A276712 Decimal expansion of zeta(3)/8.

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

Offset

0,2

References

James Dodson, The Mathematical Repository Containing Analytical Solutions of Five Hundred Questions: Mostly Selected from Scarce and Valuable Authors, (1748), page 375.

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

R. Barbieri, J. A. Mignaco and E. Remiddi, Electron form factors up to fourth order. I., Il Nuovo Cim. 11A (4) (1972) 824-864, table II (5)

Nick Lord, Problen 89.D, Problem Corner, The Mathematical Gazette, Vol. 89, No. 514 (2005), pp. 115-119; Solution, ibid., Vol. 89, No. 516 (2005), pp. 539-542.

Michael Penn, The solution is an important constant, YouTube video, 2021.

Formula

Equals Sum_{n>=1} 1/(2n)^3 = 1/8 + 1/64 + 1/216 + 1/512 + ...

Equals A002117/8.

zeta(3)/8 + A233091 = Sum_{n>=1} 1/(2n+1)^3 + Sum_{n>=1} 1/(2n)^3 = zeta(3).

Equals Sum_{k>=1} (-1)^(k+1) * H(k)/(k+1)^2, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Jul 22 2020

Equals Integral_{x=0..Pi/4} log(sin(x))*log(cos(x))/(sin(x)*cos(x)) dx (Lord, 2005). - Amiram Eldar, Jun 23 2023

Equals -integral_{x=0..1} log(x) log(1+x)/(1+x). [Barbieri] - R. J. Mathar, Jun 07 2024

Example

0.150257112894949285674967270188...

Mathematica

RealDigits[(Zeta[3])/8, 10, 100][[1]]

Prog

(Sage) (zeta(3)/8).n(100)
(PARI) zeta(3)/8 \\ Michel Marcus, Sep 16 2016
(Magma) SetDefaultRealField(RealField(120)); L:=RiemannZeta();  Evaluate(L, 3)/8; // G. C. Greubel, Nov 24 2021

Crossrefs

Cf. A001008, A002117, A002805, A013661, A111003, A222171, A233091.

Sequence in context: A222597 A021999 A021203 * A199387 A195720 A198883

Adjacent sequences: A276709 A276710 A276711 * A276713 A276714 A276715

Keyword

nonn,cons

Author

Terry D. Grant, Sep 15 2016

Status

approved