A340127 -id:A340127 - cp's OEIS frontend

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

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

A340628 Decimal expansion of Product_{primes p == 4 (mod 5)} (p^2+1)/(p^2-1).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 13 2021

			1.009935934829401027349038488241778167715858547548801013...

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 products, arXiv:1908.06808 [math.NT], 2019 p.20 (70 digits precision data).
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, On the constant in the Mertens product for arithmetic progressions. I. Identities, Funct. Approx. Comment. Math. Volume 42, Number 1 (2010), 17-27, (preliminary version).
Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Cf. A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340577, A340578, A340629.

evalf(Re(2*Pi^2/(5*sqrt(13*((I*Pi^2*(1/150)-I*polylog(2, (-1)^(2/5)))^2+((1/150)*(11*I)*Pi^2+I*polylog(2, (-1)^(4/5)))^2)))), 120) # Vaclav Kotesovec, Jan 20 2021, after formula by Pascal Sebah

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; PrintTemporary["iteration = ", w, ", difference = ", N[difz, digits]]; w++]; Exp[sumz]);
$MaxExtraPrecision = 1000; digits = 121; Chop[N[1/(Z[5,4,4]/Z[5,4,2]^2), digits]] (* Vaclav Kotesovec, Jan 15 2021, took over 20 minutes *)
digits = 121; digitize[c_] := RealDigits[Chop[N[c, digits]], 10, digits][[1]];
cl[x_] :=I(PolyLog[2,(-1)^x] - PolyLog[2,-(-1)^(1-x)]);
A340628 :=(4 Pi^2)/(5 Sqrt[13])/ Sqrt[cl[2/5]^2 + cl[4/5]^2];
digitize[A340628] (* Peter Luschny, Jan 23 2021 *)

Equals 6*sqrt(5)/(13*A340629).
Equals 6*sqrt(13)*Pi^2/(195*g) where g = sqrt(Cl2(2*Pi/5)^2 + Cl2(4*Pi/5)^2) = 1.0841621352693895..., and Cl2 is the Clausen function of order 2. Formula by Pascal Sebah (personal communication). - Artur Jasinski, Jan 20 2021
Equals A340127^2/A340809. - R. J. Mathar, Jan 22 2021
Equals Sum_{q in A004618} 2^A001221(q)/q^2. - R. J. Mathar, Jan 27 2021

Corrected and more terms from Vaclav Kotesovec, Jan 15 2021

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

Original entry on oeis.org

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

A340665 Decimal expansion of Product_{primes p == 3 (mod 5)} p^2/(p^2-1).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 15 2021

			1.135764878668921626868643009472082289511936413...

Vaclav Kotesovec, Table of n, a(n) for n = 1..501
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions..., arXiv:1008.2547 Zeta_{m=5,n=3}(s=2).
For links see A340628.

Cf. A004617, A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340577, A340578, A340628, A340628, A340004, A340127.

(* Using Vaclav Kotesovec's function Z from A301430. *)
$MaxExtraPrecision = 100; digits = 50; (* Adjust as needed. *)
digitize[c_] := RealDigits[Chop[N[c, digits+10]], 10, digits][[1]];
digitize[Z[5, 3, 2]]

Equals Sum_{k>=1} 1/A004617(k)^2. - Amiram Eldar, Jan 24 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

A340839 Decimal expansion of Mertens constant C(5,1).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 23 2021

Data taken from Alessandro Languasco and Alessandro Zaccagnini 2007.

			1.225238438539084580057609774749220527540595509391649938767...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.2 Meissel-Mertens constants (pp. 94-95)

Vaclav Kotesovec, Table of n, a(n) for n = 1..499
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, Computation of the Mertens constants - more than 100 correct digits, (2007), 1-134 (digital data relative to the previous paper). [in this table on page 4, the last correct digit is a(109), beyond the level there certified. - _Vaclav Kotesovec_, Jan 26 2021]
Alessandro Languasco and Alessandro Zaccagnini, Computation of the Mertens constants mod q; 3 <= q <= 100, (2007) (GP-PARI procedure 100 digits accuracy).
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.
For other links see A340711.

Cf. A077761, A083343, A091589, A138312, A161529, A230767, A238114, A271971, A340127, A340794, A340866.

