A306016 Decimal expansion of -Zeta'(2)/Zeta(2).
5, 6, 9, 9, 6, 0, 9, 9, 3, 0, 9, 4, 5, 3, 2, 8, 0, 6, 3, 9, 9, 8, 6, 4, 3, 6, 0, 0, 1, 9, 7, 3, 0, 0, 0, 2, 4, 0, 3, 4, 8, 2, 2, 8, 0, 8, 0, 6, 9, 3, 0, 9, 7, 9, 5, 5, 8, 1, 9, 7, 3, 6, 0, 4, 3, 7, 9, 1, 7, 2, 7, 7, 3, 6, 6, 7, 4, 0, 6, 0, 6, 7, 8, 7, 8, 6, 7
Offset: 0
Examples
Equals 0.569960993094532806399864360019730002403482280806930979558...
Links
- Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 179.
- J. B. Rosser, L. Schoenfeld, Approximate formulas for some functions of prime numbers, Ill. J. Math. 6 (1) (1962) 64-94, Table IV.
Programs
-
Maple
-(6/Pi^2)*Zeta(1,2): evalf(%,100);
-
Mathematica
RealDigits[- Zeta'[2] / Zeta[2], 10, 87][[1]]
-
PARI
-zeta'(2)/zeta(2) \\ Michel Marcus, Jan 11 2019
Formula
Equals -(6/Pi^2)*Zeta'(2).
Equals 1 - 12*Zeta'(-1) - gamma - log(2*Pi).
From Amiram Eldar, Aug 14 2020: (Start)
Equals Sum_{k>=1} Lambda(k)/k^2, where Lambda is the Mangoldt function.
Equals Sum_{p prime} log(p)/(p^2 - 1). (End)