A080729 Decimal expansion of the infinite product of zeta functions for even arguments.
1, 8, 2, 1, 0, 1, 7, 4, 5, 1, 4, 9, 9, 2, 9, 2, 3, 9, 0, 4, 0, 6, 7, 2, 5, 1, 3, 2, 2, 2, 6, 0, 0, 6, 8, 4, 8, 5, 7, 8, 2, 6, 8, 0, 2, 8, 6, 4, 8, 2, 7, 1, 7, 5, 5, 0, 0, 2, 0, 9, 3, 8, 0, 0, 2, 8, 6, 0, 6, 5, 8, 8, 6, 7, 7, 0, 5, 4, 8, 8, 9, 3, 6, 3, 9, 6, 0, 2, 4, 9, 7, 5, 2, 1, 4, 5, 2, 9, 7, 6, 6, 1, 0, 9, 9
Offset: 1
Examples
1.82101745149929239040672513222600684857...
Links
- Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 658.
- Bernd C. Kellner, On asymptotic constants related to products of Bernoulli numbers and factorials, Integers, Vol. 9 (2009), Article #A08, pp. 83-106; alternative link; arXiv:0604505 [math.NT], 2006.
- Eric Weisstein's World of Mathematics, Abelian group.
Programs
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Mathematica
RealDigits[Product[Zeta[2n],{n,500}],10,110][[1]] (* Harvey P. Dale, Jan 31 2012 *) -
PARI
prodinf(k=1, zeta(2*k)) \\ Vaclav Kotesovec, Jan 29 2024
Formula
Decimal expansion of zeta(2)*zeta(4)*...*zeta(2k)*...
If u(k) denotes the number of Abelian groups with group order k (A000688), then Product_{k>=1} zeta(2*k) = Sum_{k>=1} u(k)/k^2. - Benoit Cloitre, Jun 25 2003
This constant C is connected with the product of values of the Dedekind eta function on the upper imaginary axis. The product runs over the primes, where i is the imaginary unit: 1/C = Product_{prime p} (p^(1/12) * eta(i * log(p) / Pi)). - Bernd C. Kellner, May 18 2024
Extensions
More terms from Benoit Cloitre, Mar 08 2003
Comments