A337608 -id:A337608 - cp's OEIS frontend

A337606 Decimal expansion of the Gaussian twin prime constant: the Hardy-Littlewood constant for A096012.

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

Offset: 0

Amiram Eldar, Sep 04 2020

The name of this constant was suggested by Finch (2003).
Gaussian twin primes on the line x + i in the complex plane are Gaussian primes pair of the form (m - 1 + i, m + 1 + i). The numbers m are numbers such that (m-1)^2 + 1 and (m+1)^2 + 1 are both primes (A096012 plus 1).
Shanks (1960) conjectured that the number of these pairs 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 and evaluated it by 0.4876.
The first 100 digits of 4*c were calculated by Ettahri et al. (2019).

			0.487622778111571768611646391452388423131677124429735...

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

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 (Section 8).
Daniel Shanks, , A note on Gaussian twin primes, Mathematics of Computation, Vol. 14, No. 70 (1960), pp. 201-203.

Cf. A002496, A005574, A096012, A206328.

Similar constants: A005597, A331941, A337607, A337608.

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^2/8 * Zs[4, 1, 4]/Z[4, 1, 2]^2, digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *)

Equals (Pi^2/8) * Product_{primes p == 1 (mod 4)} (1 - 4/p)*((p + 1)/(p - 1))^2.

More digits from Vaclav Kotesovec, Jan 15 2021

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

Original entry on oeis.org

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

Amiram Eldar, Sep 04 2020

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).

			0.669740969937071220538922431571764406688370157436482...

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

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.

Cf. A000068, A037896, A088367, A334826.

Similar constants: A005597, A331941, A337606, A337608.

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 *)

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.

More digits from Vaclav Kotesovec, Jan 15 2021

cp's OEIS Frontend

A337606 Decimal expansion of the Gaussian twin prime constant: the Hardy-Littlewood constant for A096012.

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