A383289 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} ({x/y}*{y/z}*{z/x})^2 dx dy dz, where {w} is the fractional part of w.
0, 2, 3, 4, 0, 9, 6, 1, 8, 2, 3, 1, 5, 8, 0, 8, 7, 2, 6, 8, 0, 2, 0, 0, 9, 3, 8, 5, 5, 0, 0, 6, 9, 8, 0, 6, 7, 5, 8, 4, 0, 4, 4, 2, 5, 8, 2, 7, 1, 4, 8, 3, 8, 5, 1, 5, 9, 3, 8, 7, 1, 0, 0, 9, 6, 3, 8, 8, 8, 3, 3, 5, 9, 5, 8, 3, 1, 8, 0, 5, 9, 4, 1, 0, 4, 1, 5, 6, 4, 9, 6, 6, 8, 0, 3, 9, 4, 0, 0, 5, 3, 8, 9, 4, 0, 0, 1
Offset: 0
Examples
0.02340961823158087268020093855006980675840442582714...
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), p. viii.
Crossrefs
Programs
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Mathematica
RealDigits[1 - Zeta[2]/2 - Zeta[3]/2 + 7*Zeta[6]/48 + Zeta[2]*Zeta[3]/18 + Zeta[3]^2/18 + Zeta[3]*Zeta[4]/12, 10, 120, -1][[1]] RealDigits[With[{m = 2}, 1 - 3*Sum[Zeta[j + 1], {j, 1, m}]/(2*(m + 1)) + Sum[Zeta[j + 1], {j, 1, m}] * Sum[(j + 1)*Zeta[j + 2], {j, 1, m}]/((m + 1)^2*(m + 2))], 10, 106][[1]] (* Vaclav Kotesovec, Jul 26 2025, following the general formula found by the solvers *) -
PARI
1 - zeta(2)/2 - zeta(3)/2 + 7*zeta(6)/48 + zeta(2)*zeta(3)/18 + zeta(3)^2/18 + zeta(3)*zeta(4)/12
Formula
Equals 1 - zeta(2)/2 - zeta(3)/2 + 7*zeta(6)/48 + zeta(2)*zeta(3)/18 + zeta(3)^2/18 + zeta(3)*zeta(4)/12.
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)).