A080729 - OEIS

%I #44 Nov 03 2025 13:12:56

%S 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,

%T 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,

%U 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

%N Decimal expansion of the infinite product of zeta functions for even arguments.

%C By elementary estimates, the constant lies in the open interval (Pi/6, exp(3/4)). - _Bernd C. Kellner_, May 18 2024

%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 658.

%H Bernd C. Kellner, <a href="https://doi.org/10.1515/INTEG.2009.009">On asymptotic constants related to products of Bernoulli numbers and factorials</a>, Integers, Vol. 9 (2009), Article #A08, pp. 83-106; <a href="http://www.integers-ejcnt.org/j8/j8.Abstract.html">alternative link</a>; arXiv:<a href="https://arxiv.org/abs/math/0604505">0604505</a> [math.NT], 2006.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AbelianGroup.html">Abelian group</a>.

%F Decimal expansion of zeta(2)*zeta(4)*...*zeta(2k)*...

%F 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

%F Equals A021002/A080730. - _Amiram Eldar_, Jan 31 2024

%F 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

%e 1.82101745149929239040672513222600684857...

%t RealDigits[Product[Zeta[2n],{n,500}],10,110][[1]] (* _Harvey P. Dale_, Jan 31 2012 *)

%o (PARI) prodinf(k=1, zeta(2*k)) \\ _Vaclav Kotesovec_, Jan 29 2024

%Y Cf. A000688, A021002, A076813, A080730, A369634.

%K cons,nonn

%O 1,2

%A Deepak R. N (deepak_rn(AT)safe-mail.net), Mar 08 2003

%E More terms from _Benoit Cloitre_, Mar 08 2003