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%I #58 Mar 19 2024 06:50:39
%S 2,2,9,4,8,5,6,5,9,1,6,7,3,3,1,3,7,9,4,1,8,3,5,1,5,8,3,1,3,4,4,3,1,1,
%T 2,8,8,7,1,3,1,6,3,7,9,9,4,4,1,6,6,8,6,7,3,2,7,5,8,1,4,0,3,0,0,0,1,3,
%U 9,7,0,1,2,0,1,1,3,2,3,1,5,7,5,0,1,7,9,6,8,0,4,5,2,3,2,7,2,4,9,0,8,1,3,8,4
%N Decimal expansion of zeta(2)*zeta(3)*zeta(4)*...
%C A very good approximation is 2e-Pi = ~2.29497100332829723225793155942... - _Marco Matosic_, Nov 16 2005
%C This constant is equal to the asymptotic mean of number of Abelian groups of order n (A000688). - _Amiram Eldar_, Oct 16 2020
%D R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963, p. 198-9.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, p. 274.
%H Steven R. Finch, <a href="http://web.archive.org/web/20010603070928/http://www.mathsoft.com/asolve/constant/abel/abel.html">Abelian Group Enumeration Constants</a>. [From the Wayback machine]
%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. 86.
%H Felix Fontein and Pawel Wocjan, <a href="http://arxiv.org/abs/1111.1348">Quantum Algorithm for Computing the Period Lattice of an Infrastructure</a>, arXiv preprint arXiv:1111.1348 [quant-ph], 2011.
%H Felix Fontein and Pawel Wocjan, <a href="https://doi.org/10.1016/j.jsc.2013.12.002">On the probability of generating a lattice</a>, Journal of Symbolic Computation, Vol. 64 (2014), pp. 3-15, <a href="http://arxiv.org/abs/1211.6246">arXiv preprint</a>, arXiv:1211.6246 [math.CO], 2012-2013. - From _N. J. A. Sloane_, Jan 03 2013
%H Bernd C. Kellner, <a href="http://arxiv.org/abs/math.NT/0604505">On asymptotic constants related to products of Bernoulli numbers and factorials</a>, arXiv:math/0604505 [math.NT], 2006-2009.
%H B. R. Srinivasan, <a href="http://doi.org/10.4064/aa-23-2-195-205">On the Number of Abelian Groups of a Given Order</a>, Acta Arithmetica, Vol. 23, No. 2 (1973), pp. 195-205, <a href="https://eudml.org/doc/205185">alternative link</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AbelianGroup.html">Abelian Group</a>.
%H <a href="/wiki/Index_to_constants#Start_of_section_Z">Index entries for constants related to zeta</a>
%F Product of A080729 and A080730. - _R. J. Mathar_, Feb 16 2011
%e 2.2948565916733137941835158313443112887131637994416686732758140300...
%p evalf(product(Zeta(n), n=2..infinity), 200);
%t p = Product[ N[ Zeta[n], 256], {n, 2, 1000}]; RealDigits[p, 10, 111][[1]] (* _Robert G. Wilson v_, Nov 22 2005 *)
%o (PARI) prodinf(n=2,zeta(n)) \\ _Charles R Greathouse IV_, May 27 2015
%Y Cf. A068982 (reciprocal), A082868 (continued fraction).
%Y Cf. A002117, A000688, A063966, A080729, A080730.
%K cons,nonn
%O 1,1
%A Andre Neumann Kauffman (ank(AT)nlink.com.br)
%E More terms from _Simon Plouffe_, Jan 07 2002
%E Further terms from _Robert G. Wilson v_, Nov 22 2005
%E Mathematica program fixed by _Vaclav Kotesovec_, Sep 20 2014
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