%I #18 Feb 16 2025 08:33:12

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

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

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

%N Decimal expansion of Product_{n>=2} (1-n^(-8)).

%H E. Weisstein, <a href="https://mathworld.wolfram.com/InfiniteProduct.html">Infinite Product</a>, Mathworld.

%F Equals (cosh(sqrt(2)*Pi) - cos(sqrt(2)*Pi)) * sinh(Pi) / (16*Pi^3). - _Vaclav Kotesovec_, Apr 27 2020

%F Equals exp(Sum_{j>=1} (1 - zeta(8*j))/j). - _Vaclav Kotesovec_, Apr 27 2020

%e 0.9959233150... = (255/256)*(6560/6561)*(65535/65536)*...

%p t := Pi/sqrt(2) ; sinh(Pi)*((sin(t)*cosh(t))^2+(cos(t)*sinh(t))^2)/8/Pi^3 ; evalf(%) ;

%t RealDigits[ -Sin[(-1)^(1/4)*Pi]*Sin[(-1)^(3/4)*Pi]*Sinh[Pi] / (8*Pi^3) // Re, 10, 105] // First(* _Jean-François Alcover_, Feb 12 2013 *)

%o (PARI) exp(suminf(j=1, (1 - zeta(8*j))/j)) \\ _Vaclav Kotesovec_, Apr 27 2020

%o (PARI) prodnumrat(1-x^-8, 2) \\ _Charles R Greathouse IV_, Feb 04 2025

%Y Cf. A109219, A175615, A175617, A144667, A334411.

%K cons,easy,nonn,changed

%O 0,1

%A _R. J. Mathar_, Jul 26 2010