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%I #67 Mar 03 2024 04:18:43
%S 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,
%T 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,
%U 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
%N Decimal expansion of Product_{primes p == 1 (mod 5)} (p^2+1)/(p^2-1).
%H Vaclav Kotesovec, <a href="/A340629/b340629.txt">Table of n, a(n) for n = 1..500</a>
%H Salma Ettahri, Olivier Ramaré, and Léon Surel, <a href="https://arxiv.org/abs/1908.06808">Fast multi-precision computation of some Euler products</a>, arXiv:1908.06808 [math.NT], 2019, p. 20 (70-digit-precision data).
%H Steven Finch, Greg Martin, and Pascal Sebah, <a href="http://www.math.ubc.ca/~gerg/papers/downloads/RUNM.pdf">Roots of unity and nullity modulo n</a>, Proc. Amer. Math. Soc. Volume 138, Number 8, August 2010, pp. 2729-2743.
%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 (formulas).
%H Alessandro Languasco and Alessandro Zaccagnini, <a href="https://projecteuclid.org/euclid.facm/1269437065">On the constant in the Mertens product for arithmetic progressions. I. Identities</a>, Funct. Approx. Comment. Math. Volume 42, Number 1 (2010), 17-27, (<a href="https://arxiv.org/abs/0706.2807">preliminary version</a>).
%H Richard J. Mathar, <a href="https://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.
%F Equals 6*sqrt(5)/(13*A340628).
%F Equals A340004^2/A340808. - _R. J. Mathar_, Jan 15 2021
%F 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
%F Equals Sum_{q in A004615} 2^A001221(q)/q^2. - _R. J. Mathar_, Jan 27 2021
%e 1.0218780604187566757444489146002708261704607377325...
%p 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.
%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 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]);
%t $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 *)
%t digits = 121; digitize[c_] := RealDigits[Chop[N[c, digits]], 10, digits][[1]];
%t cl[x_] := I (PolyLog[2, (-1)^x] - PolyLog[2, -(-1)^(1 - x)]);
%t A340629 := (15 Sqrt[65]/(26 Pi^2)) Sqrt[cl[2/5]^2 + cl[4/5]^2];
%t digitize[A340629] (* _Peter Luschny_, Jan 23 2021 *)
%Y Cf. A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340577, A340578, A340628.
%K nonn,cons
%O 1,3
%A _Artur Jasinski_, Jan 13 2021
%E Corrected and more terms from _Vaclav Kotesovec_, Jan 15 2021
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