A = C(5,1)=1.225238438539084580057609774749220527540595509391649938767...
B = C(5,2)=0.546975845411263480238301287430814037751996324100819295153...
C = C(5,3)=0.805951040448267864057376860278430932081288114939010897934...
D = C(5,4)=1.299364547914977988160840014964265909502574970408329662016...
A*B*C*D = 0.70182435445860646228... = (5/4)*exp(-gamma), where gamma is the Euler-Mascheroni constant A001620.
Formula from the article by Languasco and Zaccagnini, 2010, p.9:
A = ((13*sqrt(5)*Pi^2*exp(-gamma))/(150*log((1+sqrt(5))/2))*A340628/A340808)^(1/4).

Last 11 digits corrected by Vaclav Kotesovec, Jan 25 2021

More digits from Vaclav Kotesovec, Jan 26 2021

A340710 Decimal expansion of Product_{primes p == 2 (mod 5)} (p^2+1)/(p^2-1).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 16 2021

			1.7551738411687377766074721228405237...

Vaclav Kotesovec, Table of n, a(n) for n = 1..500
For links see A340711.

Cf. A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340577, A340578, A340628, A340629, A340004, A340127, 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[1/(Z[5, 2, 4]/Z[5, 2, 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) = this constant.
G = Product_{primes p == 3 (mod 5)} (p^2+1)/(p^2-1) = A340711.
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.
Formulas by Pascal Sebah, Jan 20 2021: (Start)
Let g = sqrt(Cl2(2*Pi/5)^2+Cl2(4*Pi/5)^2) = 1.0841621352693895..., where Cl2 is the Clausen function of order 2.
E = 15*sqrt(65)*g/(13*Pi^2).
H = 6*sqrt(13)*Pi^2/(195*g). (End)
Equals Sum_{q in A004616} 2^A001221(q)/q^2. - R. J. Mathar, Jan 27 2021

A340809 Decimal expansion of Product_{primes p == 4 (mod 5)} 1/(1-p^(-4)).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 22 2021

Equals also the same product over the primes p==9 (mod 10).

			1.0000092261581025751416880414384715475110104...

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=4}(s=4).

Cf. A004618, A030433, A340127 (s=2), A340628, A340808, A340926, A340927.

Equals A340127 ^2/A340628

Equals Sum_{k>=1} 1/A004618(k)^4. - Amiram Eldar, Jan 24 2021

More digits from Vaclav Kotesovec, Jan 22 2021

A340866 Decimal expansion of the Mertens constant C(5,4).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 24 2021

Data taken from Alessandro Languasco and Alessandro Zaccagnini 2007 p. 4.

			1.299364547914977988160840014964265909502574970408329662016...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.2 Meissel-Mertens constants (pp. 94-95).

Vaclav Kotesovec, Table of n, a(n) for n = 1..497
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, Computation of the Mertens constants - more than 100 correct digits, (2007). [in this table on page 4, the last correct digit is a(108), beyond the level there certified. - _Vaclav Kotesovec_, Jan 26 2021]
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.
For other links see A340711.

Cf. A340004, A340127, A340839, 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[(13*Pi^2 / (24*Sqrt[5] * Exp[EulerGamma] * Log[(1 + Sqrt[5])/2]) * Z[5, 1, 2]^2 / (Z[5, 1, 4] * Z[5, 4, 4]))^(1/4)]

Equals A340839*5^(1/4)*sqrt(A340004/(2*A340127)).

Equals (13*Pi^2/(24*sqrt(5)*exp(gamma)*log((1+sqrt(5))/2))*A340629/A340809)^(1/4). - Vaclav Kotesovec, Jan 25 2021

Corrected by Vaclav Kotesovec, Jan 25 2021

More digits from Vaclav Kotesovec, Jan 26 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

cp's OEIS Frontend

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A340628 Decimal expansion of Product_{primes p == 4 (mod 5)} (p^2+1)/(p^2-1).

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A340665 Decimal expansion of Product_{primes p == 3 (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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A340839 Decimal expansion of Mertens constant C(5,1).

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A340710 Decimal expansion of Product_{primes p == 2 (mod 5)} (p^2+1)/(p^2-1).

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A340809 Decimal expansion of Product_{primes p == 4 (mod 5)} 1/(1-p^(-4)).

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