OFFSET
0,1
COMMENTS
Double inequality: Sum_{m>1, n>1} 1/(m^2*n^2+1) = this constant = 0.409... < Sum_{m>1, n>1} 1/(m^2*n^2) = (zeta(2)-1)^2 = 0.415... < Sum_{m>1, n>1} 1/(m^2*n^2-1) = A338106 = 0.423...
REFERENCES
Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.25, p. 277.
FORMULA
Equals Sum_{k>0} (-1)^(k-1) * (zeta(2*k) - 1)^2.
Equals 3/2 - Pi*coth(Pi) + Sum_{k>=1} (Pi*coth(Pi/k)/(2*k) - 1/2). - Vaclav Kotesovec, Oct 14 2020
EXAMPLE
0.40944792489076040575341901269025385039506836638... (with help of Amiram Eldar).
MATHEMATICA
RealDigits[Sum[(-1)^(k - 1)*(Zeta[2*k] - 1)^2, {k, 1, 100}], 10, 90][[1]] (* Amiram Eldar, Oct 10 2020 *)
PROG
(PARI) sumalt(k=1, (-1)^(k-1) * (zeta(2*k) - 1)^2) \\ Michel Marcus, Oct 10 2020
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Bernard Schott, Oct 10 2020
STATUS
approved