A088367 -id:A088367 - cp's OEIS frontend

A088375 Decimal expansion of a postulated upper estimate for the complex Grothendieck constant.

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

Offset: 1

Eric W. Weisstein, Sep 28 2003

			1.404575934663742032773958471548143743234611830652711936118...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 3.11, p. 237.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Grothendieck's Constant.

Cf. A088367, A088373, A088374.

Re(evalf(1/(2*EllipticK(I)-EllipticE(I)), 120)); # Vaclav Kotesovec, Apr 22 2015

First[ RealDigits[ N[1/(2*EllipticK[-1] - EllipticE[-1] ), 120], 10, 102]](* Jean-François Alcover, Jun 07 2012, after Eric W. Weisstein *)
RealDigits[(Sqrt[8 Pi] Gamma[3/4]^2)/(Pi^2 - 2 Gamma[3/4]^4), 10, 102][[1]] (* Jan Mangaldan, Nov 23 2020 *)

magm(a, b)=my(eps=10^-(default(realprecision)-5), c); while(abs(a-b)>eps, my(z=sqrt((a-c)*(b-c))); [a, b, c] = [(a+b)/2, c+z, c-z]); (a+b)/2
E(x)=Pi/2/agm(1,sqrt(1-x))*magm(1,1-x)
K(x)=Pi/2/agm(1,sqrt(1-x))
1/(2*K(-1)-E(-1)) \\ Charles R Greathouse IV, Aug 02 2018

Equals (sqrt(8*Pi)*Gamma(3/4)^2)/(Pi^2 - 2*Gamma(3/4)^4). - Jan Mangaldan, Nov 23 2020

A088373 Decimal expansion of a constant related to the postulated upper estimate for the complex Grothendieck constant.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Sep 28 2003

			0.8125578588...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Grothendieck's Constant

Cf. A088367, A088374, A088375.

psi[x_] := (Sqrt[1 - x^2]*(EllipticE[-x^2/(1 - x^2)] - EllipticK[-x^2/(1 - x^2)]))/x; x0 = x /. FindRoot[psi[x] == 1/8*Pi*(x + 1), {x, 1/2}, WorkingPrecision -> 110]; RealDigits[x0, 10, 102] // First (* Jean-François Alcover, Feb 06 2013 *)

A088374 Decimal expansion of a postulated upper estimate for the complex Grothendieck constant.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Sep 28 2003

			1.4049091327357955...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Grothendieck's Constant

Cf. A088367, A088373, A088375.

psi[x_] := (Sqrt[1 - x^2]*(EllipticE[-x^2/(1 - x^2)] - EllipticK[-x^2/(1 - x^2)]))/x; x0 = x /. FindRoot[psi[x] == 1/8*Pi*(x + 1), {x, 1/2}, WorkingPrecision -> 110]; RealDigits[8/(Pi*(x0 + 1)), 10, 102] // First (* Jean-François Alcover, Feb 06 2013 *)

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

A337608 Decimal expansion of Lal's constant: the Hardy-Littlewood constant for A217795.

Original entry on oeis.org

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, 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, 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

Offset: 0

Amiram Eldar, Sep 04 2020

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.

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

			0.792208238167541668775455566579024101128932250986221...

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

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 1.9).
Mohan Lal, Primes of the form n^4 + 1, Mathematics of Computation, Vol. 21, No. 98 (1967), pp. 245-247.
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, A210630, A217795, A217796.

Similar constants: A005597, A331941, A337606, A337607.

$MaxExtraPrecision = 1000; digits = 121;
f[p_] := (p-8)*(p+1)^4/((p-1)^4*p);
coefs = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, 1000}], x]];
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}];
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++];
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 *)

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.

More terms from Vaclav Kotesovec, Jan 16 2021

A348669 Decimal expansion of 2sqrt(2)log(1 + sqrt(2))/(3*Pi).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Oct 29 2021

The average length of a random line segment in a unit square defined as follows. A line that is making a random angle with a given edge of the square is chosen, and a random distance of this line from a given vertex of this edge is chosen uniformly between 0 and the distance to the opposite vertex in the square. The segment is then being chosen by picking at random two points between the two intersection points of the line with the perimeter of the square.

			0.26450500700786984551577520129722526936340009096805...

Eric Rosenberg, The expected length of a random line segment in a rectangle, Operations Research Letters, Vol. 32, No. 2 (2004), pp. 99-102.

Cf. A088367, A091505, A091648, A244920.

evalf(sqrt(8/9)*arcsinh(1)/Pi, 120);  # Alois P. Heinz, Oct 29 2021

RealDigits[2*Sqrt[2]*Log[1 + Sqrt[2]]/(3*Pi), 10, 100][[1]]

2*sqrt(2)*log(1 + sqrt(2))/(3*Pi) \\ Michel Marcus, Oct 29 2021

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A088375 Decimal expansion of a postulated upper estimate for the complex Grothendieck constant.

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A088373 Decimal expansion of a constant related to the postulated upper estimate for the complex Grothendieck constant.

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A088374 Decimal expansion of a postulated upper estimate for the complex Grothendieck constant.

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

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A337608 Decimal expansion of Lal's constant: the Hardy-Littlewood constant for A217795.

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A348669 Decimal expansion of 2*sqrt(2)*log(1 + sqrt(2))/(3*Pi).

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A348669 Decimal expansion of 2sqrt(2)log(1 + sqrt(2))/(3*Pi).