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

1,1

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

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

Table of n, a(n) for n=1..105.

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, arXiv:math/0604505 [math.NT], 2006-2009.

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

Formula

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

Example

2.2948565916733137941835158313443112887131637994416686732758140300...

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 *)

Prog

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

Crossrefs

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

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

Sequence in context: A271524 A157216 A020776 * A103710 A178236 A093589

Adjacent sequences: A020999 A021000 A021001 * A021003 A021004 A021005

Keyword

cons,nonn

Author

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

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

Status

approved