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%I #17 Jan 31 2023 08:26:33
%S 4,3,5,7,5,7,0,7,6,7,7,2,6,4,5,5,9,3,7,3,7,6,2,2,9,7,0,1,2,0,9,4,1,8,
%T 6,3,4,9,6,8,6,4,1,7,4,9,2,4,3,6,8,0,3,8,1,7,5,4,6,0,9,8,9,0,9,2,3,0,
%U 0,2,3,6,0,1,6,1,0,3,0,5,3,1,8,8,0,4,3,9,7,9,5,9,7,7,2,3,4,0,6,5,3,7,6,9
%N Decimal expansion of the limit of the product of a modified Zeta function.
%C The "modified Zeta function" Zetam(n) = sum(mu(k)/k^n) may be helpful when searching for a closed form for Apery's constant.
%H Melanie Matchett Wood, <a href="https://arxiv.org/abs/2301.09687">Probability theory for random groups arising in number theory</a>, arXiv:2301.09687 [math.NT], 2023. See Theorem 3.6 at p. 21.
%F Equals Product_{k=1..oo} Sum_{n=2..oo} mu(k)/k^n.
%F Equals 1/A021002. - _R. J. Mathar_, Jan 31 2009
%e 0.43575707...
%p with(numtheory); evalf(Product(Sum('mobius(k)/k^n','k'=1..infinity),n=2..infinity),40); Note: For practical reasons you should change "infinity" to some finite value.
%p evalf(product(1/Zeta(n), n=2..infinity), 120); # _Vaclav Kotesovec_, Oct 22 2014
%t digits = 104; 1/NProduct[ Zeta[n], {n, 2, Infinity}, WorkingPrecision -> digits+10, NProductFactors -> 1000] // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Feb 15 2013 *)
%Y Cf. A021002, A002117.
%K cons,nonn
%O 0,1
%A Andre Neumann Kauffman (andrekff(AT)hotmail.com), Apr 01 2002
%E Corrected and extended by _R. J. Mathar_, Jan 31 2009
%E Example corrected by _R. J. Mathar_, Jul 23 2009
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