A340577 -id:A340577

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A340004

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

+10
15

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,5

COMMENTS

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

LINKS

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).

FORMULA

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

EXAMPLE

1.01091516060101952260495658428951492...

MATHEMATICA

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 *)

CROSSREFS

Cf. A340127, A340628, A340629, A340665, A340794, A340839.

Cf. A175646, A301429, A333240.

Cf. A175647, A248930, A248938, A335963.

Cf. A004615, A340576, A340577, A340578, A334826.

KEYWORD

nonn,cons

AUTHOR

Artur Jasinski, Jan 15 2021

STATUS

approved

A340127

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

+10
15

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 1..501

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.

FORMULA

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

EXAMPLE

1.0049603239222975589937496248102521847955102941880228801995283785215071277...

MATHEMATICA

(* 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]]

CROSSREFS

Cf. A004618, A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340577, A340578, A340628, A340628, A340004.

KEYWORD

nonn,cons

AUTHOR

Artur Jasinski, Jan 15 2021

STATUS

approved

A340628

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

+10
15

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

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.

FORMULA

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

EXAMPLE

1.009935934829401027349038488241778167715858547548801013...

MAPLE

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

MATHEMATICA

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 *)

CROSSREFS

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

KEYWORD

nonn,cons

AUTHOR

Artur Jasinski, Jan 13 2021

EXTENSIONS

Corrected and more terms from Vaclav Kotesovec, Jan 15 2021

STATUS

approved

A340576

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

+10
13

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

The four similar sequences for products of primes mod 6 are these:

A175646 for primes p == 1 (mod 6)} 1/(1-1/p^2),

A340576 for primes p == 5 (mod 6)} 1/(1-1/p^2),

A340577 for primes p == 1 (mod 6)} 1/(1+1/p^2),

A340578 for primes p == 5 (mod 6)} 1/(1+1/p^2).

LINKS

Table of n, a(n) for n=1..105.

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.

FORMULA

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

EXAMPLE

1.06054829316911072817412636430987203493077130204487163127994372...

MAPLE

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

MATHEMATICA

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 *)

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 *)

CROSSREFS

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

Cf. A259548.

KEYWORD

nonn,cons

AUTHOR

Jean-François Alcover, Jan 12 2021

STATUS

approved

A340578

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

+10
12

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Table of n, a(n) for n=0..104.

EXAMPLE

0.94450093450470098673429109419144434254611078086906676955735771...

MATHEMATICA

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 *)

(* -------------------------------------------------------------------------- *)

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 *)

CROSSREFS

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

KEYWORD

nonn,cons

AUTHOR

Jean-François Alcover, Jan 12 2021

STATUS

approved

A340629

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

+10
12

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

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).

Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

FORMULA

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

EXAMPLE

1.0218780604187566757444489146002708261704607377325...

MAPLE

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.

MATHEMATICA

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}];

$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 *)

CROSSREFS

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

KEYWORD

nonn,cons

AUTHOR

Artur Jasinski, Jan 13 2021

EXTENSIONS

Corrected and more terms from Vaclav Kotesovec, Jan 15 2021

STATUS

approved

A340711

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

+10
11

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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).

FORMULA

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

EXAMPLE

1.273986613206833925158...

MATHEMATICA

(* 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)]

CROSSREFS

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

KEYWORD

nonn,cons

AUTHOR

Artur Jasinski, Jan 16 2021

STATUS

approved

A340665

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

+10
8

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

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.

FORMULA

Equals Sum_{k>=1} 1/A004617(k)^2. - Amiram Eldar, Jan 24 2021

EXAMPLE

1.135764878668921626868643009472082289511936413...

MATHEMATICA

(* 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]]

CROSSREFS

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

KEYWORD

nonn,cons

AUTHOR

Artur Jasinski, Jan 15 2021

STATUS

approved

A340794

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

+10
8

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 1..501

For links see A340711.

FORMULA

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

EXAMPLE

1.36857205387664908586076389048310999017020782888589520500850402955633118881...

MATHEMATICA

(* 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]]

CROSSREFS

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

KEYWORD

nonn,cons

AUTHOR

Artur Jasinski, Jan 21 2021

STATUS

approved

A340710

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

+10
6

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 1..500

For links see A340711.

FORMULA

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

EXAMPLE

1.7551738411687377766074721228405237...

MATHEMATICA

(* 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)]

CROSSREFS

Cf. A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340577, A340578, A340628, A340629, A340004, A340127, A340711.

KEYWORD

nonn,cons

AUTHOR

Artur Jasinski, Jan 16 2021

STATUS

approved

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