).

6, 3, 5, 1, 8, 1, 4, 2, 2, 7, 3, 0, 7, 3, 9, 0, 8, 5, 0, 1, 1, 8, 7, 2, 1, 0, 5, 7, 7, 0, 2, 8, 9, 4, 9, 9, 5, 5, 8, 8, 2, 9, 7, 3, 5, 1, 5, 0, 0, 8, 9, 4, 2, 6, 4, 6, 3, 2, 2, 3, 6, 2, 2, 1, 8, 9, 1, 3, 0, 6, 7, 4, 3, 7, 3, 6, 7, 9, 6, 9, 3, 2, 7, 1

Offset: 0

Author

R. J. Mathar, Aug 31 2013

Keywords

• nonn
• cons

Comments

The complement (A239097) is gamma(0,2) = lim_{x->oo} ((Sum_{n=1..x, n even} 1/n) - log(x)/2) = (A001620 - A002162)/2 = -0.05796575... - R. J. Mathar, Sep 06 2013

Examples

			0.63518142273073908501187210577028949955882973515008942646322...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• M. L. Glasser, A note on Beukers's and related integrals, Amer. Math. Monthly 126(4) (2019), 361-363.
• J. C. Lagarias, Euler's constant: Euler's work and modern developments, arXiv:1303.1856 [math.NT], 2013. See Section 3.8.
• D. H. Lehmer, Euler constants for arithmetic progressions, Acta Arith. 27 (1975), 125-142.

Crossrefs

Cf. A001620, A002162, A020759, A239097.

Programs

Magma

SetDefaultRealField(RealField(100)); R:= RealField();
(EulerGamma + Log(2))/2; // G. C. Greubel, Aug 27 2018

Maple

(gamma+log(2))/2 ; evalf(%) ;

Mathematica

RealDigits[(EulerGamma+Log[2])/2,10,120][[1]] (* Harvey P. Dale, Dec 26 2013 *)

PARI

(Euler+log(2))/2 \\ Charles R Greathouse IV, Jul 21 2015

Formula

Equals lim_{x->oo} ((Sum_{n=1..x, n odd} 1/n) - log(x)/2).

Equals (A001620 + A002162)/2.

From Amiram Eldar, Jun 30 2020: (Start)

Equals -Integral_{x=0..1} log(log(1/x))*x dx.

Equals -Integral_{x=0..oo} exp(-2*x)*log(x) dx. (End)

Equals Integral_{x=0..1, y=0..1} log(-log(x*y))*x*y/log(x*y) dx dy. (Apply Theorem 1 or Theorem 2 of Glasser (2019) to one of Amiram Eldar's integrals.) - Petros Hadjicostas, Jun 30 2020

Equals -(psi(1/2) + log(2))/2 = (A020759 - A002162)/2. - Amiram Eldar, Jan 07 2024