A338106 - OEIS

%I #23 Oct 14 2020 13:54:09

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

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

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

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

%C For p>1, q>1 in R, Sum_{m >1, n>1} 1/(m^p*n^q-1) = Sum_{k>0} (zeta(k*p) - 1) * (zeta(k*q) - 1) [Proof in References]. This sequence corresponds to p = q = 2.

%C Double inequality: Sum_{m>1, n>1} 1/(m^2*n^2+1) = A338107 = 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) = this constant = 0.423...

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

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

%F Equals -3/4 + Sum_{k>=2} (1/2 - Pi*cot(Pi/k)/(2*k)). - _Vaclav Kotesovec_, Oct 14 2020

%e 0.4230355257613131597420971016391038628995464... (with help of _Amiram Eldar_).

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

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

%Y Cf. A098198, A333972, A338107.

%K nonn,cons

%O 0,1

%A _Bernard Schott_, Oct 10 2020