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 A080729 Decimal expansion of the infinite product of zeta functions for even arguments. 5
 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 (list; constant; graph; refs; listen; history; text; internal format)
 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 Table of n, a(n) for n=1..105. 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 Equals A021002/A080730. - Amiram Eldar, Jan 31 2024 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 Cf. A000688, A021002, A076813, A080730, A369634. Sequence in context: A098829 A190404 A243433 * A262080 A164800 A011008 Adjacent sequences: A080726 A080727 A080728 * A080730 A080731 A080732 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

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Last modified August 8 03:39 EDT 2024. Contains 375018 sequences. (Running on oeis4.)