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%I #26 Apr 17 2018 17:30:04
%S 1,9,4,1,0,8,9,3,5,0,9,2,1,8,2,0,4,9,7,3,9,1,4,9,2,4,4,9,2,8,1,9,4,7,
%T 2,6,6,3,5,3,2,0,5,5,2,6,3,4,0,4,7,8,1,5,4,0,2,3,9,8,3,7,6,6,0,9,5,6,
%U 6,6,8,3,7,2,6,2,5,5,4,7,6,4,0,0,6,5,3,1,8,9,6,4,9,6,5,5,2,4,7,0,1,2,2,6,8,3,5,1,9
%N Decimal expansion of Sum_{n >= 1} cos(n)/sqrt(n), negated.
%C A slowly convergent series. It may be efficiently computed via the Hurwitz zeta-function (see formula below).
%H G. C. Greubel, <a href="/A263192/b263192.txt">Table of n, a(n) for n = 0..10000</a>
%H Iaroslav V. Blagouchine, <a href="http://dx.doi.org/10.1016/j.jnt.2014.08.009">A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations</a>, Journal of Number Theory (Elsevier), vol. 148, pp. 537-592 & vol. 151, pp. 276-277, 2015. <a href="http://arxiv.org/abs/1401.3724">arXiv version</a>, arXiv:1401.3724 [math.NT].
%F (Zeta(1/2, 1/(2*Pi)) + Zeta(1/2, 1-1/(2*Pi)))/2, see formula (26) in the reference.
%e -0.1941089350921820497391492449281947266353205526340478...
%p evalf(1/2*(Zeta(0, 1/2, 1/(2*Pi)) + Zeta(0, 1/2, 1-1/(2*Pi))), 120);
%t N[(Zeta[1/2, 1/(2*Pi)] + Zeta[1/2, 1 - 1/(2*Pi)])/2, 200]
%t RealDigits[Re[(1/2)*(PolyLog[1/2, E^(-I)] + PolyLog[1/2, E^I])], 10, 109][[1]] (* _Vaclav Kotesovec_, Oct 31 2015 *)
%o (PARI) zetahurwitz(1/2, 1/Pi/2)/2 + zetahurwitz(1/2, 1-1/Pi/2)/2 \\ _Charles R Greathouse IV_, Jan 30 2018
%Y Cf. A113024, A121225, A263193.
%K nonn,cons
%O 0,2
%A _Iaroslav V. Blagouchine_, Oct 11 2015
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