A340576 - OEIS

A340576

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

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

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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 *)

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 *)

CROSSREFS

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

Cf. A259548.

Sequence in context: A274419 A131329 A165954 * A104288 A198995 A168218

Adjacent sequences: A340573 A340574 A340575 * A340577 A340578 A340579

KEYWORD

nonn,cons

AUTHOR

Jean-François Alcover, Jan 12 2021

STATUS

approved