A386564 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} ({x/y}*{y/z}*{z/x})^3 dx dy dz, where {w} is the fractional part of w.
0, 0, 7, 7, 8, 8, 9, 5, 5, 0, 8, 4, 0, 9, 6, 6, 5, 2, 0, 5, 4, 2, 8, 3, 6, 0, 9, 6, 5, 9, 9, 2, 7, 1, 4, 1, 1, 9, 0, 1, 7, 1, 9, 6, 4, 8, 9, 2, 6, 6, 3, 2, 0, 8, 4, 1, 9, 1, 0, 2, 4, 4, 6, 9, 5, 8, 0, 0, 5, 3, 5, 9, 8, 6, 8, 2, 9, 2, 3, 4, 1, 2, 0, 4, 2, 2, 4, 9, 6, 9, 2, 9, 8, 5, 4, 8, 5, 7, 6, 5, 9, 9, 1, 7, 6
Offset: 0
Examples
0.00778895508409665205428360965992714119017196489266...
Links
- Cornel Ioan Vălean, Problem 11902, Problems and Solutions, The American Mathematical Monthly, Vol. 123, No. 4 (2016), p. 399; A Row of Zetas, Solution to Problem 11902 by Rituraj Nandan, ibid., Vol. 125, No. 2 (2018), pp. 182-184.
- Cornel Ioan Vălean, (Almost) Impossible Integrals, Sums, and Series, Springer (2019), section 1.48 The Calculation of a Beautiful Triple Fractional Part Integral with a Cubic Power, p. 31.
Crossrefs
Programs
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Mathematica
RealDigits[1 - 3*(Zeta[2]+Zeta[3]+Zeta[4])/8 + 21*Zeta[6]/320 + 7*Zeta[8]/160 + Zeta[3]^2/40 + Zeta[2]*Zeta[3]/40 + Zeta[2]*Zeta[5]/20 + Zeta[3]*Zeta[4]/16 + Zeta[3]*Zeta[5]/20 + Zeta[4]*Zeta[5]/20, 10, 120, -1][[1]]
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PARI
1 - 3*(zeta(2)+zeta(3)+zeta(4))/8 + 21*zeta(6)/320 + 7*zeta(8)/160 + zeta(3)^2/40 + zeta(2)*zeta(3)/40 + zeta(2)*zeta(5)/20 + zeta(3)*zeta(4)/16 + zeta(3)*zeta(5)/20 + zeta(4)*zeta(5)/20
Formula
Equal 1 - 3*(zeta(2)+zeta(3)+zeta(4))/8 + 21*zeta(6)/320 + 7*zeta(8)/160 + zeta(3)^2/40 + zeta(2)*zeta(3)/40 + zeta(2)*zeta(5)/20 + zeta(3)*zeta(4)/16 + zeta(3)*zeta(5)/20 + zeta(4)*zeta(5)/20.
In general, Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} ({x/y}*{y/z}*{z/x})^m dx dy dz = 1 - 3*Sum_{j=1..m} zeta(j+1)/(2*(m+1)) + (Sum_{j=1..m} zeta(j+1))*(Sum_{j=1..m} (j+1)*zeta(j+2))/((m+1)^2*(m+2)).