A340710 -id:A340710 - cp's OEIS frontend

A340711 Decimal expansion of Product_{primes p == 3 (mod 5)} (p^2+1)/(p^2-1).

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

Offset: 1

Artur Jasinski, Jan 16 2021

			1.273986613206833925158...

Vaclav Kotesovec, Table of n, a(n) for n = 1..500
Salma Ettahri, Olivier Ramaré, and Léon Surel, Fast multi-precision computation of some Euler product, arXiv:1908.06808 [math.NT], 2019, p. 20.
Steven Finch, Quartic and Octic Characters Modulo n, arXiv:1008.2547 [math.NT], 2007-2010 p. 11 (formula on kappa(5) and kappa(-5)).
Steven Finch, Greg Martin and Pascal Sebah, Roots of unity and nullity modulo n, Proc. Amer. Math. Soc. Volume 138, Number 8, August 2010, pp. 2729-2743.
Steven Finch and Pascal Sebah, Residue of a Mod 5 Euler Product, arXiv:0912.3677 [math.NT], 2009 (formulas).
Alessandro Languasco and Alessandro Zaccagnini, A note on Mertens' formula for arithmetic progressions, Journal of Number Theory Volume 127, Issue 1, (2007), 37-46.
Alessandro Languasco and Alessandro Zaccagnini, On the constant in the Mertens product for arithmetic progressions. II: Numerical values, Math. Comp. 78 (2009), 315-326.
Alessandro Languasco and Alessandro Zaccagnini, On the constant in the Mertens product for arithmetic progressions. I. Identities, Funct. Approx. Comment. Math. Volume 42, Number 1 (2010), 17-27.
Alessandro Languasco and Alessandro Zaccagnini, Computation of the Mertens constants - more than 100 correct digits, (2007), 1-134 (digital data).
Alessandro Languasco and Alessandro Zaccagnini, Computation of the Mertens constants mod q; 3 <= q <= 100, (2007) (GP-PARI procedure 100 digits accuracy).
Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.
G. Niklasch, Some number-theoretical constants arising as products of rational functions of p over primes
Doug S. Phillips and Peter Zvengrowski, Convergence of Dirichlet Series and Euler Products, Contributions Section of Natural Mathematical and Biotechnical Sciences 38(2):153 (2017).

Cf. A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340577, A340578, A340628, A340629, A340004, A340127, A340710.

(* Using Vaclav Kotesovec's function Z from A301430. *)
$MaxExtraPrecision = 1000; digits = 121;
digitize[c_] := RealDigits[Chop[N[c, digits]], 10, digits - 1][[1]];
digitize[1/(Z[5, 3, 4]/Z[5, 3, 2]^2)]

D = Product_{primes p == 0 (mod 5)} (p^2+1)/(p^2-1) = 13/12.
E = Product_{primes p == 1 (mod 5)} (p^2+1)/(p^2-1) = A340629.
F = Product_{primes p == 2 (mod 5)} (p^2+1)/(p^2-1) = A340710.
G = Product_{primes p == 3 (mod 5)} (p^2+1)/(p^2-1) = this constant.
H = Product_{primes p == 4 (mod 5)} (p^2+1)/(p^2-1) = A340628.
D*E*F*G*H = 5/2.
E*F*G*H = 30/13.
D*E*H = sqrt(5)/2.
D*F*G = 13*sqrt(5)/12.
F*G = sqrt(5).
E*H = 6*sqrt(5)/13.
Equals Sum_{q in A004617} 2^A001221(q)/q^2. - R. J. Mathar, Jan 27 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].

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A340711 Decimal expansion of Product_{primes p == 3 (mod 5)} (p^2+1)/(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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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).