A091812 -id:A091812 - 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

A242494 Decimal expansion of Sum_{k > 0} (-1)^k*log(k)^2/k.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 16 2014

			0.065372592558898599146207399388201...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, chapter 2.21, p. 168.

Cf. A091812.

-Log[2]^3/3 + EulerGamma*Log[2]^2 +  StieltjesGamma[1]*Log[4] // RealDigits[#, 10, 100]& // First

sumalt(k=1,(-1)^k*log(k)^2/k) \\ Charles R Greathouse IV, Mar 10 2016

sum(k>0, (-1)^k*log(k)^2/k) = -log(2)^3/3 + gamma*log(2)^2 + gamma(1)*log(4).

A265162 Decimal expansion of Sum_{k>=1} (-1)^k*log(k)/sqrt(k).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Dec 03 2015

Differentiation of Sum_{k>=1} (-1)^k/k^s = -(2^s-2)*zeta(s)/2^s with respect to s gives -Sum_{k>=1} (-1)^k*log(k)/k^s = -2^(1-s)*log(2)*zeta(s) - (1-2^(1-s))*zeta'(s), where zeta(.) and zeta'(.) are the Riemann zeta function and its derivative. - R. J. Mathar, Apr 17 2019, typo in the first formula corrected by Vaclav Kotesovec, Jan 11 2024

			0.1932888316392827389646154593552381142952702225292219922936048103344...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Henri Cohen, Fernando Rodriguez Villegas, and Don Zagier, Convergence Acceleration of Alternating Series, Exp. Math. 9 (1) (2000) 3-12.
Eric Weisstein's World of Mathematics, Dirichlet Eta Function.

Cf. A091812, A113024.

evalf(sum((-1)^k*log(k)/sqrt(k), k=1..infinity), 120);

RealDigits[((3-Sqrt[2])*Log[2]/2 - (Sqrt[2]-1)*(2*EulerGamma + Pi + 2*Log[Pi])/4) * Zeta[1/2], 10, 106][[1]]
RealDigits[DirichletEta'[1/2], 10, 110][[1]] (* Eric W. Weisstein, Jan 08 2024 *)

((3-sqrt(2))*log(2)/2 - (sqrt(2)-1)*(2*Euler + Pi + 2*log(Pi))/4)* zeta(1/2) \\ G. C. Greubel, Apr 15 2018

Equals ((3-sqrt(2))*log(2)/2 - (sqrt(2)-1)*(2*gamma + Pi + 2*log(Pi))/4) * zeta(1/2), where gamma is the Euler-Mascheroni constant A001620.

A265162/A113024 = gamma/2 + Pi/4 - (1/2 + sqrt(2))*log(2) + log(Pi)/2.

A349220 Decimal expansion of Sum_{k>=1} (-1)^k * log(k) / k^3.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Nov 11 2021

First derivative of the Dirichlet eta function at 3.

			0.0597059061601953583634292662879256783169268731565...

Eric Weisstein's World of Mathematics, Dirichlet Eta Function

Cf. A002117, A091812, A197070, A210593, A244115, A256358.

Flatten[{0, RealDigits[(Log[2] Zeta[3] + 3 Zeta'[3])/4, 10, 120][[1]]}]

sumalt(k=1, (-1)^k * log(k) / k^3) \\ Michel Marcus, Nov 11 2021

Equals (log(2) * zeta(3) + 3 * zeta'(3)) / 4.

A354295 Decimal expansion of Sum_{k>=1} (-1)^k * log(k) / (k+1).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, May 23 2022

			0.096614934732226887013571828677661550056338319496835535473156803861323...

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 1 (Overseas Publishers Association, Amsterdam, 1986), p. 746, section 5.5.1, formula 4.

Cf. A091812, A257812.

evalf(Sum((-1)^k*log(k)/(k + 1), k = 1..infinity), 105);

N[NSum[(-1)^k*Log[k]/(k+1), {k, 1, Infinity}, WorkingPrecision -> 120, Method -> "AlternatingSigns"], 105]

PARI
```
sumalt(k=1, (-1)^k*log(k)/(k+1))
```

A354450 Decimal expansion of Sum_{k>=1} (1 - log(k)/k)^(2*k).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, May 30 2022

Convergence of this sum is slow, but (1 - log(k)/k)^(2*k) can be expanded as 1/k^2 - log(k)^2/k^3 + (-4*log(k)^3 + 3*log(k)^4)/(6*k^4) + (-3*log(k)^4 + 4*log(k)^5 - log(k)^6)/(6*k^5) + ... and each of these summands can be evaluated exactly because Sum_{k>=1} log(k)^r/k^s is equal to the r-th derivative of zeta(s) * (-1)^r. For example, Sum_{k>=1} log(k)^2/k^3 = zeta''(3).

			1.40710442743517658735368769650782855052127407144777551479405092825455...

Mathematica Stack Exchange, Converges or diverges?, 2021.

Cf. A091812, A354592, A354593.

