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A386738 Decimal expansion of Integral_{x=0..1} {1/x}^4 dx, where {} denotes fractional part.

%I A386738 #9 Aug 14 2025 02:57:59
%S A386738 1,4,5,5,3,2,8,9,4,8,7,9,1,3,2,8,7,1,9,7,7,4,5,5,9,6,4,9,4,7,2,2,4,4,
%T A386738 0,1,6,6,5,6,6,6,4,6,3,7,9,5,1,4,2,5,5,0,1,6,6,9,0,0,5,9,5,7,3,2,9,9,
%U A386738 9,1,4,2,9,3,8,3,6,0,2,9,7,5,2,7,9,2,6,6,1,2,4,9,9,1,2,5,5,9,2,8,2,3,8,5,9
%N A386738 Decimal expansion of Integral_{x=0..1} {1/x}^4 dx, where {} denotes fractional part.
%H A386738 Ovidiu Furdui, <a href="https://cms.math.ca/wp-content/uploads/crux-pdfs/CRUXv34n6.pdf">Problem 3366</a>, Crux Mathematicorum, Vol. 34, No. 6 (2008), pp. 362 and 365; <a href="https://cms.math.ca/wp-content/uploads/crux-pdfs/CRUXv35n6.pdf">Solution to Problem 3366</a>, by Chip Curtis, ibid., Vol. 35, No. 6 (2009), pp. 403-405.
%H A386738 Huizeng Qin and Youmin Lu, <a href="https://m-hikari.com/ijcms-2011/13-16-2011/luyouminIJCMS13-16-2011.pdf">Integrals of Fractional Parts and Some New Identities on Bernoulli Numbers</a>, Int. J. Contemp. Math. Sciences, Vol. 6, No. 15 (2011), pp. 745-761.
%F A386738 Equals log(2*Pi) - 2*gamma - 1/3 + 3*zeta(3)/Pi^2 + 6*zeta'(2)/Pi^2.
%F A386738 In general, for m >= 2, Integral_{x=0..1} {1/x}^m dx = log(2*Pi) - m*gamma/2 - 1/(m-1) - Sum_{k=1..floor((m-2)/2)} (-1)^k * (m!/(m-2*k-1)!) * zeta(2*k+1) / (2^(2*k+1) * Pi^(2*k)) + 2 * Sum_{k=1..floor((m-1)/2)} (-1)^(k-1) * (m!/(m-2*k)!) * zeta'(2*k) / (2*Pi)^(2*k).
%e A386738 0.14553289487913287197745596494722440166566646379514...
%t A386738 RealDigits[Log[2*Pi] - 2*EulerGamma - 1/3 + (Zeta[3]/2 + Zeta'[2])/Zeta[2], 10, 120][[1]]
%o A386738 (PARI) log(2*Pi) - 2*Euler - 1/3 + (zeta(3)/2 + zeta'(2))/zeta(2)
%Y A386738 Cf. A001620 (gamma), A002117, A061444, A073002, A306016.
%Y A386738 Cf. A153810 (m=1), A345208 (m=2), A345208 (m=3), this constant (m=4).
%K A386738 nonn,cons
%O A386738 0,2
%A A386738 _Amiram Eldar_, Aug 01 2025