A254331 Decimal expansion of gamma_1(1/3), the first generalized Stieltjes constant at 1/3 (negated).

3, 2, 5, 9, 5, 5, 7, 5, 1, 5, 9, 1, 7, 9, 1, 0, 1, 9, 5, 2, 5, 0, 8, 7, 4, 5, 8, 2, 6, 7, 6, 5, 5, 9, 2, 5, 7, 9, 7, 6, 4, 7, 2, 2, 0, 4, 3, 9, 9, 4, 3, 0, 0, 4, 8, 1, 1, 7, 9, 7, 4, 8, 6, 7, 3, 8, 9, 7, 9, 3, 7, 0, 1, 4, 9, 5, 4, 6, 8, 7, 9, 2, 4, 7, 8, 9, 6, 5, 2, 5, 8, 8, 2, 0, 0, 8, 6, 7, 3, 8, 0, 4, 3, 2

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

Cf. A001620 (gamma), A082633 (gamma_1), A254327 (gamma_1(1/2)), A254345 (gamma_1(2/3)), A254347 (gamma_1(1/4)), A254348 (gamma_1(3/4)), A254349 (gamma_1(1/6)), A254350 (gamma_1(5/6)), A251866 (gamma_1(1/5)), A255188 (gamma_1(1/8)), A255189 (gamma_1(1/12)).

Sequence in context: A280875 A173701 A116627 * A210742 A175056 A320274

Adjacent sequences: A254328 A254329 A254330 * A254332 A254333 A254334

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Jan 28 2015

Status

approved