A301429 -id:A301429 - cp's OEIS frontend

A175646 Decimal expansion of the Product_{primes p == 1 (mod 3)} 1/(1 - 1/p^2).

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

Offset: 1

R. J. Mathar, Aug 01 2010

The Euler product of the Riemann zeta function at 2 restricted to primes in A002476, which is the inverse of the infinite product (1-1/7^2)*(1-1/13^2)*(1-1/19^2)*...
There is a complementary Product_{primes p == 2 (mod 3)} 1/(1-1/p^2) = A333240 = 1.4140643908921476375655018190798... such that (this constant here)*1.4140643.../(1-1/3^2) = zeta(2) = A013661.
Because 1/(1-p^(-2)) = 1+1/(p^2-1), the complementary 1.414064... also equals Product_{primes p == 2 (mod 3)} (1+1/(p^2-1)), which appears in Eq. (1.8) of [Dence and Pomerance]. - R. J. Mathar, Jan 31 2013

			1.03401487541434188053903064441304762857896...

Peter Luschny, Table of n, a(n) for n = 1..1000 (terms 1..105 from Vaclav Kotesovec).
Thomas Dence and Carl Pomerance, Euler's Function in Residue Classes, Raman. J., Vol. 2 (1998) pp. 7-20, alternative link.
S. Ettahri, O. Ramare, L. Surel, Fast multi-precision computation of some Euler products, arxiv:1908.06808 (2019), Section 9.
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, p. 26.

Cf. A002476, A004611, A007528, A175647, A301429, A333240, A334478, A334480.

z := n -> Zeta(n)/Im(polylog(n, (-1)^(2/3))):
x := n -> (z(2^n)*(3^(2^n)-1)*sqrt(3)/2)^(1/2^n) / 3:
evalf(4*Pi^2 / (27*mul(x(n), n=1..8)), 106); # Peter Luschny, Jan 17 2021

digits = 105;
precision = digits + 5;
prodeuler[p_, a_, b_, expr_] := Product[If[a <= p <= b, expr, 1], {p, Prime[Range[PrimePi[a], PrimePi[b]]]}];
Lv3[s_] := prodeuler[p, 1, 2^(precision/s), 1/(1 - KroneckerSymbol[-3, p]*p^-s)] // N[#, precision]&;
Lv4[s_] := 2*Im[PolyLog[s, Exp[2*I*Pi/3]]]/Sqrt[3];
Lv[s_] := If[s >= 10000, Lv3[s], Lv4[s]];
gv[s_] := (1 - 3^(-s))*Zeta[s]/Lv[s];
pB = (3/4)*Product[gv[2^n*2]^(2^-(n+1)), {n, 0, 11}] // N[#, precision]&;
pA = Pi^2/9/pB ;
RealDigits[pA, 10, digits][[1]]
(* Jean-François Alcover, Jan 11 2021, after PARI code due to Artur Jasinski *)
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[3,1,2], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *)
z[n_] := Zeta[n] / Im[PolyLog[n, (-1)^(2/3)]];
x[n_] := (z[2^n] (3^(2^n) - 1) Sqrt[3]/2)^(1/2^n) / 3;
N[4 Pi^2 / (27 Product[x[n], {n, 8}]), 106] (* Peter Luschny, Jan 17 2021 *)

Equals 2*Pi^2 / (3^(7/2) * A301429^2). - Vaclav Kotesovec, May 12 2020

Equals Sum_{k>=1} 1/A004611(k)^2. - Amiram Eldar, Sep 27 2020

More digits from Vaclav Kotesovec, May 12 2020 and Jun 27 2020

A333240 Decimal expansion of Product_{primes p == 2 (mod 3)} 1/(1 - 1/p^2).

Original entry on oeis.org

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

Offset: 1

Peter Luschny, May 13 2020

The range of product are the primes of the form 3*k - 1 (A003627).

See a comment of R. J. Mathar in A175646.

