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%I #63 Jan 26 2021 16:09:10
%S 2,6,2,6,5,2,1,8,8,7,2,0,5,3,6,7,6,6,6,7,5,9,6,2,0,1,1,4,7,2,0,8,8,3,
%T 4,6,5,3,0,2,0,4,3,9,3,0,6,4,7,4,4,7,3,9,1,0,6,8,2,5,5,1,0,5,8,7,0,9,
%U 2,6,6,8,3,8,6,9,0,2,2,7,4,1,7,9,4,1,9,3,8,3,6,5,5,2,3,5,0,0,2,0,1,0,0,8,9,1
%N Decimal expansion of constant K5 = 29*log(2+sqrt(5))*(Product_{primes p == 1 (mod 5)} (1-4*(2*p-1)/(p*(p+1)^2)))/(15*Pi^2).
%C Finch and Sebah, 2009, p. 7 (see link) call this constant K_5. K_5 is related to the Mertens constant C(5,1) (see A340839). For more references see the links in A340711. Finch and Sebah give the following definition:
%C Consider the asymptotic enumeration of m-th order primitive Dirichlet characters mod n. Let b_m(n) denote the count of such characters. There exists a constant 0 < K_m < oo such that Sum_{n <= N} b_m(n) ∼ K_m*N*log(N)^(d(m) - 2) as N -> oo, where d(m) is the number of divisors of m.
%H Steven Finch and Pascal Sebah, <a href="https://arxiv.org/abs/0912.3677">Residue of a Mod 5 Euler Product</a>, arXiv:0912.3677 [math.NT], 2009 p. 10.
%F Equals (29/25)*(Product_{primes p} (1-1/p)^2*(1+gcd(p-1,5)/(p-1))) [Finch and Sebah, 2009, p. 10].
%e 0.262652188720536766675962011472088346530204393064744739106825510587...
%t $MaxExtraPrecision = 1000; digits = 121; f[p_] := (1 - 4*(2*p-1)/(p*(p+1)^2));
%t coefs = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, 1000}], x]];
%t 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 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 m = 2; sump = 0; difp = 1; While[Abs[difp] > 10^(-digits - 5) || difp == 0, difp = coefs[[m]]*P[5, 1, m]; sump = sump + difp; PrintTemporary[m]; m++];
%t RealDigits[Chop[N[29*Log[2+Sqrt[5]]/(15*Pi^2) * Exp[sump], digits]], 10, digits-1][[1]] (* _Vaclav Kotesovec_, Jan 25 2021, took over 50 minutes *)
%Y Cf. A340878 (K3), A340879 (K4).
%Y Cf. A340004, A340794, A340665, A340127.
%Y Cf. A340629, A340710, A340711, A340628.
%Y Cf. A175646, A301429, A333240.
%Y Cf. A175647, A248930, A248938, A335963.
%Y Cf. A340576, A340577, A340578, A334826.
%K nonn,cons
%O 0,1
%A _Artur Jasinski_, Jan 24 2021
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