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Decimal expansion of Lal's constant: the Hardy-Littlewood constant for A217795.
2

%I #12 May 31 2024 22:10:31

%S 7,9,2,2,0,8,2,3,8,1,6,7,5,4,1,6,6,8,7,7,5,4,5,5,5,6,6,5,7,9,0,2,4,1,

%T 0,1,1,2,8,9,3,2,2,5,0,9,8,6,2,2,1,1,1,7,2,2,7,9,7,3,4,5,2,5,6,9,5,1,

%U 4,1,5,4,9,4,4,1,2,4,9,0,6,6,0,2,9,5,3,8,8,3,9,8,0,2,7,5,2,9,2,7,8,7,3,9,7,3

%N Decimal expansion of Lal's constant: the Hardy-Littlewood constant for A217795.

%C Shanks (1967) conjectured that the number of primes of the form (m + 1)^4 + 1 such that (m - 1)^4 + 1 is also a prime (A217795 plus 1), with m <= x, is asymptotic to c * li_2(x), where li_2(x) = Integral_{t=2..n} (1/log(t)^2) dt, and c is this constant. He defined c as in the formula section, evaluated it by 0.79220 and named it after the mathematician Mohan Lal, who conjectured the asymptotic formula without evaluating this constant.

%C The first 100 digits of this constant were calculated by Ettahri et al. (2019).

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, pp. 90-91.

%H Keith Conrad, <a href="https://doi.org/10.1007/978-1-4615-0304-0_15">Hardy-Littlewood constants</a> in: Mathematical properties of sequences and other combinatorial structures, Jong-Seon No et al. (eds.), Kluwer, Boston/Dordrecht/London, 2003, pp. 133-154, <a href="https://kconrad.math.uconn.edu/articles/hlconst.pdf">alternative link</a>.

%H Salma Ettahri, Olivier Ramaré, 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 (Corollary 1.9).

%H Mohan Lal, <a href="https://doi.org/10.1090/S0025-5718-1967-0222007-9">Primes of the form n^4 + 1</a>, Mathematics of Computation, Vol. 21, No. 98 (1967), pp. 245-247.

%H Daniel Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1967-0223315-8">Lal's constant and generalizations</a>, Mathematics of Computation, Vol. 21, No. 100 (1967), pp. 705-707.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LalsConstant.html">Lal's Constant</a>.

%F Equals (Pi^4/(2^7 * log(1+sqrt(2))^2)) * Product_{primes p == 1 (mod 8)} (1 - 4/p)^2 * ((p + 1)/(p - 1))^4 * p*(p-8)/(p-4)^2 = (Pi^2/32) * A088367^2 * A334826^2 * A210630 = 2 * A337607^2 * A210630.

%e 0.792208238167541668775455566579024101128932250986221...

%t $MaxExtraPrecision = 1000; digits = 121;

%t f[p_] := (p-8)*(p+1)^4/((p-1)^4*p);

%t coefs = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, 1000}], x]];

%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 m = 2; sump = 0; difp = 1; While[Abs[difp] > 10^(-digits - 5) || difp == 0, difp = coefs[[m]]*(P[8, 1, m] - 1/17^m); sump = sump + difp; m++];

%t RealDigits[Chop[N[f[17] * Pi^4/(2^7 * Log[1+Sqrt[2]]^2) * Exp[sump], digits]], 10, digits - 1][[1]] (* _Vaclav Kotesovec_, Jan 16 2021 *)

%Y Cf. A000068, A037896, A088367, A210630, A217795, A217796.

%Y Similar constants: A005597, A331941, A337606, A337607.

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, Sep 04 2020

%E More terms from _Vaclav Kotesovec_, Jan 16 2021