A068982 Decimal expansion of the limit of the product of a modified Zeta function.

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, 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, 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

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

Table of n, a(n) for n=0..103.

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

Cf. A021002, A002117.

Sequence in context: A000211 A059902 A304225 * A317530 A171021 A035427

Adjacent sequences: A068979 A068980 A068981 * A068983 A068984 A068985

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