			1.414064390892147637565501819079829379907695069393162175039924962423928106992...

Peter Luschny, Table of n, a(n) for n = 1..1000
Thomas Dence and Carl Pomerance, Euler's Function in Residue Classes, Raman. J., Vol. 2 (1998) pp. 7-20, c_3 in formula (1.8) and (5.6), alternative link.
S. Ettahri, O. Ramare, L. Surel, Fast multi-precision computation of some Euler products, arxiv:1908.06808 (2019), Section 9.
Jerzy Kaczorowski, Waldemar Ratajczak, Peter Nijkamp, Krzysztof Górnisiewicz, Economic hierarchical spatial systems - new properties of Löschian numbers, Applied Mathematics and Computation, Volume 461 (2024) 128319. (The product appears in Theorem 3 (12)).
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, Zeta_{3,2}(2) in section 3.2.

Cf. A003627, A004612, A175646, A214549, A301429, A333239.

z := n -> Zeta(n)/Im(polylog(n, (-1)^(2/3))):
x := n -> (z(2^n)*(3^(2^n)-1)*sqrt(3)/2)^(1/2^n)/3:
evalf(mul(x(n), n=1..8), 105); # Peter Luschny, Jan 17 2021

digits = 104; precision = digits + 10;
prodeuler[p_, a_, b_, expr_] := Product[If[a <= p <= b, expr, 1], {p, Prime[Range[PrimePi[a], PrimePi[b]]]}];
Lv3[s_] := prodeuler[p, 1, 2^(precision/s), 1/(1 - KroneckerSymbol[-3, p]*p^-s)] // N[#, precision]&;
Lv4[s_] := 2*Im[PolyLog[s, Exp[2*I*Pi/3]]]/Sqrt[3];
Lv[s_] := If[s >= 10000, Lv3[s], Lv4[s]];
gv[s_] := (1 - 3^(-s))*Zeta[s]/Lv[s];
pgv = Product[gv[2^n*2]^(2^-(n + 1)), {n, 0, 11}] // N[#, precision]&;
RealDigits[pgv, 10, digits][[1]]
(* Jean-François Alcover, Jan 12 2021, after PARI code due to Artur Jasinski *)
z[n_] := Zeta[n]/Im[PolyLog[n, (-1)^(2/3)]];
x[n_] := (z[2^n] (3^(2^n) - 1) Sqrt[3]/2)^(1/2^n)/3;
N[Product[x[n], {n, 8}], 105] (* Peter Luschny, Jan 17 2021 *)

A333240 * A175646 = (4*Pi^2)/27 = A214549.
A301429 = sqrt(A333240) / 12^(1/4).
Equals Sum_{k>=1} 1/A004612(k)^2. - Amiram Eldar, Sep 27 2020

Last 5 digits corrected by Jean-François Alcover, Jan 12 2021

Better name by Peter Luschny, Jan 17 2021

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

A340127 Decimal expansion of Product_{primes p == 4 (mod 5)} p^2/(p^2-1).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 15 2021

			1.0049603239222975589937496248102521847955102941880228801995283785215071277...

Vaclav Kotesovec, Table of n, a(n) for n = 1..501
S. Ettahri, O. Ramare, L. Surel, Fast multi-precision computation of some Euler products, arxiv:1908.06808 (2019), Section 9.
Steven Finch and Pascal Sebah, Residue of a Mod 5 Euler Product, arXiv:0912.3677 [math.NT], 2009 (C(5,n) = mu(n,5) formulas p.2).
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. A004618, A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340577, A340578, A340628, A340628, A340004.

