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Decimal expansion of Sum_{m>1, n>1} 1/(m^2*n^2+1).
1

%I #22 Oct 14 2020 13:54:29

%S 4,0,9,4,4,7,9,2,4,8,9,0,7,6,0,4,0,5,7,5,3,4,1,9,0,1,2,6,9,0,2,5,3,8,

%T 5,0,3,9,5,0,6,8,3,6,6,3,8,8,3,3,8,6,3,3,3,7,0,9,7,0,1,8,2,8,0,1,7,2,

%U 8,5,3,9,7,7,8,8,1,2,5,4,8,5,1,1,5,0,7,0,6,4

%N Decimal expansion of Sum_{m>1, n>1} 1/(m^2*n^2+1).

%C 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...

%D Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.25, p. 277.

%F Equals Sum_{k>0} (-1)^(k-1) * (zeta(2*k) - 1)^2.

%F Equals 3/2 - Pi*coth(Pi) + Sum_{k>=1} (Pi*coth(Pi/k)/(2*k) - 1/2). - _Vaclav Kotesovec_, Oct 14 2020

%e 0.40944792489076040575341901269025385039506836638... (with help of _Amiram Eldar_).

%t RealDigits[Sum[(-1)^(k - 1)*(Zeta[2*k] - 1)^2, {k, 1, 100}], 10, 90][[1]] (* _Amiram Eldar_, Oct 10 2020 *)

%o (PARI) sumalt(k=1, (-1)^(k-1) * (zeta(2*k) - 1)^2) \\ _Michel Marcus_, Oct 10 2020

%Y Cf. A098198, A333972, A338106.

%K nonn,cons

%O 0,1

%A _Bernard Schott_, Oct 10 2020