OFFSET
1,2
COMMENTS
By elementary estimates, the constant lies in the open interval (Pi/6, exp(3/4)). - Bernd C. Kellner, May 18 2024
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.
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
EXAMPLE
1.82101745149929239040672513222600684857...
MATHEMATICA
RealDigits[Product[Zeta[2n], {n, 500}], 10, 110][[1]] (* Harvey P. Dale, Jan 31 2012 *)
PROG
(PARI) prodinf(k=1, zeta(2*k)) \\ Vaclav Kotesovec, Jan 29 2024
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Deepak R. N (deepak_rn(AT)safe-mail.net), Mar 08 2003
EXTENSIONS
More terms from Benoit Cloitre, Mar 08 2003
STATUS
approved