A375901 Decimal expansion of the triple integral of {x/y}{y/z}{z/x} over the unit cube.
0, 9, 5, 8, 5, 0, 1, 7, 4, 9, 1, 3, 3, 7, 9, 5, 2, 5, 6, 7, 8, 5, 3, 6, 1, 9, 8, 5, 9, 6, 3, 3, 5, 3, 7, 0, 0, 9, 9, 4, 7, 9, 4, 8, 5, 2, 0, 4, 9, 2, 3, 5, 3, 9, 8, 1, 4, 3, 0, 1, 7, 0, 7, 4, 8, 1, 6, 1, 3, 5, 6, 9, 5, 5, 5, 3, 6, 6, 7, 1, 2, 5, 7, 5, 1, 7, 5
Offset: 0
Examples
0.095850174913379525678536198596335370099479485204923539814301707481613569555....
Links
- Cornel Ioan Vălean, (Almost) Impossible Integrals, Sums, and Series, Springer (2019), p. viii.
Programs
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Mathematica
RealDigits[1 + (Zeta[3]/6 - 3/4) * Zeta[2], 10, 120, -1][[1]] (* Amiram Eldar, Sep 03 2024 *)
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PARI
1-.75*zeta(2)+zeta(2)*zeta(3)/6
Formula
Integral_{0..1} Integral_{0..1} Integral_{0..1} {x/y}*{y/z}*{z/x} dx dy dz, where {w} is the fractional part of w.
Vălean shows that this is equal to 1 - 3/4 * zeta(2) + 1/6 * zeta(2) * zeta(3).