OFFSET
1,1
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..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] (3*(-2*arctan(3*x) + 3*x*log(1/9 + x^2)))/((-1 + exp(2*Pi*x))*(1 + 9*x^2)) dx - 3*log(3)/2 - log(3)^2/2.
EXAMPLE
-3.25955751591791019525087458267655925797647220439943...
MAPLE
evalf(int((3*(-2*arctan(3*x)+3*x*log(1/9+x^2)))/((-1+exp(2*Pi*x))*(9*x^2+1)), x = 0..infinity)-3*log(3)*(1/2)-(1/2)*log(3)^2, 120); # Vaclav Kotesovec, Jan 29 2015
MATHEMATICA
gamma1[1/3] = (1/6)*((-Sqrt[3])*Pi*(EulerGamma + Log[(24*Pi^(5/2))/Gamma[1/6]^3]) - 3*(Log[3]^2 + EulerGamma*Log[27] + 2*Log[3]*Log[2*Pi] + 2*Log[2*Pi]^2 + Log[3/(4*Pi^2)]*Log[6*Pi] - 2*StieltjesGamma[1] + Derivative[2, 0][Zeta][0, 1/3] + Derivative[2, 0][Zeta][0, 2/3])) // Re; RealDigits[gamma1[1/3], 10, 104] // First
(* Or, from Mma version 7 up: *) RealDigits[StieltjesGamma[1, 1/3], 10, 104] // First
CROSSREFS
KEYWORD
AUTHOR
Jean-François Alcover, Jan 28 2015
STATUS
approved