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 A301430 Decimal expansion of an analog of the Landau-Ramanujan constant for Loeschian numbers which are sums of two squares. 12
 3, 0, 2, 3, 1, 6, 1, 4, 2, 3, 5, 7, 0, 6, 5, 6, 3, 7, 9, 4, 7, 7, 6, 9, 9, 0, 0, 4, 8, 0, 1, 9, 9, 7, 1, 5, 6, 0, 2, 4, 1, 2, 7, 9, 5, 1, 8, 9, 3, 6, 9, 6, 4, 5, 4, 5, 8, 8, 6, 7, 8, 4, 1, 2, 8, 8, 8, 6, 5, 4, 4, 8, 7, 5, 2, 4, 1, 0, 5, 1, 0, 8, 9, 9, 4, 8, 7, 4, 6, 7, 8, 1, 3, 9, 7, 9, 2, 7, 2, 7, 0, 8, 5, 6, 7, 7 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS This is the decimal expansion of the number alpha such that the number of positive integers <= N which are sums of two squares and are also represented by the quadratic form x^2 + xy + y^2 is asymptotic to alpha*N*(log(N))^(-3/4). Based on the constants Zeta(m=12,n=5,s=2) = 1.0482019036007..., Zeta(m=12,n=7,s=2) = 1.0262021468... and Zeta(m=12,n=11,s=2) = 1.01177863 ... read from arXiv:1008.2547 we have Product_{p == 5, 7, 11(mod 12)} (1-1/p^2)^(-1/2) = sqrt( Zeta(m=12,n=5,s=2) * Zeta(m=12,n=7,s=2) * Zeta(m=12,n=11,s=2) ) as a factor in the formulas. - R. J. Mathar, Feb 04 2021 LINKS Table of n, a(n) for n=0..105. Salma Ettahri, Olivier Ramaré, and Léon Surel, Fast multi-precision computation of some Euler products, arXiv:1908.06808 [math.NT], 2019. 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. Alessandro Languasco and Pieter Moree, Euler constants from primes in arithmetic progression, arXiv:2406.16547 [math.NT], 2024. See p. 17. Olivier Ramaré, S. Ettahri, and L. Surel, Fast multi-precision computation of some Euler products, Mathematics of Computation (2021) hal-03381427. FORMULA Equals (3^(1/4)/2^(5/4)) * Pi^(1/2) * (log(2 + sqrt(3)))^(1/4) / Gamma(1/4) * Product_{p == 5, 7, 11 (mod 12), p prime} (1 - 1/p^2)^(-1/2). One can base the definition on p(n) = A167135(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, 3/2, 25/16, 1225/768, 29645/18432, ... -> L. Then A301430 = sqrt(L)*M with M = ((arccosh(2)/6)^(1/4)*Gamma(3/4))/(2*sqrt(Pi)). - Peter Luschny, Mar 29 2018 EXAMPLE 0.30231614235706563794776990048019971560241279... MAPLE Digits:= 1000: with(numtheory): B:= evalf(3^(1/4)*Pi^(1/2)*log(2+sqrt(3))^(1/4)/(2^(5/4)*GAMMA(1/4))): for t to 500 do p:=ithprime(t): if `or`(`or`(`mod`(p, 12) = 5, `mod`(p, 12) = 7), `mod`(p, 12) = 11) then B:= evalf(B/(1-1/p^2)^(1/2)) end if end do: B; MATHEMATICA prec := 200; B = N[(Sqrt[Pi] ((3 Log[2 + Sqrt[3]])/2)^(1/4))/(2 Gamma[1/4]), prec]; For[n = 3, n < 50000, n++, p = Prime[n]; If[Mod[p, 12] != 1, B = B / Sqrt[(1 - 1/p) (1 + 1/p)]]] Print[B] (* Peter Luschny, Mar 23 2018 *) (* -------------------------------------------------------------------------- *) 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[(3^(1/4)/2^(5/4)) * Pi^(1/2) * (Log[2 + Sqrt[3]])^(1/4) / Gamma[1/4] * Sqrt[Z[12, 5, 2] * Z[12, 7, 2] * Z[12, 11, 2]], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *) CROSSREFS Cf. A003136, A064533, A167135, A301429, A340552. Sequence in context: A355344 A224317 A032531 * A328311 A143394 A112455 Adjacent sequences: A301427 A301428 A301429 * A301431 A301432 A301433 KEYWORD nonn,cons AUTHOR Michel Waldschmidt, Mar 21 2018 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, Jan 15 2021 STATUS approved

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Last modified September 8 02:36 EDT 2024. Contains 375749 sequences. (Running on oeis4.)