Digits := 120: ser := sort(convert(series((1-log(n)/n)^(2*n), n = infinity, 300), polynom), n): s := evalf(sum(op(1, ser), n = 1..infinity) + sum(op(2, ser), n = 1..infinity), 120): for k from 3 to nops(ser) do serx := expand(op(k, ser)): for j to nops(serx) do s := s + evalf(sum(op(j, serx), n = 1..infinity), 120) end do: print(k, s) end do:

NSum[(1 - Log[n]/n)^(2*n), {n, 1, Infinity}, WorkingPrecision -> 100, NSumTerms -> 1000] (* only 74 digits are correct *)

default(realprecision, 120); sumpos(k=1, (1 - log(k)/k)^(2*k))

A358789 Decimal expansion of Sum_{p prime, p>=3} (-1)^((p-1)/2)*log(p)/p, negated.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Jan 03 2023

Sum_{p prime} log(p)/p is divergent.

			-0.54568127279512790148953238338...

Cf. A086239, A091812, A136271, A354295.

alfa[s_]:= 1/(1 + 1/2^s) * DirichletBeta[s] * Zeta[s] / Zeta[2*s]; beta[s_]:= (1 - 1/2^s) * Zeta[s] / DirichletBeta[s]; Do[Print[N[-1/2*Sum[MoebiusMu[2*n + 1]/(2*n + 1) * Limit[D[Log[alfa[(2*n + 1)*s]/beta[(2*n + 1)*s]], s], s -> 1], {n, 0, m}], 120]], {m, 20, 200, 20}] (* Vaclav Kotesovec, Jan 25 2023 *)

Limit_{N->oo} ((Sum_{p<=N prime == 3 (mod 4)} log(p)/p) - (Sum_{p<=N prime == 1 (mod 4)} log(p)/p)).

A271533 Decimal expansion of the derivative of the Dirichlet function eta(z) at z = -1.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Apr 09 2016

This entry completes the values of the derivatives eta'(z) at z = 0,1,i,-1,-i (see crossrefs).

			0.265214370914704351169348273575616405600275762885520266292673582574...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Eric Weisstein's World of Mathematics, Dirichlet Eta Function

Cf. A074962, A256358 (eta'(0)), A091812 (eta'(1)), A271525 (real(eta'(i))), A271526 (-imag(eta'(i))) .

RealDigits[3*Log[Glaisher] - Log[2]/3 - 1/4, 10, 120][[1]] (* G. C. Greubel, Apr 09 2016 *)
RealDigits[DirichletEta'[-1], 10, 110][[1]] (* Eric W. Weisstein, Jan 06 2024 *)

\\ Derivative of Dirichlet eta function (fails for z=1):
derdireta(z)=2^(1-z)*log(2)*zeta(z)+(1-2^(1-z))*zeta'(z);
derdireta(-1) \\ Evaluation

eta'(-1) = 3*log(A) - log(2)/3 - 1/4, where A = A074962 is the Glaisher-Kinkelin constant.

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.

A373207 Decimal expansion of Product_{k>=1} f(2k)^2/(f(2k-1) * f(2*k+1)), where f(k) = k^(1/k).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 28 2024

			(2^(1/2)/1^1) * (2^(1/2)/3^(1/3)) * (4^(1/4)/3^(1/3)) * (4^(1/4)/5^(1/5)) * ...
1.37676673907488822612716594825041629908712439037992...

Dirk Huylebrouck, Generalizing Wallis' formula, The American Mathematical Monthly, Vol. 122, No. 4 (2015), pp. 371-372; alternative link; arXiv preprint, arXiv:1402.6577 [math.HO], 2014.
Eric Weisstein's World of Mathematics, Dirichlet Eta Function.
Wikipedia, Dirichlet eta function.

Cf. A001620, A002162, A091812, A373208.

RealDigits[2^(2*EulerGamma - Log[2]), 10, 120][[1]]

PARI
```
2^(2*Euler - log(2))
```

Equals exp(2*eta'(1)) = exp(2*A091812), where eta is the Dirichlet eta function.

Equals 2^(2*gamma - log(2)), where gamma is Euler's constant (A001620).

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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A242494 Decimal expansion of Sum_{k > 0} (-1)^k*log(k)^2/k.

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A265162 Decimal expansion of Sum_{k>=1} (-1)^k*log(k)/sqrt(k).

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A349220 Decimal expansion of Sum_{k>=1} (-1)^k * log(k) / k^3.

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A354295 Decimal expansion of Sum_{k>=1} (-1)^k * log(k) / (k+1).

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A354450 Decimal expansion of Sum_{k>=1} (1 - log(k)/k)^(2*k).

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A358789 Decimal expansion of Sum_{p prime, p>=3} (-1)^((p-1)/2)*log(p)/p, negated.

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A373207 Decimal expansion of Product_{k>=1} f(2k)^2/(f(2k-1) * f(2*k+1)), where f(k) = k^(1/k).