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

%I A175646 #47 Dec 04 2024 09:43:25
%S A175646 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,
%T A175646 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,
%U A175646 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
%N A175646 Decimal expansion of the Product_{primes p == 1 (mod 3)} 1/(1 - 1/p^2).
%C A175646 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)*...
%C A175646 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.
%C A175646 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
%H A175646 Peter Luschny, <a href="/A175646/b175646.txt">Table of n, a(n) for n = 1..1000</a> (terms 1..105 from Vaclav Kotesovec).
%H A175646 Thomas Dence and Carl Pomerance, <a href="http://dx.doi.org/10.1023/A:1009753405498">Euler's Function in Residue Classes</a>, Raman. J., Vol. 2 (1998) pp. 7-20, <a href="https://math.dartmouth.edu/~carlp/PDF/paper116.pdf">alternative link</a>.
%H A175646 S. Ettahri, O. Ramare, L. Surel, <a href="https://arxiv.org/abs/1908.06808">Fast multi-precision computation of some Euler products</a>, arxiv:1908.06808 (2019), Section 9.
%H A175646 R. J. Mathar, <a href="http://arxiv.org/abs/1008.2547">Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli</a>, arXiv:1008.2547 [math.NT], 2010-2015, p. 26.
%F A175646 Equals 2*Pi^2 / (3^(7/2) * A301429^2). - _Vaclav Kotesovec_, May 12 2020
%F A175646 Equals Sum_{k>=1} 1/A004611(k)^2. - _Amiram Eldar_, Sep 27 2020
%e A175646 1.03401487541434188053903064441304762857896...
%p A175646 z := n -> Zeta(n)/Im(polylog(n, (-1)^(2/3))):
%p A175646 x := n -> (z(2^n)*(3^(2^n)-1)*sqrt(3)/2)^(1/2^n) / 3:
%p A175646 evalf(4*Pi^2 / (27*mul(x(n), n=1..8)), 106); # _Peter Luschny_, Jan 17 2021
%t A175646 digits = 105;
%t A175646 precision = digits + 5;
%t A175646 prodeuler[p_, a_, b_, expr_] := Product[If[a <= p <= b, expr, 1], {p, Prime[Range[PrimePi[a], PrimePi[b]]]}];
%t A175646 Lv3[s_] := prodeuler[p, 1, 2^(precision/s), 1/(1 - KroneckerSymbol[-3, p]*p^-s)] // N[#, precision]&;
%t A175646 Lv4[s_] := 2*Im[PolyLog[s, Exp[2*I*Pi/3]]]/Sqrt[3];
%t A175646 Lv[s_] := If[s >= 10000, Lv3[s], Lv4[s]];
%t A175646 gv[s_] := (1 - 3^(-s))*Zeta[s]/Lv[s];
%t A175646 pB = (3/4)*Product[gv[2^n*2]^(2^-(n+1)), {n, 0, 11}] // N[#, precision]&;
%t A175646 pA = Pi^2/9/pB ;
%t A175646 RealDigits[pA, 10, digits][[1]]
%t A175646 (* _Jean-François Alcover_, Jan 11 2021, after PARI code due to _Artur Jasinski_ *)
%t A175646 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);
%t A175646 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}];
%t A175646 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]);
%t A175646 $MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Z[3,1,2], digits]], 10, digits-1][[1]] (* _Vaclav Kotesovec_, Jan 15 2021 *)
%t A175646 z[n_] := Zeta[n] / Im[PolyLog[n, (-1)^(2/3)]];
%t A175646 x[n_] := (z[2^n] (3^(2^n) - 1) Sqrt[3]/2)^(1/2^n) / 3;
%t A175646 N[4 Pi^2 / (27 Product[x[n], {n, 8}]), 106] (* _Peter Luschny_, Jan 17 2021 *)
%Y A175646 Cf. A002476, A004611, A007528, A175647, A301429, A333240, A334478, A334480.
%K A175646 cons,nonn
%O A175646 1,3
%A A175646 _R. J. Mathar_, Aug 01 2010
%E A175646 More digits from _Vaclav Kotesovec_, May 12 2020 and Jun 27 2020