A301429 Decimal expansion of an analog of the Landau-Ramanujan constant for Loeschian numbers.

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

Offset: 0

Author

Michel Waldschmidt, Mar 21 2018

Keywords

• nonn
• cons

Comments

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)).

Examples

			0.638909405445343882254942674928245093754975508...
		

References

• S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, p. 99 (K3).

Links

• 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.
• Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 204.
• Étienne 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.
• Olivier Ramaré, S. Ettahri, and L. Surel, Fast multi-precision computation of some Euler products, Mathematics of Computation (2021) hal-03381427.
• StackExchange, Iterative calculation of a number-theoretical constant, Mar 24 2018.

Crossrefs

Cf. A003136, A003627, A064533, A301430, A333240.

Programs

Maple

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

Mathematica

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 *)

Formula

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

Extensions

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