A256921 -id:A256921 - cp's OEIS frontend

A256922 Decimal expansion of Sum_{k>=2} (-1)^kzeta(k)/(k2^k).

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

Offset: 0

Jean-François Alcover, Apr 13 2015

			0.167825594815521207957737599259554003269226940067362331039...

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

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

Cf. A001620, A002162, A053510, A094640, A256921.

SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/2 + (1/2)*Log(Pi(R)) - Log(2); // G. C. Greubel, Sep 05 2018

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

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

Equals A001620/2 + (1/2)*log(Pi) - log(2).
Equals Sum_{k>=1} (1/(2*k) - log(1 + 1/(2*k))). - Amiram Eldar, Jul 22 2020
Equals (A001620 - A094640)/2. - Ruud H.G. van Tol, Apr 26 2025

A374775 Decimal expansion of (2 + gamma - log(Pi))/2.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Jul 19 2024

			0.71624288952606634323154236936467185969743226151231...

Stephane Louboutin, Majorations explicites de |L(1, χ)| (Suite), C. R. Acad. Sci. Paris. 323, pp. 443-446 (1996). (In French). See Theorem 2 at p. 444.

Cf. A374774, A374776.

Cf. A000796, A001620, A053510, A155739, A256921.

RealDigits[(2+EulerGamma-Log[Pi])/2,10,100][[1]]

Equals 2*A374774.

Equals 1 - A256921. - Hugo Pfoertner, Jul 19 2024

A365959 Decimal expansion of Sum_{k>=2} zeta(k)/k^2.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Sep 23 2023

			0.835998332700964322970911198696029096427042168093233248329556349257701895253...

Mathematics Stack Exchange, On a log-gamma definite integral, 2020.
Michael I. Shamos, A catalog of the real numbers, 2011, p. 596.

Cf. A231132, A256921.

PARI
```
sumpos(k=2,zeta(k)/k^2)
```

Equals Sum_{k>=1} (polylog(2, 1/k) - 1/k).
From Velin Yanev, Jul 30 2025: (Start)
Equals Integral_{x=0..1} log(Gamma(1 - x))/x dx - A001620. [Proved by Paul Enta, 2020]
Conjecture: Equals 2 - A001620 - Pi^2/12 + Integral_{x=0..oo} (2*x*arccot(x) - log(1/x^2 + 1))*log(1 - exp(-2*Pi*x))/(2*Pi*(x^2 + 1)) dx. (End)

A307671 Decimal expansion of the alternating convergent series S = Sum_{k>=0} (-1)^kf(k), where f(k) = harmonic(2^k) - klog(2) - gamma, harmonic(m) is the Sum_{j=1..m} 1/j, and gamma is Euler-Mascheroni constant.

Original entry on oeis.org

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

Offset: 0

Luis H. Gallardo, Apr 20 2019

			0.272343587707596764784070676923955578748225108064395871645389620412837597...

Cf. A001620 (Euler-Mascheroni), A001008/A002805 (harmonic), A002162 (log(2)), A094640 (alternate Euler's constant), A256921 (a similar constant).

evalf(Sum((-1)^k*(harmonic(2^k) - k*log(2) - gamma), k=0..infinity), 120); # Vaclav Kotesovec, Apr 30 2019

digits = 104; s = NSum[(-1)^k*(HarmonicNumber[2^k] - k*Log[2] - EulerGamma), {k, 0, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> digits+10]; RealDigits[s, 10, digits][[1]] (* Jean-François Alcover, Apr 28 2019 *)

default(realprecision, 120); sumalt(k=0, (-1)^k*(psi(2^k+1) - k*log(2))) \\ Vaclav Kotesovec, Apr 30 2019

cp's OEIS Frontend

A256922 Decimal expansion of Sum_{k>=2} (-1)^k*zeta(k)/(k*2^k).

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A374775 Decimal expansion of (2 + gamma - log(Pi))/2.

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A365959 Decimal expansion of Sum_{k>=2} zeta(k)/k^2.

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A307671 Decimal expansion of the alternating convergent series S = Sum_{k>=0} (-1)^k*f(k), where f(k) = harmonic(2^k) - k*log(2) - gamma, harmonic(m) is the Sum_{j=1..m} 1/j, and gamma is Euler-Mascheroni constant.

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A256922 Decimal expansion of Sum_{k>=2} (-1)^kzeta(k)/(k2^k).

A307671 Decimal expansion of the alternating convergent series S = Sum_{k>=0} (-1)^kf(k), where f(k) = harmonic(2^k) - klog(2) - gamma, harmonic(m) is the Sum_{j=1..m} 1/j, and gamma is Euler-Mascheroni constant.