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

Author

Andre Neumann Kauffman (andrekff(AT)hotmail.com), Apr 01 2002

Keywords

• cons
• nonn

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.

Examples

			0.43575707...
		

Links

• Robert Price, Table of n, a(n) for n = 0..999
• 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.

Crossrefs

Cf. A021002, A002117.

Programs

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 *)

Formula

Equals Product_{k=1..oo} Sum_{n=2..oo} mu(k)/k^n.

Equals 1/A021002. - R. J. Mathar, Jan 31 2009

Extensions

Corrected and extended by R. J. Mathar, Jan 31 2009

Example corrected by R. J. Mathar, Jul 23 2009