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Decimal expansion of Landau-Ramanujan constant.
41

%I #130 Sep 27 2024 07:48:37

%S 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,

%T 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,

%U 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

%N Decimal expansion of Landau-Ramanujan constant.

%C Named after the German mathematician Edmund Georg Hermann Landau (1877-1938) and the Indian mathematician Srinivasa Ramanujan (1887-1920). - _Amiram Eldar_, Jun 20 2021

%D Bruce C. Berndt, Ramanujan's notebook part IV, Springer-Verlag, 1994, pp. 52, 60-66; MR 95e: 11028.

%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 98-104.

%D G. H. Hardy, "Ramanujan, Twelve lectures on subjects suggested by his life and work", Chelsea, 1940, pp. 60-63; MR 21 # 4881.

%D 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.

%H David E. G. Hare, <a href="/A064533/b064533.txt">Table of n, a(n) for n = 0..125078</a>

%H Bruce C. Berndt and Pieter Moree, <a href="https://arxiv.org/abs/2409.03428">Sums of two squares and the tau-function: Ramanujan's trail</a>, arXiv:2409.03428 [math.NT], 2024. See p. 30.

%H Alexandru Ciolan, Alessandro Languasco and Pieter Moree, <a href="https://doi.org/10.1016/j.jmaa.2022.126854">Landau and Ramanujan approximations for divisor sums and coefficients of cusp forms</a>, section 10, Journal of Mathematical Analysis and Applications, 2022; see also preprint on <a href="https://arxiv.org/abs/2109.03288">arXiv</a>, arXiv:2109.03288 [math.NT], 2021.

%H Alessandro Languasco, <a href="https://www.dei.unipd.it/~languasco/CLM.html">Programs and numerical results</a>, providing 130000 digits. [Note: information ancillary to above link.]

%H Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/lr/lr.html">Landau-Ramanujan Constant</a>. [Broken link]

%H Steven R. Finch, <a href="http://web.archive.org/web/20010605004309/http://www.mathsoft.com/asolve/constant/lr/lr.html">Landau-Ramanujan Constant</a>. [From the Wayback machine]

%H Steven R. Finch, <a href="http://web.archive.org/web/20010605004309/http://www.mathsoft.com/asolve/constant/lr/lr.html">Landau-Ramanujan Constant</a>. [From the Wayback Machine]

%H Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/csolve/fermat.pdf">On a Generalized Fermat-Wiles Equation</a>. [Broken link]

%H Steven R. Finch, <a href="http://web.archive.org/web/20010602030546/http://www.mathsoft.com/asolve/fermat/fermat.html">On a Generalized Fermat-Wiles Equation</a>. [From the Wayback Machine]

%H Philippe Flajolet and Ilan Vardi, <a href="https://web.archive.org/web/20221130162351/http://algo.inria.fr/flajolet/Publications/landau.ps">Zeta function expansions of some classical constants</a>, Feb 18 1996.

%H Étienne Fouvry, Claude Levesque and Michel Waldschmidt, <a href="https://arxiv.org/abs/1712.09019">Representation of integers by cyclotomic binary forms</a>, arXiv:1712.09019 [math.NT], 2017.

%H Xavier Gourdon and Pascal Sebah, <a href="http://numbers.computation.free.fr/Constants/Miscellaneous/Records.html">Constants and records of computation</a>.

%H David E. G. Hare, <a href="/A064533/a064533_1.txt">125,079 digits of the Landau-Ramanujan constant</a>.

%H David E. G. Hare, <a href="http://www.plouffe.fr/simon/constants/LandauRamanujan.txt">Landau-Ramanujan constant up to 10000 digits</a>.

%H Institute of Physics, <a href="http://scenta.co.uk/tcaep/science/constant/details/landauramanujanconstant.xml">Constants - Landau-Ramanujan Constant</a>.

%H Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap51.html">Landau Ramanujan constant</a>.

%H Daniel Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1964-0159174-9">The second-order term in the asymptotic expansion of B(x)</a>, Mathematics of Computation, Vol. 18, No. 85 (1964), pp. 75-86.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Landau-RamanujanConstant.html">Ramanujan constant</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Landau-Ramanujan_constant">Landau-Ramanujan constant</a>.

%H Robert G. Wilson v, <a href="/A064533/a064533.txt">The first 15584 digits of the Landau-Ramanujan constant</a>.

%F From _Amiram Eldar_, Mar 08 2024: (Start)

%F Equals (Pi/4) * Product_{primes p == 1 (mod 4)} (1 - 1/p^2)^(1/2).

%F Equals (1/sqrt(2)) * Product_{primes p == 3 (mod 4)} (1 - 1/p^2)^(-1/2).

%F 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)

%e 0.76422365358922066299069873125009232811679054139340951472168667374...

%t 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 *)

%t (* 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]];

%t (* 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 *)

%Y Cf. A125776 = Continued fraction. - _Harry J. Smith_, May 13 2009

%Y Cf. A000692, A001481, A009003, A075880, A090735, A090736, A227158.

%K cons,nonn

%O 0,1

%A _N. J. A. Sloane_, Oct 08 2001

%E More references needed! Hardy and Wright? Gruber and Lekkerkerker?

%E More terms from _Vladeta Jovovic_, Oct 08 2001