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Decimal expansion of Pi^6/540 = zeta(2) * zeta(4).
3

%I #45 Jan 25 2024 02:46:33

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

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

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

%N Decimal expansion of Pi^6/540 = zeta(2) * zeta(4).

%C Compare 1st formula with Sum_{m>0, q>0} 1/(m^2*q^2) = Pi^4/36 = (zeta(2))^2 = A098198.

%D Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.22, p. 275.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%F Equals Sum_{m>0, q>0, m | q} 1/(m^2*q^2).

%F Equals A013661 * A068447.

%F Equals Sum_{k>=1} sigma_2(k)/k^4. - _Amiram Eldar_, Sep 30 2020

%F Equals Sum_{k>=1} A046951(k)/k^2. - _Amiram Eldar_, Jan 25 2024

%e 1.78035035847278599450040637713411092382818060755749373322421516...

%p evalf(Pi^6/540,120);

%t RealDigits[Pi^6/540, 10, 100][[1]] (* _Amiram Eldar_, Sep 29 2020 *)

%o (PARI) Pi^6/540 \\ _Michel Marcus_, Sep 30 2020

%Y Cf. A001157, A013661, A046951, A068447, A098198.

%K nonn,cons

%O 1,2

%A _Bernard Schott_, Sep 29 2020