A340004 -id:A340004 - cp's OEIS frontend

A340857 Decimal expansion of constant K5 = 29log(2+sqrt(5))(Product_{primes p == 1 (mod 5)} (1-4(2p-1)/(p(p+1)^2)))/(15Pi^2).

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

Offset: 0

Artur Jasinski, Jan 24 2021

Finch and Sebah, 2009, p. 7 (see link) call this constant K_5. K_5 is related to the Mertens constant C(5,1) (see A340839). For more references see the links in A340711. Finch and Sebah give the following definition:

Consider the asymptotic enumeration of m-th order primitive Dirichlet characters mod n. Let b_m(n) denote the count of such characters. There exists a constant 0 < K_m < oo such that Sum_{n <= N} b_m(n) ∼ K_m*N*log(N)^(d(m) - 2) as N -> oo, where d(m) is the number of divisors of m.

			0.262652188720536766675962011472088346530204393064744739106825510587...

Steven Finch and Pascal Sebah, Residue of a Mod 5 Euler Product, arXiv:0912.3677 [math.NT], 2009 p. 10.

Cf. A340878 (K3), A340879 (K4).
Cf. A340004, A340794, A340665, A340127.
Cf. A340629, A340710, A340711, A340628.
Cf. A175646, A301429, A333240.
Cf. A175647, A248930, A248938, A335963.
Cf. A340576, A340577, A340578, A334826.

$MaxExtraPrecision = 1000; digits = 121; f[p_] := (1 - 4*(2*p-1)/(p*(p+1)^2));
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[5, 1, m]; sump = sump + difp; PrintTemporary[m]; m++];
RealDigits[Chop[N[29*Log[2+Sqrt[5]]/(15*Pi^2) * Exp[sump], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 25 2021, took over 50 minutes *)

Equals (29/25)*(Product_{primes p} (1-1/p)^2*(1+gcd(p-1,5)/(p-1))) [Finch and Sebah, 2009, p. 10].

A335576 Decimal expansion of Mertens constant C(5,2).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Jan 26 2021

First 100 digits from Alessandro Languasco and Alessandro Zaccagnini 2007 p. 4.

			0.546975845411263480238301287430814...

Alessandro Languasco and Alessandro Zaccagnini, Computation of the Mertens constants - more than 100 correct digits, (2007), 1-134.

Cf. A001620, A336798, A340004, A340127, A340213, A340665, A340794, A340839, A340866.

A = C(5,1)=1.2252384385390845800576097747492205... see A340839.
B = C(5,2)=0.5469758454112634802383012874308140... this constant.
C = C(5,3)=0.8059510404482678640573768602784309... see A336798.
D = C(5,4)=1.2993645479149779881608400149642659... see A340866.
A*B*C*D = 0.70182435445860646228... = (5/4)*exp(-gamma), where gamma is the Euler-Mascheroni constant A001620.
B = sqrt(2)*5^(3/4)*sqrt(A340127)*exp(-gamma)/(4*sqrt(A340004)*A^2*C).
B = 2*A*D*log((1+sqrt(5))/2)/(C*sqrt(5)*A340794*A340665).
B = A*D*log((1+sqrt(5))/2)^2/(C*Pi*A340213^2).
From Vaclav Kotesovec, Jan 27 2021: (Start)
B*C = 5^(1/4) * exp(-gamma/2) * sqrt(log((1+sqrt(5))/2) / (2 * A340665 * A340794)).
A*D = 5^(3/4) * exp(-gamma/2) * sqrt(A340665 * A340794 / (8 * log((1+sqrt(5))/2))).
(End)

A340213 Decimal expansion of the constant kappa(-5) = (1/2)sqrt(sqrt(5)log(9+4sqrt(5))/(3Pi))sqrt(A340794A340665).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Jan 26 2021

For general definition of the constants kappa(n) see Steven Finch 2009 p. 7, for this particular case kappa(-5) see p. 11.

			0.51593948227965348495312501394...

Steven Finch, Quartic and Octic Characters Modulo n, arXiv:0907.4894 [math.NT], 2009 p. 7-11.

Cf. A340004, A340127, A340665, A340794, A340839, A340866.

Equals exp(-gamma/2)*log((1+sqrt(5))/2)*sqrt(5/Pi)/(2*C(5,2)*C(5,3)), where C(5,2) and C(5,3) are Mertens constants see A340839.
Equals 2*A340866*exp(gamma/4)*((1/5)*log((1+sqrt(5))/2))^(3/4)/sqrt(A340004).
Equals 2*A340866*exp(gamma/4)*log((1+sqrt(5))/2)/(sqrt(5*Pi)*A340884^(1/4)).
Equals 2*A340839*A340866*exp(gamma/2)*log((1+sqrt(5))/2)/sqrt(5*Pi).
Equals sqrt((1/3)*Pi*log(9+4*sqrt(5)))/(sqrt(5^(3/2)*A340004*A340127)). [Finch 2009 p. 11]

A340884 Decimal expansion of the constant rho(1,5).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Jan 25 2021

From definition Steven Finch and Pascal Sebah 2009 p. 1:

rho(n,m) = lim_{s->1} (s-1) Product_{primes p==n (mod m)} (1-1/p^s)^phi(m), where phi(n) = A000010(n) is the Euler totient function.

			0.249135702764931424659963795...

Steven Finch and Pascal Sebah, Residue of a Mod 5 Euler Product, arXiv:0912.3677 [math.NT], 2009 p. 1-2.

Cf. A340628, A340808, A340004.

Equals 1/(exp(gamma)*A340839^4).
Formulas by Steven Finch and Pascal Sebah 2009 p. 2.
Equals 5*log(2 + sqrt(5))*A340004^2/(3*Pi^2).
Equals 50*log(2 + sqrt(5))*A340808/(13*Pi^2*sqrt(5)*A340628).

cp's OEIS Frontend

A340857 Decimal expansion of constant K5 = 29log(2+sqrt(5))(Product_{primes p == 1 (mod 5)} (1-4(2p-1)/(p(p+1)^2)))/(15Pi^2).

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A335576 Decimal expansion of Mertens constant C(5,2).

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A340213 Decimal expansion of the constant kappa(-5) = (1/2)sqrt(sqrt(5)log(9+4sqrt(5))/(3Pi))sqrt(A340794A340665).

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A340884 Decimal expansion of the constant rho(1,5).

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cp's OEIS Frontend

A340857 Decimal expansion of constant K5 = 29*log(2+sqrt(5))*(Product_{primes p == 1 (mod 5)} (1-4*(2*p-1)/(p*(p+1)^2)))/(15*Pi^2).

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A335576 Decimal expansion of Mertens constant C(5,2).

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A340213 Decimal expansion of the constant kappa(-5) = (1/2)*sqrt(sqrt(5)*log(9+4*sqrt(5))/(3*Pi))*sqrt(A340794*A340665).

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A340884 Decimal expansion of the constant rho(1,5).

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A340857 Decimal expansion of constant K5 = 29log(2+sqrt(5))(Product_{primes p == 1 (mod 5)} (1-4(2p-1)/(p(p+1)^2)))/(15Pi^2).

A340213 Decimal expansion of the constant kappa(-5) = (1/2)sqrt(sqrt(5)log(9+4sqrt(5))/(3Pi))sqrt(A340794A340665).