A256921 - OEIS

%I #29 Sep 08 2022 08:46:12

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

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

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

%N Decimal expansion of Sum_{k>=2} zeta(k)/(k*2^k).

%D H. M. Srivastava and Junesang Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Insights (2011) p. 272.

%H G. C. Greubel, <a href="/A256921/b256921.txt">Table of n, a(n) for n = 0..10000</a>

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/RiemannZetaFunction.html">Riemann Zeta Function</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Riemann_zeta_function">Riemann Zeta Function</a>

%F Equals (1/2)*log(Pi) - EulerGamma/2.

%F Equals Sum_{k>0} (-1)^(k+1)*(H(k)-log(k)-EulerGamma), where H(k) is the k-th harmonic number.

%F Equals -Sum_{k>=1} (1/(2*k) + log(1 - 1/(2*k))). - _Amiram Eldar_, Jul 22 2020

%e 0.2837571104739336567684576306353281403025677384876939863539279...

%t RealDigits[(1/2)*Log[Pi] - EulerGamma/2, 10, 105] // First

%o (PARI) log(Pi)/2 - Euler/2 \\ _Michel Marcus_, Apr 13 2015

%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); (Log(Pi(R)) - EulerGamma(R))/2; // _G. C. Greubel_, Sep 04 2018

%Y Cf. A001620, A053510, A256922.

%K nonn,cons,easy

%O 0,1

%A _Jean-François Alcover_, Apr 13 2015