A340665 -id:A340665 - cp's OEIS frontend

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

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]

A340004 Decimal expansion of Product_{primes p == 1 (mod 5)} p^2/(p^2-1).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 15 2021

This constant is called Euler product 2==1 modulo 5 (see Mathar's Definition 5 formula (38)) or equivalently zeta 2==1 modulo 5.

			1.01091516060101952260495658428951492...

Vaclav Kotesovec, Table of n, a(n) for n = 1..501
Salma Ettahri, Olivier Ramaré, and Léon Surel, Fast multi-precision computation of some Euler products, arXiv:1908.06808 [math.NT], 2019 p.20 (100 digits precision data).
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2014-2015, Section 3.3. zeta_{5,1}(2).

Cf. A340127, A340628, A340629, A340665, A340794, A340839.
Cf. A175646, A301429, A333240.
Cf. A175647, A248930, A248938, A335963.
Cf. A004615, A340576, A340577, A340578, A334826.

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]);
$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Z[5, 1, 2], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021, took 20 minutes *)

Equals Sum_{k>=1} 1/A004615(k)^2. - Amiram Eldar, Jan 24 2021

Equals exp(-gamma/2)*Pi/(A340839^2*sqrt(5*log((1 + sqrt (5))/2))). - Artur Jasinski, Jan 30 2021

A340794 Decimal expansion of Product_{primes p == 2 (mod 5)} p^2/(p^2-1).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 21 2021

			1.36857205387664908586076389048310999017020782888589520500850402955633118881...

Vaclav Kotesovec, Table of n, a(n) for n = 1..501
For links see A340711.

Cf. A004616, A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340127, A340576, A340577, A340578, A340628, A340629, A340665, A340710, A340711.

(* Using Vaclav Kotesovec's function Z from A301430. *)
$MaxExtraPrecision = 1000; digits = 121;
digitize[c_] := RealDigits[Chop[N[c, digits]], 10, digits - 1][[1]];
digitize[Z[5, 2, 2]]

I = Product_{primes p == 0 (mod 5)} p^2/(p^2-1) = 25/24.
J = Product_{primes p == 1 (mod 5)} p^2/(p^2-1) = A340004.
K = Product_{primes p == 2 (mod 5)} p^2/(p^2-1) = this constant.
L = Product_{primes p == 3 (mod 5)} p^2/(p^2-1) = A340665.
M = Product_{primes p == 4 (mod 5)} p^2/(p^2-1) = A340127.
I*J*K*L*M = Pi^2/6 = zeta(2).
J*K*L*M = 4*Pi^2/25.
M = (Pi/2)*C(5,4)^(-2)*exp(-gamma/2)*sqrt(3/log(2+sqrt(5))), where gamma is the Euler-Mascheroni constant A001620 and C(5,4) is the Mertens constant = 1.29936454791497798816084...
Equals Sum_{k>=1} 1/A004616(k)^2. - Amiram Eldar, Jan 24 2021

A340926 Decimal expansion of Product_{primes p == 2 (mod 5)} 1/(1 - 1/p^4).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jan 27 2021

			1.067124761502234255634582163136137073885091716528006051500764099869277...

Vaclav Kotesovec, Table of n, a(n) for n = 1..500
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, Zeta_{m=5,n=2}(s=4).

Cf. A340808, A340809, A340927.
Cf. A340004, A340794, A340665, A340127.
Cf. A340629, A340710, A340711, A340628.

Equals A340794^2 / A340710.
Equals 104*Pi^4 / (9375 * A340808 * A340927 * A340809).
Equals Sum_{k>=1} 1/A004616(k)^4. - Amiram Eldar, Jan 28 2021

A340927 Decimal expansion of Product_{primes p == 3 (mod 5)} 1/(1 - 1/p^4).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jan 27 2021

			1.012539571644935903522100272691152140478362802787749854800134772695303...

Vaclav Kotesovec, Table of n, a(n) for n = 1..500
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, Zeta_{m=5,n=3}(s=4).

Cf. A340808, A340809, A340926.
Cf. A340004, A340794, A340665, A340127.
Cf. A340629, A340710, A340711, A340628.

Equals A340665^2 / A340711.
Equals 104*Pi^4 / (9375 * A340808 * A340926 * A340809).
Equals Sum_{k>=1} 1/A004617(k)^4. - Amiram Eldar, Jan 28 2021

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).

Original entry on oeis.org

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)

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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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A340004 Decimal expansion of Product_{primes p == 1 (mod 5)} p^2/(p^2-1).

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A340794 Decimal expansion of Product_{primes p == 2 (mod 5)} p^2/(p^2-1).

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A340926 Decimal expansion of Product_{primes p == 2 (mod 5)} 1/(1 - 1/p^4).

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A340927 Decimal expansion of Product_{primes p == 3 (mod 5)} 1/(1 - 1/p^4).

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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+4sqrt(5))/(3Pi))sqrt(A340794A340665).

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).