A334446 Decimal expansion of Product_{k>=1} (1 - 1/A002144(k)^4).

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

Offset: 0

Author

Vaclav Kotesovec, Apr 30 2020

Keywords

• nonn
• cons

Comments

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)).

For s>1, zeta(s, 1/4) - zeta(s, 3/4) = (-1)^s*(PolyGamma(s-1, 1/4) - PolyGamma(s-1, 3/4))/(s-1)! = 2*(-1)^s * PolyGamma(s-1, 1/4) / Gamma(s) - 2^s*(2^s - 1)*zeta(s) = 4^s * DirichletBeta(s).

Examples

			0.998350495723200406499905517565541629191539407019605795046314...
		

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.
• X. Gourdon and P. Sebah, Some Constants from Number theory
• 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 4 = 1/A334446).

Crossrefs

Cf. A002144, A088539, A334425, A334450.

Formula

A334445 / A334446 = 35*(PolyGamma(3, 1/4)/(8*Pi^4) - 1)/34.

A334446 * A334448 = 96/Pi^4.

Extensions

More digits from Vaclav Kotesovec, Jun 27 2020