A276712 - OEIS

%I #44 Jun 07 2024 11:44:34

%S 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,

%T 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,

%U 9,7,7,5,7,2,3,2,8,9

%N Decimal expansion of zeta(3)/8.

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

%H G. C. Greubel, <a href="/A276712/b276712.txt">Table of n, a(n) for n = 0..5000</a>

%H R. Barbieri, J. A. Mignaco and E. Remiddi, <a href="https://dx.doi.org/10.1007/BF02728545">Electron form factors up to fourth order. I.</a>, Il Nuovo Cim. 11A (4) (1972) 824-864, table II (5)

%H Nick Lord, <a href="http://www.jstor.org/stable/3620652">Problen 89.D</a>, Problem Corner, The Mathematical Gazette, Vol. 89, No. 514 (2005), pp. 115-119; <a href="http://www.jstor.org/stable/3621975">Solution</a>, ibid., Vol. 89, No. 516 (2005), pp. 539-542.

%H Michael Penn, <a href="https://www.youtube.com/watch?v=mUU8ZUBa3t4">The solution is an important constant</a>, YouTube video, 2021.

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

%F Equals A002117/8.

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

%F 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

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

%F Equals -integral_{x=0..1} log(x) log(1+x)/(1+x). [Barbieri] - _R. J. Mathar_, Jun 07 2024

%e 0.150257112894949285674967270188...

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

%o (Sage) (zeta(3)/8).n(100)

%o (PARI) zeta(3)/8 \\ _Michel Marcus_, Sep 16 2016

%o (Magma) SetDefaultRealField(RealField(120)); L:=RiemannZeta();  Evaluate(L,3)/8; // _G. C. Greubel_, Nov 24 2021

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

%K nonn,cons

%O 0,2

%A _Terry D. Grant_, Sep 15 2016