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A064533
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Decimal expansion of Landau-Ramanujan constant.
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41
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7, 6, 4, 2, 2, 3, 6, 5, 3, 5, 8, 9, 2, 2, 0, 6, 6, 2, 9, 9, 0, 6, 9, 8, 7, 3, 1, 2, 5, 0, 0, 9, 2, 3, 2, 8, 1, 1, 6, 7, 9, 0, 5, 4, 1, 3, 9, 3, 4, 0, 9, 5, 1, 4, 7, 2, 1, 6, 8, 6, 6, 7, 3, 7, 4, 9, 6, 1, 4, 6, 4, 1, 6, 5, 8, 7, 3, 2, 8, 5, 8, 8, 3, 8, 4, 0, 1, 5, 0, 5, 0, 1, 3, 1, 3, 1, 2, 3, 3, 7, 2, 1, 9, 3, 7, 2, 6, 9, 1, 2, 0, 7, 9, 2, 5, 9, 2, 6, 3, 4, 1, 8, 7, 4, 2, 0, 6, 4, 6, 7, 8, 0, 8, 4, 3, 2, 3, 0, 6, 3, 3, 1, 5, 4, 3, 4, 6, 2, 9, 3, 8, 0, 5, 3, 1, 6, 0, 5, 1, 7, 1, 1, 6, 9, 6, 3, 6, 1, 7, 7, 5, 0, 8, 8, 1, 9, 9, 6, 1, 2, 4, 3, 8, 2, 4, 9, 9, 4, 2, 7, 7, 6, 8, 3, 4, 6, 9, 0, 5, 1, 6, 2, 3, 5, 1, 3, 9, 2, 1, 8, 7, 1, 9, 6, 2, 0, 5, 6, 9, 0, 5, 3, 2, 9, 5, 6, 4, 4, 6, 7, 0, 4
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OFFSET
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0,1
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COMMENTS
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Named after the German mathematician Edmund Georg Hermann Landau (1877-1938) and the Indian mathematician Srinivasa Ramanujan (1887-1920). - Amiram Eldar, Jun 20 2021
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REFERENCES
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Bruce C. Berndt, Ramanujan's notebook part IV, Springer-Verlag, 1994, pp. 52,60-66; MR 95e: 11028.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 98-104.
G. H. Hardy, "Ramanujan, Twelve lectures on subjects suggested by his life and work", Chelsea, 1940, pp. 60-63; MR 21 # 4881.
Edmund Landau, Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate. Arch. Math. Phys., 13, 1908, pp. 305-312.
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LINKS
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FORMULA
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Equals (Pi/4) * Product_{primes p == 1 (mod 4)} (1 - 1/p^2)^(1/2).
Equals (1/sqrt(2)) * Product_{primes p == 3 (mod 4)} (1 - 1/p^2)^(-1/2).
Equals (1/sqrt(2)) * Product_{k>=1} ((1 - 1/2^(2^k)) * zeta(2^k)/beta(2^k)), where beta is the Dirichlet beta function (Shanks, 1964). (End)
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EXAMPLE
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0.76422365358922066299069873125009232811679054139340951472168667374...
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MATHEMATICA
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First@ RealDigits@ N[1/Sqrt@2 Product[((1 - 2^(-2^k)) 4^(2^k) Zeta[2^k]/(Zeta[2^k, 1/4] - Zeta[2^k, 3/4]))^(2^(-k - 1)), {k, 8}], 2^8] (* Robert G. Wilson v, Jul 01 2007 *)
(* Victor Adamchik calculated 5100 digits of the Landau-Ramanujan constant using Mathematica (from Mathematica 4 demos): *) LandauRamanujan[n_] := With[{K = Ceiling[Log[2, n*Log[3, 10]]]}, N[Product[(((1 - 2^(-2^k))*4^2^k*Zeta[2^k])/(Zeta[2^k, 1/4] - Zeta[2^k, 3/4]))^2^(-k - 1), {k, 1, K}]/Sqrt[2], n]];
(* The code reported here is the code at https://library.wolfram.com/infocenter/Demos/120/. Looking carefully at the outputs reported there one sees that: the last 8 digits of the 500-digit output ("74259724") are not the same as those listed in the 1000-digit output ("94247095"); the same happens with the last 18 digits of the 1000-digit output ("584868265713856413") and the corresponding ones in the 5100-digit output ("852514327407923660"). - Alessandro Languasco, May 07 2021 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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More references needed! Hardy and Wright? Gruber and Lekkerkerker?
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STATUS
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approved
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