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

%I A265162 #30 Jan 11 2024 09:20:44
%S A265162 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,
%T A265162 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,
%U A265162 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
%N A265162 Decimal expansion of Sum_{k>=1} (-1)^k*log(k)/sqrt(k).
%C A265162 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
%H A265162 G. C. Greubel, <a href="/A265162/b265162.txt">Table of n, a(n) for n = 0..10000</a>
%H A265162 Henri Cohen, Fernando Rodriguez Villegas, and Don Zagier, <a href="https://dx.doi.org/10.1080/10586458.2000.10504632">Convergence Acceleration of Alternating Series</a>, Exp. Math. 9 (1) (2000) 3-12.
%H A265162 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DirichletEtaFunction.html">Dirichlet Eta Function</a>.
%F A265162 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.
%F A265162 A265162/A113024 = gamma/2 + Pi/4 - (1/2 + sqrt(2))*log(2) + log(Pi)/2.
%e A265162 0.1932888316392827389646154593552381142952702225292219922936048103344...
%p A265162 evalf(sum((-1)^k*log(k)/sqrt(k), k=1..infinity), 120);
%t A265162 RealDigits[((3-Sqrt[2])*Log[2]/2 - (Sqrt[2]-1)*(2*EulerGamma + Pi + 2*Log[Pi])/4) * Zeta[1/2], 10, 106][[1]]
%t A265162 RealDigits[DirichletEta'[1/2], 10, 110][[1]] (* _Eric W. Weisstein_, Jan 08 2024 *)
%o A265162 (PARI) ((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
%Y A265162 Cf. A091812, A113024.
%K A265162 nonn,cons
%O A265162 0,2
%A A265162 _Vaclav Kotesovec_, Dec 03 2015