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

0,1

Links

Table of n, a(n) for n=0..86.

Mark W. Coffey and Nicholas Lubbers, On generalized harmonic number sums, Appl. Math. Comput., Vol. 217, No. 2 (2010), pp. 689-698. See Corollary 6, eq. (52), p. 696.

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 179.

J. Barkley Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Ill. J. Math. 6 (1) (1962), 64-94, Table IV.

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)

Equals Sum_{k>=1} (Lambda(k)/k - Lambda(k+1)/(k+1)) * H(k), where Lambda is the Mangoldt function, and H(k) = A001008(k)/A002805(k) is the k-th harmonic number (Coffey and Lubbers, 2010). - Amiram Eldar, Oct 13 2025

Example

0.569960993094532806399864360019730002403482280806930979558...

Maple

-(6/Pi^2)*Zeta(1, 2): evalf(%, 100);

Mathematica

RealDigits[- Zeta'[2] / Zeta[2], 10, 87][[1]]

Prog

(PARI) -zeta'(2)/zeta(2) \\ Michel Marcus, Jan 11 2019

Crossrefs

Cf. A001008, A001620, A002805, A013661, A059956, A061444, A073002.

Sequence in context: A021641 A332327 A260635 * A087176 A227006 A139454

Adjacent sequences: A306013 A306014 A306015 * A306017 A306018 A306019

Keyword

nonn,cons

Author

Peter Luschny, Jun 17 2018

Status

approved