A334450 Decimal expansion of Product_{k>=1} (1 - 1/A002144(k)^5).

9, 9, 9, 6, 7, 6, 5, 2, 7, 0, 7, 9, 6, 2, 6, 6, 6, 2, 0, 1, 8, 2, 4, 6, 1, 8, 0, 8, 7, 3, 0, 8, 3, 7, 0, 1, 5, 0, 0, 7, 5, 1, 5, 7, 4, 3, 7, 9, 5, 5, 4, 4, 3, 0, 5, 6, 8, 4, 3, 2, 8, 4, 0, 4, 2, 4, 9, 7, 5, 9, 8, 1, 9, 2, 1, 2, 1, 9, 1, 3, 2, 9, 9, 7, 0, 4, 0, 0, 3, 0, 2, 9, 1, 9, 3, 0, 4, 4, 5, 3, 7, 5, 2, 8, 3, 9

Offset: 0

Author

Vaclav Kotesovec, Apr 30 2020

Keywords

• nonn
• cons

Comments

In general, for s>0, Product_{k>=1} (1 + 1/A002144(k)^(2*s+1))/(1 - 1/A002144(k)^(2*s+1)) = Pi^(2*s+1) * A000364(s) * zeta(2*s+1) / ((2^(2*s+2) + 2) * (2*s)! * zeta(4*s+2)). - Dimitris Valianatos, May 01 2020

In general, for s>1, Product_{k>=1} (1 + 1/A002144(k)^s)/(1 - 1/A002144(k)^s) = (zeta(s, 1/4) - zeta(s, 3/4)) * zeta(s) / (2^s * (2^s + 1) * zeta(2*s)).

Examples

			0.999676527079626662018246180873083701500751574379554430568432840424975981921219...
		

References

• B. C. Berndt, Ramanujan's notebook part IV, Springer-Verlag, 1994, p. 64-65.

Links

• Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants, Feb 18 1996, p. 7-8.
• R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, p. 26 (case 4 1 5 = 1/A334450).

Crossrefs

Cf. A002144, A088539, A334446, A334450.

Formula

A334449 / A334450 = 4725*zeta(5)/(16*Pi^5).

A334450 * A334452 = 32/(31*zeta(5)).

Extensions

More digits from Vaclav Kotesovec, Jun 27 2020