A091812 -id:A091812 - cp's OEIS frontend

A373564 Decimal expansion of Sum_{k>=0} (-1)^k * log(4k+3)/(4k+3) .

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

Offset: 0

R. J. Mathar, Jun 10 2024

			0.2084772946845770695428937666547886...

Cf. A091812, A373563, A242011.

sumalt(k = 0, (-1)^k * log(4*k+3)/(4*k+3)) \\ Amiram Eldar, Aug 20 2024

this - A373563 = A242011.

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).

A373562 Decimal expansion of Sum_{k>=0} (-1)^(k+1)*( log(2k+1)/(2k+1) + log(2k+2)/(2k+2) ).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jun 10 2024

			0.03260926170902720290811243602...

Cf. A091812, A078127, A366349.

RealDigits[Pi*(EulerGamma + 2*Log[2] + 3*Log[Pi] - 4*Log[Gamma[1/4]])/4 - (3 Log[2]^2 - 2 EulerGamma Log[2])/4, 10, 120, -1][[1]] (* Amiram Eldar, Jun 10 2024 *)

Equals A078127 - A366349.

cp's OEIS Frontend

A373564 Decimal expansion of Sum_{k>=0} (-1)^k * log(4k+3)/(4k+3) .

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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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A373562 Decimal expansion of Sum_{k>=0} (-1)^(k+1)*( log(2k+1)/(2k+1) + log(2k+2)/(2k+2) ).

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