OFFSET
0,1
COMMENTS
The "modified Zeta function" Zetam(n) = sum(mu(k)/k^n) may be helpful when searching for a closed form for Apery's constant.
LINKS
Melanie Matchett Wood, Probability theory for random groups arising in number theory, arXiv:2301.09687 [math.NT], 2023. See Theorem 3.6 at p. 21.
FORMULA
Equals Product_{k=1..oo} Sum_{n=2..oo} mu(k)/k^n.
Equals 1/A021002. - R. J. Mathar, Jan 31 2009
EXAMPLE
0.43575707...
MAPLE
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.
evalf(product(1/Zeta(n), n=2..infinity), 120); # Vaclav Kotesovec, Oct 22 2014
MATHEMATICA
digits = 104; 1/NProduct[ Zeta[n], {n, 2, Infinity}, WorkingPrecision -> digits+10, NProductFactors -> 1000] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 15 2013 *)
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Andre Neumann Kauffman (andrekff(AT)hotmail.com), Apr 01 2002
EXTENSIONS
Corrected and extended by R. J. Mathar, Jan 31 2009
Example corrected by R. J. Mathar, Jul 23 2009
STATUS
approved