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A334446
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Decimal expansion of Product_{k>=1} (1 - 1/A002144(k)^4).
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8
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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
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OFFSET
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0,1
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COMMENTS
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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).
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REFERENCES
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B. C. Berndt, Ramanujan's notebook part IV, Springer-Verlag, 1994, p. 64-65.
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LINKS
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FORMULA
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EXAMPLE
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0.998350495723200406499905517565541629191539407019605795046314...
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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