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%I #133 Apr 06 2024 14:56:57
%S 6,3,8,9,0,9,4,0,5,4,4,5,3,4,3,8,8,2,2,5,4,9,4,2,6,7,4,9,2,8,2,4,5,0,
%T 9,3,7,5,4,9,7,5,5,0,8,0,2,9,1,2,3,3,4,5,4,2,1,6,9,2,3,6,5,7,0,8,0,7,
%U 6,3,1,0,0,2,7,6,4,9,6,5,8,2,4,6,8,9,7,1,7,9,1,1,2,5,2,8,6,6,4,3,8,8,1,4,1,6
%N Decimal expansion of an analog of the Landau-Ramanujan constant for Loeschian numbers.
%C This is the decimal expansion of the number alpha such that the number of positive integers <= N which are represented by the quadratic form x^2 + xy + y^2 is asymptotic to alpha*N/sqrt(log(N)).
%D S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, p. 99 (K3).
%H Peter Luschny, <a href="/A301429/b301429.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..105 from Vaclav Kotesovec).
%H Salma Ettahri, Olivier Ramaré, and Léon Surel, <a href="https://arxiv.org/abs/1908.06808">Fast multi-precision computation of some Euler products</a>, arXiv:1908.06808 [math.NT], 2019.
%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 204.
%H Étienne Fouvry, Claude Levesque, and Michel Waldschmidt, <a href="http://arxiv.org/abs/1712.09019">Representation of integers by cyclotomic binary forms</a>, arXiv:1712.09019 [math.NT], 2017 and <a href="https://doi.org/10.4064/aa171012-24-12">Acta Arithmetica</a>, online 15 March 2018.
%H Olivier Ramaré, S. Ettahri, and L. Surel, <a href="https://hal.science/hal-03381427">Fast multi-precision computation of some Euler products</a>, Mathematics of Computation (2021) hal-03381427.
%H StackExchange, <a href="https://mathematica.stackexchange.com/questions/169665/iterative-calculation-of-a-number-theoretical-constant">Iterative calculation of a number-theoretical constant</a>, Mar 24 2018.
%F Equals 2^(-1/2)*3^(-1/4)*Product_{p == 2 (mod 3), p prime} (1 - p^(-2))^(-1/2).
%F One can base the definition on p(n) = A003627(n). Setting r(n) = (Product_{k=1..n} p(k)^2) / (Product_{k=1..n} (p(k)^2 - 1)) the rational sequence r(n) starts 4/3, 25/18, 605/432, 174845/124416, ... -> L. Then A301429 = sqrt(L)/12^(1/4). - _Peter Luschny_, Mar 29 2018 [This L is now A333240. - _Peter Luschny_, Jan 14 2021]
%F Equals Pi*sqrt(2) / (3^(7/4) * sqrt(A175646)). - _Vaclav Kotesovec_, May 12 2020
%F Equals 12^(-1/4)*Product_{n>=0} a(-n-2)*b(2^(n+1))^(2^(-n-2)) where a(n) = 3^(2^(n-1))*(1/2-3^(-2^(-n-1))/2)^(2^n) and b(n) = zeta(n)/Im(polylog(n, (-1)^(2/3))). - _Peter Luschny_, Jan 14 2021
%e 0.638909405445343882254942674928245093754975508...
%p Digits:= 1000: A:= 2^(-1/2)*3^(-1/4):
%p for t to 40000 do p:= ithprime(t): if `mod`(p, 3) = 2 then
%p A:= evalf(A/(1-1/p^2)^(1/2)) end if end do: A;
%p # Alternative:
%p z := n -> Zeta(n)/Im(polylog(n, (-1)^(2/3))):
%p x := n -> (z(2^n)*(3^(2^n)-1)*sqrt(3)/2)^(1/2^n)/3:
%p evalf(sqrt(mul(x(n), n=1..8))/12^(1/4), 110); # _Peter Luschny_, Jan 17 2021
%t digits = 106;
%t precision = digits + 10;
%t prodeuler[p_, a_, b_, expr_] := Product[If[a <= p <= b, expr, 1], {p, Prime[Range[PrimePi[a], PrimePi[b]]]}];
%t Lv3[s_] := prodeuler[p, 1, 2^(precision/s), 1/(1 - KroneckerSymbol[-3, p]*p^-s)] // N[#, precision]&;
%t Lv4[s_] := 2*Im[PolyLog[s, Exp[2*I*Pi/3]]]/Sqrt[3];
%t Lv[s_] := If[s >= 10000, Lv3[s], Lv4[s]];
%t gv[s_] := (1 - 3^(-s))*Zeta[s]/Lv[s];
%t pgv = Product[gv[2^n*2]^(2^-(n + 1)), {n, 0, 11}] // N[#, precision]&;
%t RealDigits[Sqrt[pgv]/12^(1/4), 10, digits][[1]]
%t (* _Jean-François Alcover_, Jan 12 2021, after PARI code due to _Artur Jasinski_ *)
%t S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);
%t P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]] * S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];
%t Z[m_, n_, s_] := (w = 1; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = P[m, n, s*w]/w; sumz = sumz + difz; w++]; Exp[sumz]);
%t $MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Pi * Sqrt[2] / (3^(7/4) * Sqrt[Z[3, 1, 2]]), digits]], 10, digits-1][[1]]
%t (* _Vaclav Kotesovec_, Jan 15 2021 *)
%t z[n_] := Zeta[n]/Im[PolyLog[n, (-1)^(2/3)]];
%t x[n_] := (z[2^n] (3^(2^n) - 1) Sqrt[3]/2)^(1/2^n)/3;
%t N[Sqrt[Product[x[n], { n, 8}]]/12^(1/4), 110] (* _Peter Luschny_, Jan 17 2021 *)
%Y Cf. A003136, A003627, A064533, A301430, A333240.
%K nonn,cons,changed
%O 0,1
%A _Michel Waldschmidt_, Mar 21 2018
%E Offset corrected by _Vaclav Kotesovec_, Mar 25 2018
%E a(6)-a(10) from _Peter Luschny_, Mar 29 2018
%E More digits from Ettahri article added by _Vaclav Kotesovec_, May 12 2020
%E More digits from _Vaclav Kotesovec_, Jun 27 2020
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