OFFSET
0,1
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..5000
Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments, arXiv:1401.3724 [math.NT], 2015.
Iaroslav V. Blagouchine, A theorem ... (same title), Journal of Number Theory Volume 148, March 2015, Pages 537-592.
Iaroslav V. Blagouchine, Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results, The Ramanujan Journal October 2014, Volume 35, Issue 1, pp 21-110.
Iaroslav V. Blagouchine, Rediscovery of Malmsten’s integrals: Full PDF text.
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function.
Eric Weisstein's World of Mathematics, Stieltjes Constants.
Wikipedia, Stieltjes constants
FORMULA
Equals integral_[0..infinity] (6*(-10*arctan((6*x)/5) + 6*x*log(25/36 + x^2)))/((-1 + e^(2*Pi*x))*(25 + 36*x^2)) dx -(3/5 + (1/2)*log(6/5))*log(6/5).
EXAMPLE
-0.24616900381139073314849171532749069577086909012844...
MATHEMATICA
gamma1[5/6] = (1/2)*((-Log[6])*Log[24] - EulerGamma*Log[432] - 2*Log[2]*Log[2*Pi^2] + Log[(2*Pi)/Sqrt[3]]*Log[144*Pi^2] + Log[Pi]*Log[4/Gamma[1/6]^2] - *Log[12] * Log[Gamma[1/6]] - 2*Log[12*Pi]*Log[Gamma[5/6]] + Sqrt[3]*Pi*(EulerGamma + Log[(12*2^(2/3)*Pi^(3/2)*Gamma[5/6])/Gamma[1/6]^2]) + 2*StieltjesGamma[1] + Derivative[2, 0][Zeta][0, 1/6] - Derivative[2, 0][Zeta][0, 1/3] -
2*Derivative[2, 0][Zeta][0, 1/2] - Derivative[2, 0][Zeta][0, 2/3] + Derivative[2, 0][Zeta][0, 5/6]) // Re; RealDigits[gamma1[5/6], 10, 104] // First
(* Or, from Mma version 7 up: *) RealDigits[StieltjesGamma[1, 5/6], 10, 104] // First
CROSSREFS
KEYWORD
AUTHOR
Jean-François Alcover, Jan 29 2015
STATUS
approved