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A337607 Decimal expansion of Shanks's constant: the Hardy-Littlewood constant for A000068.

%I A337607 #11 Jan 15 2021 10:20:18
%S A337607 6,6,9,7,4,0,9,6,9,9,3,7,0,7,1,2,2,0,5,3,8,9,2,2,4,3,1,5,7,1,7,6,4,4,
%T A337607 0,6,6,8,8,3,7,0,1,5,7,4,3,6,4,8,2,4,1,8,5,7,3,2,9,8,5,2,2,8,4,5,2,4,
%U A337607 6,7,9,9,9,5,6,4,5,7,1,4,7,2,7,3,1,5,0,6,2,1,0,2,1,4,3,5,9,3,7,3,5,0,2,7,3,2
%N A337607 Decimal expansion of Shanks's constant: the Hardy-Littlewood constant for A000068.
%C A337607 Named by Finch (2003) after the American mathematician Daniel Shanks (1917 - 1996).
%C A337607 Shanks (1961) conjectured that the number of primes of the form m^4 + 1 (A037896) with m <= x is asymptotic to c * li(x), where li(x) is the logarithmic integral function and c is this constant. He defined c as in the formula section and evaluated it by 0.66974.
%C A337607 The first 100 digits of this constant were calculated by Ettahri et al. (2019).
%D A337607 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 90.
%H A337607 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 A337607 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 1.8).
%H A337607 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 A337607 Daniel Shanks, <a href="https://doi.org/10.1090/S0025-5718-1961-0120184-6">On Numbers of the form n^4 + 1</a>, Mathematics of Computation, Vol. 15, No. 74 (1961), pp. 186-189.
%H A337607 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 A337607 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LalsConstant.html">Lal's Constant</a>.
%F A337607 Equals (Pi^2/(16*log(1+sqrt(2)))) * Product_{primes p == 1 (mod 8)} (1 - 4/p)*((p + 1)/(p - 1))^2 = (Pi/8) * A088367 * A334826.
%e A337607 0.669740969937071220538922431571764406688370157436482...
%t A337607 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 A337607 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 A337607 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 A337607 Zs[m_, n_, s_] := (w = 2; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = (s^w - s) * P[m, n, w]/w; sumz = sumz + difz; w++]; Exp[-sumz]);
%t A337607 $MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Pi^2/(16*Log[1+Sqrt[2]]) * Zs[8, 1, 4]/Z[8, 1, 2]^2, digits]], 10, digits-1][[1]] (* _Vaclav Kotesovec_, Jan 15 2021 *)
%Y A337607 Cf. A000068, A037896, A088367, A334826.
%Y A337607 Similar constants: A005597, A331941, A337606, A337608.
%K A337607 nonn,cons
%O A337607 0,1
%A A337607 _Amiram Eldar_, Sep 04 2020
%E A337607 More digits from _Vaclav Kotesovec_, Jan 15 2021