OFFSET
0,2
COMMENTS
A slowly convergent series. It may be efficiently computed via the Hurwitz zeta-function (see formula below).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..10000
Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations, Journal of Number Theory (Elsevier), vol. 148, pp. 537-592 & vol. 151, pp. 276-277, 2015. arXiv version, arXiv:1401.3724 [math.NT].
FORMULA
(Zeta(1/2, 1/(2*Pi)) + Zeta(1/2, 1-1/(2*Pi)))/2, see formula (26) in the reference.
EXAMPLE
-0.1941089350921820497391492449281947266353205526340478...
MAPLE
evalf(1/2*(Zeta(0, 1/2, 1/(2*Pi)) + Zeta(0, 1/2, 1-1/(2*Pi))), 120);
MATHEMATICA
N[(Zeta[1/2, 1/(2*Pi)] + Zeta[1/2, 1 - 1/(2*Pi)])/2, 200]
RealDigits[Re[(1/2)*(PolyLog[1/2, E^(-I)] + PolyLog[1/2, E^I])], 10, 109][[1]] (* Vaclav Kotesovec, Oct 31 2015 *)
PROG
(PARI) zetahurwitz(1/2, 1/Pi/2)/2 + zetahurwitz(1/2, 1-1/Pi/2)/2 \\ Charles R Greathouse IV, Jan 30 2018
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Iaroslav V. Blagouchine, Oct 11 2015
STATUS
approved