A368551 Decimal expansion of 6*gamma/Pi^2 - 72*zeta'(2)/Pi^4.

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

Offset: 1

Author

Artur Jasinski, Dec 29 2023

Keywords

• nonn
• cons

Comments

Also the Wolf-Kawalec constant of index 0.

For the Wolf-Kawalec constant of index 1 see A368547.

For the Wolf-Kawalec constant of index 2 see A368568.

Let g(n) be the Wolf-Kawalec constant of index n; then the function

zeta(x)/zeta(2*x) - 6/(Pi^2*(x-1))

has the expansion

Sum_{n>=0} (-1)^n*(g(n)/n!)*(x-1)^n

at x=1.

Examples

			1.0438945157119382974...
		

Links

• Artur Kawalec, On the series expansion of a square-free zeta series, arXiv:2312.16811 [math.NT], 2023.
• Marek Wolf, Numerical Determination of a Certain Mathematical Constant Related to the Mobius Function, Computational Methods in Science and Technology, Volume 29 (1-4) 2023, 17-20 see formulas (26) and (27).

Crossrefs

Cf. A000796, A001620, A008683, A073002, A074962, A368547, A368568.

Programs

Mathematica

RealDigits[6 EulerGamma/Pi^2 - 72 Zeta'[2]/Pi^4, 10, 105][[1]]

Formula

Equals (6/Pi^2)*(24*Glaisher - gamma - 2*log(2*Pi)) where Glaisher is A074962.

Equals lim_{x->oo} {(Sum_{n=1..x} abs(mu(n))/n) - 6*log(x)/Pi^2}.