A197070 Decimal expansion of the Dirichlet eta-function at 3.

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

Offset

0,1

Comments

This constant is irrational by Apéry's theorem. - Charles R Greathouse IV, Feb 11 2024

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Seán Stewart, Problem 12206, The American Mathematical Monthly, Vol. 127, No. 8 (2020), p. 752.

Wikipedia, Dirichlet eta function.

Formula

Equals 3*zeta(3)/4 = 3*A002117/4.

Also equals the integral over the unit cube [0,1]x[0,1]x[0,1] of 1/(1+x*y*z) dx dy dz. - Jean-François Alcover, Nov 24 2014

Equals Sum_{n>=1} (-1)^(n+1)/n^3. - Terry D. Grant, Aug 03 2016

Equals Lim_{n -> infinity} A136675(n)/A334582(n). - Petros Hadjicostas, May 07 2020

Equals Sum_{n>=1} AH(2*n)/n^2, where AH(n) = Sum_{k=1..n} (-1)^(k+1)/k = A058313(n)/A058312(n) is the n-th alternating harmonic number (Stewart, 2020). - Amiram Eldar, Oct 04 2021

Example

0.9015426773696957140498036211335874930737...

Maple

3*Zeta(3)/4 ; evalf(%) ;

Mathematica

RealDigits[3(Zeta[3])/4, 10, 75][[1]] (* Bruno Berselli, Dec 20 2011 *)

Prog

(PARI) -polylog(3, -1) \\ Charles R Greathouse IV, Mar 28 2012
(PARI) 3/4*zeta(3) \\ Charles R Greathouse IV, Mar 28 2012

Crossrefs

Cf. A002117 (zeta(3)), A058312, A058313, A072691, A136675, A233090 (5*zeta(3)/8), A233091 (7*zeta(3)/8), A334582.

Sequence in context: A361060 A364830 A345738 * A197333 A226120 A244593

Adjacent sequences: A197067 A197068 A197069 * A197071 A197072 A197073

Keyword

cons,easy,nonn

Author

R. J. Mathar, Oct 09 2011

Status

approved