OFFSET
0,2
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.
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)).
EXAMPLE
0.02340961823158087268020093855006980675840442582714...
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 *)
PROG
(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
CROSSREFS
KEYWORD
AUTHOR
Amiram Eldar, Jul 26 2025
STATUS
approved
