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A175647
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Decimal expansion of the Product_{primes p == 1 (mod 4)} 1/(1-1/p^2).
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26
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1, 0, 5, 6, 1, 8, 2, 1, 2, 1, 7, 2, 6, 8, 1, 6, 1, 4, 1, 7, 3, 7, 9, 3, 0, 7, 6, 5, 3, 1, 6, 2, 1, 9, 8, 9, 0, 5, 8, 7, 5, 8, 0, 4, 2, 5, 4, 6, 0, 7, 0, 8, 0, 1, 2, 0, 0, 4, 3, 0, 6, 1, 9, 8, 3, 0, 2, 7, 9, 2, 8, 1, 6, 0, 6, 2, 2, 2, 6, 9, 3, 0, 4, 8, 9, 5, 1, 2, 9, 5, 8, 3, 7, 2, 9, 1, 5, 9, 7, 1, 8, 4, 7, 5, 0
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OFFSET
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1,3
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COMMENTS
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The Euler product of the Riemann zeta function at 2 restricted to primes in A002144, which is the inverse of the infinite product (1-1/5^2)*(1-1/13^2)*(1-1/17^2)*(1-1/29^2)*...
There is a complementary Product_{primes p == 3 (mod 4)} 1/(1-1/p^2) = 1.16807558541051428866969673706404040136467... such that (this constant here)*1.16807.../(1-1/2^2) = zeta(2) = A013661.
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LINKS
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FORMULA
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The complementary product equals Sum_{k>=1} 1/A004614(k)^2. (End)
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EXAMPLE
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1.0561821217268161417379307653162198905...
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MATHEMATICA
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digits = 105;
LandauRamanujanK = 1/Sqrt[2]*NProduct[((1 - 2^(-2^n))*Zeta[2^n]/ DirichletBeta[2^n])^(1/2^(n+1)), {n, 1, 24}, WorkingPrecision -> digits+5];
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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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