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%I #11 Jul 28 2021 17:00:06
%S 4,5,2,9,3,6,4,5,8,6,8,1,0,1,1,7,9,1,2,8,9,9,2,2,1,4,3,8,3,9,1,4,2,0,
%T 1,0,6,9,2,9,2,6,4,2,8,1,5,1,5,4,8,0,5,7,4,2,1,9,7,8,2,4,1,9,3,3,3,0,
%U 1,1,5,2,9,7,1,4,2,5,1,9,9,9,4,9,8,8,1
%N Decimal expansion of Integral_{x=0..1} log(G(x+1)) dx, where G(x) is the Barnes G-function.
%D H. M. Srivastava and Junesang Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier, 2012. See p. 53.
%H Ernest William Barnes, <a href="https://gdz.sub.uni-goettingen.de/id/PPN600494829_0031">The theory of the G-function</a>, Quart. J. Math., Vol. 31 (1900), pp. 264-314. See p. 288.
%H Junesang Choi and Hari M. Srivastava, <a href="https://doi.org/10.1006/jmaa.1997.5198">Sums associated with the zeta function</a>, Journal of Mathematical Analysis and Applications, Vol. 206, No. 1 (1997), pp. 103-120. See eq. (2.55), p. 114.
%H Junesang Choi and H. M. Srivastava, <a href="https://doi.org/10.1016/S0377-0427(00)00311-3">Certain classes of series associated with the Zeta function and multiple Gamma functions</a>, Journal of Computational and Applied Mathematics, Vol. 118, No. 1-2 (2000), pp. 87-109. See eq. (5.10), p. 97.
%H Junesang Choi, H. M. Srivastava and J. R. Quine, <a href="https://doi.org/10.1017/S0004972700014210">Some series involving the zeta function</a>, Bulletin of the Australian Mathematical Society, Vol. 51, No. 3 (1995), pp. 383-393. See p. 386.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BarnesG-Function.html">Barnes G-Function</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Barnes_G-function">Barnes G-function</a>.
%F Equals 1/12 + log(2*Pi)/4 - 2*log(A), where A is the Glaisher-Kinkelin constant (A074962) (Barnes, 1899).
%F Equals 1/12 + (1/4) * A061444 - 2 * A225746.
%F Equals 2*zeta'(-1) - zeta'(0)/2 - 1/12. - _Vaclav Kotesovec_, Jun 19 2021
%e 0.04529364586810117912899221438391420106929264281515480574...
%t RealDigits[1/12 + Log[2*Pi]/4 - 2*Log[Glaisher], 10, 100][[1]]
%Y Cf. A061444, A074962, A225746.
%Y Similar constant: A110544.
%Y Cf. A087013, A087014, A087015, A087016, A087017, A252798, A252799.
%K nonn,cons
%O -1,1
%A _Amiram Eldar_, Jun 10 2021
%E Offset corrected by _Georg Fischer_, Jul 28 2021
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