A021002 Decimal expansion of zeta(2)*zeta(3)*zeta(4)*...

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, 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, 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

Offset: 1

Author

Andre Neumann Kauffman (ank(AT)nlink.com.br)

Keywords

• cons
• nonn

Comments

A very good approximation is 2e-Pi = ~2.29497100332829723225793155942... - Marco Matosic, Nov 16 2005

This constant is equal to the asymptotic mean of number of Abelian groups of order n (A000688). - Amiram Eldar, Oct 16 2020

Examples

			2.2948565916733137941835158313443112887131637994416686732758140300...
		

References

• R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963, p. 198-9.
• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, p. 274.

Links

• Robert Price, Table of n, a(n) for n = 1..1000
• Steven R. Finch, Abelian Group Enumeration Constants. [From the Wayback machine]
• Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 86.
• Felix Fontein and Pawel Wocjan, Quantum Algorithm for Computing the Period Lattice of an Infrastructure, arXiv preprint arXiv:1111.1348 [quant-ph], 2011.
• Felix Fontein and Pawel Wocjan, On the probability of generating a lattice, Journal of Symbolic Computation, Vol. 64 (2014), pp. 3-15, arXiv preprint, arXiv:1211.6246 [math.CO], 2012-2013. - From _N. J. A. Sloane_, Jan 03 2013
• 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.
• B. R. Srinivasan, On the Number of Abelian Groups of a Given Order, Acta Arithmetica, Vol. 23, No. 2 (1973), pp. 195-205, alternative link.
• Eric Weisstein's World of Mathematics, Abelian Group.
• Index entries for constants related to zeta

Crossrefs

Cf. A068982 (reciprocal), A082868 (continued fraction).

Cf. A002117, A000688, A063966, A080729, A080730.

Programs

Maple

evalf(product(Zeta(n), n=2..infinity), 200);

Mathematica

p = Product[ N[ Zeta[n], 256], {n, 2, 1000}]; RealDigits[p, 10, 111][[1]] (* Robert G. Wilson v, Nov 22 2005 *)

PARI

prodinf(n=2,zeta(n)) \\ Charles R Greathouse IV, May 27 2015

Formula

Product of A080729 and A080730. - R. J. Mathar, Feb 16 2011

Extensions

More terms from Simon Plouffe, Jan 07 2002

Further terms from Robert G. Wilson v, Nov 22 2005

Mathematica program fixed by Vaclav Kotesovec, Sep 20 2014