(* 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, 4, 2]]

Equals (1/C(5,4))*Pi*sqrt(3*C(5,1)*C(5,2)*C(5,3)/(5*C(5,4)*log(2+sqrt(5)))).
for definitions of Mertens constants C(5,n) see A. Languasco and A. Zaccagnini 2010.
for high precision numerical values C(5,n) see A. Languasco and A. Zaccagnini 2007.
C(5,1)=1.225238438539084580057609774749220527540595509391649938767...
C(5,2)=0.546975845411263480238301287430814037751996324100819295153...
C(5,3)=0.8059510404482678640573768602784309320812881149390108979348...
C(5,4)=1.29936454791497798816084001496426590950257497040832966201678...
Equals (1/C(5,4)^2)*Pi*sqrt(3*exp(-gamma)/(4*log(2 + sqrt(5)))), where gamma is the Euler-Mascheroni constant A001620.
Equals Sum_{k>=1} 1/A004618(k)^2. - Amiram Eldar, Jan 24 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

A340576 Decimal expansion of Product_{primes p == 5 (mod 6)} 1/(1-1/p^2).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jan 12 2021

The four similar sequences for products of primes mod 6 are these:
  A175646 for Product_{primes p == 1 (mod 6)} 1/(1-1/p^2),
  A340576 for Product_{primes p == 5 (mod 6)} 1/(1-1/p^2),
  A340577 for Product_{primes p == 1 (mod 6)} 1/(1+1/p^2),
  A340578 for Product_{primes p == 5 (mod 6)} 1/(1+1/p^2).

			1.06054829316911072817412636430987203493077130204487163127994372...

R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, Zeta_{6,5}(2) in section 3.2.

Similar constants: A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340577, A340578.

Cf. A259548.

a := n -> 3^(2^(-n-2))*((1-3^(-2^(n+1)))/2)^(2^(-n-1)):
b := n -> Zeta(n)/Im(polylog(n, (-1)^(2/3))):
c := n -> a(n)*b(2^(n+1))^(1/2^(n+1)):
Digits := 107: evalf((3/4)*mul(c(n), n=0..9)); # Peter Luschny, Jan 14 2021

digits = 105;
precision = digits + 10;
prodeuler[p_, a_, b_, expr_] := Product[If[a <= p <= b, expr, 1], {p, Prime[Range[PrimePi[a], PrimePi[b]]]}];
Lv3[s_] := prodeuler[p, 1, 2^(precision/s), 1/(1 - KroneckerSymbol[-3, p]*p^-s)] // N[#, precision] &;
Lv4[s_] := 2*Im[PolyLog[s, Exp[2*I*Pi/3]]]/Sqrt[3];
Lv[s_] := If[s >= 10000, Lv3[s], Lv4[s]];
gv[s_] := (1 - 3^(-s))*Zeta[s]/Lv[s];
pB = (3/4)*Product[gv[2^n*2]^(2^-(n+1)), {n, 0, 11}] // N[#, precision]&;
RealDigits[pB, 10, digits][[1]] (* Most of this code is due to Artur Jasinski *)
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[6,5,2], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *)

g = A143298 = (9 - PolyGamma(1, 2/3) + PolyGamma(1, 4/3))/(4 sqrt(3));
h = A301429;
Equals (3*sqrt(3)*h^2)/2.
Equals (3/4)*A333240.
A340577 = Pi^4/(243*g*h^2);
A340578 = (45*g*h^2)/(2*Pi^2).
Equals Pi^2/(9*A175646). - Artur Jasinski, Jan 11 2021
Equals Sum_{k>=1} 1/A259548(k)^2. - Amiram Eldar, Jan 24 2021

A301430 Decimal expansion of an analog of the Landau-Ramanujan constant for Loeschian numbers which are sums of two squares.

Original entry on oeis.org

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

Offset: 0

Michel Waldschmidt, Mar 21 2018

This is the decimal expansion of the number alpha such that the number of positive integers <= N which are sums of two squares and are also represented by the quadratic form x^2 + xy + y^2 is asymptotic to alpha*N*(log(N))^(-3/4).

