A349220 -id:A349220 - cp's OEIS frontend

A210593 Decimal expansion of the series limit of Sum_{k>=1} (-1)^k*log(k)/k^2.

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

Offset: 0

R. J. Mathar, Mar 23 2012

First derivative of the Dirichlet eta-function eta(s) at s=2.

Phatisena et al. misspell "Euler" and provide the wrong sign and an invalid 7th digit.

			0.101316578163504501886002882212242183659384776374911163334294247196204...

G. C. Greubel, Table of n, a(n) for n = 0..10000
S. Phatisena, R. E. Amritkar, P. V. Panat, Exchange and correlation potential for a two-dimensional electron gas at finite temperatures, Phys. Rev. A 34 (1986) 5070.
Wikipedia, Dirichlet eta function

Cf. A073002, A013661, A002162, A091812 (s=1), A375506 (s=3/2), A349220 (s=3), A349252 (s=4).

1/2*log(2)*Zeta(2)+Zeta(1,2)/2 ; evalf(%) ;

N[(1/12)*Pi^2*(Log[4] - 12*Log[Glaisher] + Log[Pi] + EulerGamma), 105] // RealDigits // First (* Jean-François Alcover, Feb 05 2013 *)

(log(2)*zeta(2)+zeta'(2))/2 \\ Charles R Greathouse IV, Mar 28 2012

Decimal expansion of (log(2)*zeta(2) + zeta'(2)) / 2.

Extended to 105 digits by Jean-François Alcover, Feb 05 2013

A349252 Decimal expansion of Sum_{k>=1} (-1)^k * log(k) / k^4.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Nov 12 2021

First derivative of the Dirichlet eta function at 4.

			0.0334788045785650663859568547887377997137597304057...

Eric Weisstein's World of Mathematics, Dirichlet Eta Function

Cf. A013662, A091812, A210593, A256358, A261506, A267315, A349220.

Flatten[{0, RealDigits[(Pi^4 Log[2] + 630 Zeta'[4])/720, 10, 120][[1]]}]

sumalt(k=1, (-1)^k * log(k) / k^4) \\ Michel Marcus, Nov 12 2021

Equals (Pi^4 * log(2) + 630 * zeta'(4)) / 720.

A375506 Decimal expansion of the first derivative of the Dirichlet eta-function eta(s) at s=3/2.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Aug 18 2024

			0.12867475083035719009595292991030137571142185354249...

Cf. A091812 (at s=1), A210593 (at s=2), A349220 (at s=3), A078434 (zeta(3/2)), A375503 (zeta'(3/2)).

s :=3/2 ; 2^(1-s)*log(2)*Zeta(s)+(1-2^(1-s))*Zeta(1,s) ; evalf(%) ;

RealDigits[DirichletEta'[3/2], 10, 120][[1]] (* Amiram Eldar, Aug 19 2024 *)

Equals log(2)*zeta(3/2)/sqrt(2) +(1-1/sqrt(2))*zeta'(3/2) = Sum_{i>=1} (-1)^i*log(i)/i^(3/2).

cp's OEIS Frontend

A210593 Decimal expansion of the series limit of Sum_{k>=1} (-1)^k*log(k)/k^2.

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A349252 Decimal expansion of Sum_{k>=1} (-1)^k * log(k) / k^4.

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A375506 Decimal expansion of the first derivative of the Dirichlet eta-function eta(s) at s=3/2.

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