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A301429
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Decimal expansion of an analog of the Landau-Ramanujan constant for Loeschian numbers.
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16
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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, 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, 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
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OFFSET
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0,1
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COMMENTS
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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)).
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REFERENCES
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S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, p. 99 (K3).
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LINKS
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Peter Luschny, Table of n, a(n) for n = 0..1000 (terms 0..105 from Vaclav Kotesovec).
Salma Ettahri, Olivier Ramaré, and Léon Surel, Fast multi-precision computation of some Euler products, arXiv:1908.06808 [math.NT], 2019.
Etienne Fouvry, Claude Levesque, and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017 and Acta Arithmetica, online 15 March 2018.
StackExchange, Iterative calculation of a number-theoretical constant, Mar 24 2018.
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FORMULA
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Equals 2^(-1/2)*3^(-1/4)*Product_{p == 2 mod 3, p prime} (1 - p^(-2))^(-1/2).
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]
Equals Pi*sqrt(2) / (3^(7/4) * sqrt(A175646)). - Vaclav Kotesovec, May 12 2020
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
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EXAMPLE
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0.638909405445343882254942674928245093754975508...
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MAPLE
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Digits:= 1000: A:= 2^(-1/2)*3^(-1/4):
for t to 40000 do p:= ithprime(t): if `mod`(p, 3) = 2 then
A:= evalf(A/(1-1/p^2)^(1/2)) end if end do: A;
# Alternative:
z := n -> Zeta(n)/Im(polylog(n, (-1)^(2/3))):
x := n -> (z(2^n)*(3^(2^n)-1)*sqrt(3)/2)^(1/2^n)/3:
evalf(sqrt(mul(x(n), n=1..8))/12^(1/4), 110); # Peter Luschny, Jan 17 2021
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MATHEMATICA
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digits = 106;
precision = digits + 10;
prodeuler[p_, a_, b_, expr_] := Product[If[a <= p <= b, expr, 1], {p, Prime[Range[PrimePi[a], PrimePi[b]]]}];
Lv3[s_] := prodeuler[p, 1, 2^(precision/s), 1/(1 - KroneckerSymbol[-3, p]*p^-s)] // N[#, precision]&;
Lv4[s_] := 2*Im[PolyLog[s, Exp[2*I*Pi/3]]]/Sqrt[3];
Lv[s_] := If[s >= 10000, Lv3[s], Lv4[s]];
gv[s_] := (1 - 3^(-s))*Zeta[s]/Lv[s];
pgv = Product[gv[2^n*2]^(2^-(n + 1)), {n, 0, 11}] // N[#, precision]&;
RealDigits[Sqrt[pgv]/12^(1/4), 10, digits][[1]]
(* Jean-François Alcover, Jan 12 2021, after PARI code due to Artur Jasinski *)
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);
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}];
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]);
$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Pi * Sqrt[2] / (3^(7/4) * Sqrt[Z[3, 1, 2]]), digits]], 10, digits-1][[1]]
(* Vaclav Kotesovec, Jan 15 2021 *)
z[n_] := Zeta[n]/Im[PolyLog[n, (-1)^(2/3)]];
x[n_] := (z[2^n] (3^(2^n) - 1) Sqrt[3]/2)^(1/2^n)/3;
N[Sqrt[Product[x[n], { n, 8}]]/12^(1/4), 110] (* Peter Luschny, Jan 17 2021 *)
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CROSSREFS
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Cf. A003136, A003627, A064533, A301430, A333240.
Sequence in context: A199447 A273067 A306774 * A198836 A271179 A220085
Adjacent sequences: A301426 A301427 A301428 * A301430 A301431 A301432
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KEYWORD
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nonn,cons
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AUTHOR
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Michel Waldschmidt, Mar 21 2018
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EXTENSIONS
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Offset corrected by Vaclav Kotesovec, Mar 25 2018
a(6)-a(10) from Peter Luschny, Mar 29 2018
More digits from Ettahri article added by Vaclav Kotesovec, May 12 2020
More digits from Vaclav Kotesovec, Jun 27 2020
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STATUS
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approved
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