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A064533 Decimal expansion of Landau-Ramanujan constant. 41
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Named after the German mathematician Edmund Georg Hermann Landau (1877-1938) and the Indian mathematician Srinivasa Ramanujan (1887-1920). - Amiram Eldar, Jun 20 2021
REFERENCES
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.
LINKS
Steven R. Finch, Landau-Ramanujan Constant. [Broken link]
Steven R. Finch, Landau-Ramanujan Constant. [From the Wayback machine]
Steven R. Finch, Landau-Ramanujan Constant. [From the Wayback Machine]
Steven R. Finch, On a Generalized Fermat-Wiles Equation. [Broken link]
Steven R. Finch, On a Generalized Fermat-Wiles Equation. [From the Wayback Machine]
Philippe Flajolet and Ilan Vardi, Zeta function expansions of some classical constants, Feb 18 1996.
Etienne Fouvry, Claude Levesque and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.
Xavier Gourdon and Pascal Sebah, Constants and records of computation.
Daniel Shanks, The second-order term in the asymptotic expansion of B(x), Mathematics of Computation, Vol. 18, No. 85 (1964), pp. 75-86.
Eric Weisstein's World of Mathematics, Ramanujan constant.
FORMULA
From Amiram Eldar, Mar 08 2024: (Start)
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)
EXAMPLE
0.76422365358922066299069873125009232811679054139340951472168667374...
MATHEMATICA
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 *)
CROSSREFS
Cf. A125776 = Continued fraction. - Harry J. Smith, May 13 2009
Sequence in context: A175996 A248940 A134982 * A131184 A021933 A154730
KEYWORD
cons,nonn
AUTHOR
N. J. A. Sloane, Oct 08 2001
EXTENSIONS
More references needed! Hardy and Wright? Gruber and Lekkerkerker?
More terms from Vladeta Jovovic, Oct 08 2001
STATUS
approved

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Last modified April 18 18:58 EDT 2024. Contains 371781 sequences. (Running on oeis4.)