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

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, 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, 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

Offset: 0

Author

Amiram Eldar, Sep 04 2020

Keywords

• nonn
• cons

Comments

Named by Finch (2003) after the American mathematician Daniel Shanks (1917 - 1996).

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.

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

Examples

			0.669740969937071220538922431571764406688370157436482...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 90.

Links

• Keith Conrad, Hardy-Littlewood constants in: Mathematical properties of sequences and other combinatorial structures, Jong-Seon No et al. (eds.), Kluwer, Boston/Dordrecht/London, 2003, pp. 133-154, alternative link.
• Salma Ettahri, Olivier Ramaré, Léon Surel, Fast multi-precision computation of some Euler products, arXiv:1908.06808 [math.NT], 2019 (Corollary 1.8).
• Mohan Lal, Primes of the form n^4 + 1, Mathematics of Computation, Vol. 21, No. 98 (1967), pp. 245-247.
• Daniel Shanks, On Numbers of the form n^4 + 1, Mathematics of Computation, Vol. 15, No. 74 (1961), pp. 186-189.
• Daniel Shanks, Lal's constant and generalizations, Mathematics of Computation, Vol. 21, No. 100 (1967), pp. 705-707.
• Eric Weisstein's World of Mathematics, Lal's Constant.

Crossrefs

Cf. A000068, A037896, A088367, A334826.

Similar constants: A005597, A331941, A337606, A337608.

Programs

Mathematica

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]);
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]);
$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 *)

Formula

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.

Extensions

More digits from Vaclav Kotesovec, Jan 15 2021