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A175646 Decimal expansion of the Product_{primes p == 1 (mod 3)} 1/(1 - 1/p^2). 28
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

LINKS

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.

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.

FORMULA

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

EXAMPLE

1.03401487541434188053903064441304762857896...

MAPLE

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

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]&;

pA = Pi^2/9/pB ;

RealDigits[pA, 10, digits][[1]]

(* Jean-Fran├žois 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 *)

CROSSREFS

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

Sequence in context: A021298 A170952 A194587 * A324362 A073234 A123685

Adjacent sequences:  A175643 A175644 A175645 * A175647 A175648 A175649

KEYWORD

cons,nonn

AUTHOR

R. J. Mathar, Aug 01 2010

EXTENSIONS

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

STATUS

approved

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Last modified June 29 16:49 EDT 2022. Contains 354913 sequences. (Running on oeis4.)