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A269330 Decimal expansion of the "alternating Euler constant" beta = li(2) - gamma.

%I A269330 #42 Oct 16 2023 17:05:53
%S A269330 4,6,7,9,4,8,1,1,5,2,1,5,9,5,9,9,2,4,2,3,8,0,7,6,7,9,9,1,1,2,2,1,0,7,
%T A269330 0,5,4,8,0,4,5,6,2,4,2,2,1,1,2,7,7,9,7,7,0,2,7,1,4,1,9,0,9,1,9,0,1,4,
%U A269330 5,4,7,8,4,3,2,6,9,4,8,5,9,2,3,5,7,7,0,3,4,2,3,3,4,6,3,6,6,0,6,7,9,1,3,8
%N A269330 Decimal expansion of the "alternating Euler constant" beta = li(2) - gamma.
%C A269330 The function li(x) is the integral logarithm, gamma is Euler's constant.
%C A269330 Decimal expansion of Sum_{n>=1} G_n/n = beta, where numbers G_n are Gregory's coefficients (see A002206 and A002207). In comparison to the Fontana-Mascheroni's series Sum_{n>=1} |G_n|/n = gamma (see A195189), the constant beta may be regarded as the "alternating Euler constant". A similar analogy also exists between gamma and log(4/Pi), see A094640.
%C A269330 Another striking analogy between beta and gamma follows from the fact that beta = Integral_{x=0..1} (1/log(1+x) - 1/x) dx, while gamma = Integral_{x=0..1} (1/log(1-x) + 1/x) dx.
%C A269330 For more details, see references below.
%H A269330 Iaroslav V. Blagouchine, <a href="/A269330/b269330.txt">Table of n, a(n) for n = 0..1000</a>
%H A269330 Iaroslav V. Blagouchine, <a href="http://dx.doi.org/10.1016/j.jmaa.2016.04.032">Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/pi</a>, Journal of Mathematical Analysis and Applications (Elsevier), 2016. <a href="http://arxiv.org/abs/1408.3902">arXiv version</a>, arXiv:1408.3902 [math.NT], 2014-2016.
%H A269330 Iaroslav V. Blagouchine, <a href="https://doi.org/10.1080/00029890.2022.1992214">Another Alternating Analogue of Euler's Constant</a>. The American Mathematical Monthly, vol. 120, no. 1, pp. 24-34, 2022.
%H A269330 Steven Finch, <a href="https://arxiv.org/abs/2001.00578v3">Errata and Addenda to Mathematical Constants</a>,  arXiv:2001.00578v3 [math.HO], 2022.
%F A269330 Equals li(2) - gamma.
%F A269330 Equals Ei(log(2)) - gamma.
%F A269330 Equals Integral_{x=0..1} (1/log(1+x) - 1/x) dx.
%F A269330 Equals log(log(2)) + Sum_{k>=1} log(2)^k/(k*k!).
%e A269330 0.4679481152159599242380767991122107054804562422112779...
%p A269330 evalf(Li(2)-gamma, 120)
%p A269330 evalf(Ei(ln(2))-gamma, 120)
%p A269330 evalf(int(1/ln(1+x)-1/x, x = 0..1), 120)
%p A269330 evalf(ln(ln(2))+sum(ln(2)^k/(k*factorial(k)), k = 1..infinity), 120)
%t A269330 RealDigits[LogIntegral[2] - EulerGamma, 10, 120][[1]]
%t A269330 RealDigits[ExpIntegralEi[Log[2]] - EulerGamma, 10, 120][[1]]
%t A269330 RealDigits[Integrate[1/Log[1+x] - 1/x, {x, 0, 1}], 10, 120][[1]]
%t A269330 RealDigits[Log[Log[2]] + Sum[Log[2]^k/(k*k!), {k, 1, ∞}], 10, 120][[1]]
%o A269330 (PARI) default(realprecision, 120); -real(eint1(-log(2)))-Euler
%o A269330 (PARI) default(realprecision, 120); intnum(x=0,1,1/log(1+x)-1/x) \\ Note: PARI/GP v. 2.7.3 is able to compute only 19 digits
%o A269330 (PARI) default(realprecision,120); log(log(2))+sumpos(k=1,log(2)^k/(k*factorial(k)))
%Y A269330 Cf. A001620, A002206, A002207, A069284, A094640, A195189, A270857, A270859.
%K A269330 nonn,cons
%O A269330 0,1
%A A269330 _Iaroslav V. Blagouchine_, Feb 23 2016