A256921 Decimal expansion of Sum_{k>=2} zeta(k)/(k*2^k).

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, 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, 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

Offset: 0

Author

Jean-François Alcover, Apr 13 2015

Keywords

• nonn
• cons
• easy

Examples

			0.2837571104739336567684576306353281403025677384876939863539279...
		

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Eric Weisstein's MathWorld, Riemann Zeta Function
• Wikipedia, Riemann Zeta Function

Crossrefs

Cf. A001620, A053510, A256922.

Programs

Magma

SetDefaultRealField(RealField(100)); R:= RealField(); (Log(Pi(R)) - EulerGamma(R))/2; // G. C. Greubel, Sep 04 2018

Mathematica

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

PARI

log(Pi)/2 - Euler/2 \\ Michel Marcus, Apr 13 2015

Formula

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

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

Equals -Sum_{k>=1} (1/(2*k) + log(1 - 1/(2*k))). - Amiram Eldar, Jul 22 2020