Based on the constants Zeta(m=12,n=5,s=2) = 1.0482019036007..., Zeta(m=12,n=7,s=2) = 1.0262021468... and Zeta(m=12,n=11,s=2) = 1.01177863 ... read from arXiv:1008.2547 we have Product_{p == 5, 7, 11(mod 12)} (1-1/p^2)^(-1/2) = sqrt( Zeta(m=12,n=5,s=2) * Zeta(m=12,n=7,s=2) * Zeta(m=12,n=11,s=2) ) as a factor in the formulas. - R. J. Mathar, Feb 04 2021

			0.30231614235706563794776990048019971560241279...

Salma Ettahri, Olivier Ramaré, and Léon Surel, Fast multi-precision computation of some Euler products, arXiv:1908.06808 [math.NT], 2019.
Étienne Fouvry, Claude Levesque, and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017 and Acta Arithmetica, online 15 March 2018.
Alessandro Languasco and Pieter Moree, Euler constants from primes in arithmetic progression, arXiv:2406.16547 [math.NT], 2024. See p. 17.
Olivier Ramaré, S. Ettahri, and L. Surel, Fast multi-precision computation of some Euler products, Mathematics of Computation (2021) hal-03381427.

Cf. A003136, A064533, A167135, A301429, A340552.

Digits:= 1000: with(numtheory):
B:= evalf(3^(1/4)*Pi^(1/2)*log(2+sqrt(3))^(1/4)/(2^(5/4)*GAMMA(1/4))):
for t to 500 do p:=ithprime(t): if `or`(`or`(`mod`(p, 12) = 5, `mod`(p, 12) = 7), `mod`(p, 12) = 11) then B:= evalf(B/(1-1/p^2)^(1/2)) end if end do: B;

prec := 200; B = N[(Sqrt[Pi] ((3 Log[2 + Sqrt[3]])/2)^(1/4))/(2 Gamma[1/4]), prec];
For[n = 3, n < 50000, n++, p = Prime[n];
If[Mod[p, 12] != 1, B = B / Sqrt[(1 - 1/p) (1 + 1/p)]]]
Print[B] (* Peter Luschny, Mar 23 2018 *)
(* -------------------------------------------------------------------------- *)
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[(3^(1/4)/2^(5/4)) * Pi^(1/2) * (Log[2 + Sqrt[3]])^(1/4) / Gamma[1/4] * Sqrt[Z[12, 5, 2] * Z[12, 7, 2] * Z[12, 11, 2]], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *)

Equals (3^(1/4)/2^(5/4)) * Pi^(1/2) * (log(2 + sqrt(3)))^(1/4) / Gamma(1/4) * Product_{p == 5, 7, 11 (mod 12), p prime} (1 - 1/p^2)^(-1/2).

One can base the definition on p(n) = A167135(n). Setting r(n) = (Product_{k=1..n} p(k)^2) / (Product_{k=1..n} (p(k)^2 - 1)) the rational sequence r(n) starts 4/3, 3/2, 25/16, 1225/768, 29645/18432, ... -> L. Then A301430 = sqrt(L)*M with M = ((arccosh(2)/6)^(1/4)*Gamma(3/4))/(2*sqrt(Pi)). - Peter Luschny, Mar 29 2018

Offset corrected by Vaclav Kotesovec, Mar 25 2018
a(6)-a(10) from Peter Luschny, Mar 29 2018
More digits from Ettahri article added by Vaclav Kotesovec, May 12 2020
More digits from Vaclav Kotesovec, Jan 15 2021

A340577 Decimal expansion of Product_{primes p == 1 (mod 6)} 1/(1+1/p^2).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jan 12 2021

			0.96755040251956188660947077043906773001524912960304386356302398...

