A338107 Decimal expansion of Sum_{m>1, n>1} 1/(m^2*n^2+1).

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

Offset: 0

Author

Bernard Schott, Oct 10 2020

Keywords

• nonn
• cons

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...

Examples

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

References

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

Crossrefs

Cf. A098198, A333972, A338106.

Programs

Mathematica

RealDigits[Sum[(-1)^(k - 1)*(Zeta[2*k] - 1)^2, {k, 1, 100}], 10, 90][[1]] (* Amiram Eldar, Oct 10 2020 *)

PARI

sumalt(k=1, (-1)^(k-1) * (zeta(2*k) - 1)^2) \\ Michel Marcus, Oct 10 2020

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