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

0,1

Comments

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).

References

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

Links

Table of n, a(n) for n=0..105.

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.

Formula

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

Example

0.487622778111571768611646391452388423131677124429735...

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

Crossrefs

Cf. A002496, A005574, A096012, A206328.

Similar constants: A005597, A331941, A337607, A337608.

Sequence in context: A201658 A146501 A197574 * A019924 A021209 A247605

Adjacent sequences: A337603 A337604 A337605 * A337607 A337608 A337609

Keyword

nonn,cons

Author

Amiram Eldar, Sep 04 2020

Extensions

More digits from Vaclav Kotesovec, Jan 15 2021

Status

approved