Similar constants: A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340578.

digits = 105;
precision = digits + 5;
prodeuler[p_, a_, b_, expr_] := Product[If[a <= p <= b, expr, 1], {p, Prime[Range[PrimePi[a], PrimePi[b]]]}];
Lv3[s_] := prodeuler[p, 1, 2^(precision/s), 1/(1 - KroneckerSymbol[-3, p]*p^-s)] // N[#, precision]&;
Lv4[s_] := 2*Im[PolyLog[s, Exp[2*I*Pi/3]]]/Sqrt[3];
Lv[s_] := If[s >= 10000, Lv3[s], Lv4[s]];
gv[s_] := (1 - 3^(-s))*Zeta[s]/Lv[s];
pB = (3/4)*Product[gv[2^n*2]^(2^-(n+1)), {n, 0, 11}] // N[#, precision]&;
pC = (2*Pi^4)/(243*pB*Lv[2]);
RealDigits[pC, 10, digits][[1]](* Most of this code is due to Artur Jasinski *)
(* -------------------------------------------------------------------------- *)
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[6, 1, 4]/Z[6, 1, 2], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *)

Equals 1.0004615.../A175646, both constants taken from p 27 of arXiv:1008.2537v2. - R. J. Mathar, Aug 21 2022

a(104) corrected by Vaclav Kotesovec, Jan 15 2021

A340578 Decimal expansion of Product_{primes p == 5 (mod 6)} 1/(1+1/p^2).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jan 12 2021

			0.94450093450470098673429109419144434254611078086906676955735771...

Similar constants: A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340577.

digits = 105;
precision = digits + 5;
prodeuler[p_, a_, b_, expr_] := Product[If[a <= p <= b, expr, 1], {p, Prime[Range[PrimePi[a], PrimePi[b]]]}];
Lv3[s_] := prodeuler[p, 1, 2^(precision/s), 1/(1 - KroneckerSymbol[-3, p]*p^-s)] // N[#, precision]&;
Lv4[s_] := 2*Im[PolyLog[s, Exp[2*I*Pi/3]]]/Sqrt[3];
Lv[s_] := If[s >= 10000, Lv3[s], Lv4[s]];
gv[s_] := (1 - 3^(-s))*Zeta[s]/Lv[s];
pB = (3/4)*Product[gv[2^n*2]^(2^-(n+1)), {n, 0, 11}] // N[#, precision]&;
pD = (45*pB*Lv[2])/(4*Pi^2);
RealDigits[pD, 10, digits][[1]] (* Most of this code is due to Artur Jasinski *)
(* -------------------------------------------------------------------------- *)
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[6, 5, 4]/Z[6, 5, 2], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *)

A340629 Decimal expansion of Product_{primes p == 1 (mod 5)} (p^2+1)/(p^2-1).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 13 2021

			1.0218780604187566757444489146002708261704607377325...

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-digit-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, A340628.

evalf(Re(15*sqrt((1/13)*(5*((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)))/Pi^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[5,1,4]/Z[5,1,2]^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)]);
A340629 := (15 Sqrt[65]/(26 Pi^2)) Sqrt[cl[2/5]^2 + cl[4/5]^2];
digitize[A340629] (* Peter Luschny, Jan 23 2021 *)

Equals 6*sqrt(5)/(13*A340628).
Equals A340004^2/A340808. - R. J. Mathar, Jan 15 2021
Equals 15*sqrt(65)*g/(13*Pi^2) 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 Sum_{q in A004615} 2^A001221(q)/q^2. - R. J. Mathar, Jan 27 2021

Corrected and more terms from Vaclav Kotesovec, Jan 15 2021

cp's OEIS Frontend

A175646 Decimal expansion of the Product_{primes p == 1 (mod 3)} 1/(1 - 1/p^2).

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A333240 Decimal expansion of Product_{primes p == 2 (mod 3)} 1/(1 - 1/p^2).

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

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A340127 Decimal expansion of Product_{primes p == 4 (mod 5)} p^2/(p^2-1).

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

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A340576 Decimal expansion of Product_{primes p == 5 (mod 6)} 1/(1-1/p^2).

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A301430 Decimal expansion of an analog of the Landau-Ramanujan constant for Loeschian numbers which are sums of two squares.

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