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%I #33 Aug 10 2022 08:48:59
%S 8,9,4,8,4,1,2,2,4,5,6,2,4,8,8,1,7,0,7,2,5,6,6,1,5,0,6,9,0,8,4,3,7,3,
%T 2,1,9,8,7,5,4,7,8,0,8,9,2,0,7,1,8,9,7,2,6,0,1,7,9,9,4,2,7,6,1,6,5,6,
%U 3,8,9,2,2,1,2,0,9,1,5,5,0,2,8,8,5,9,4,2,9,1,0,5,3,9,5,8,9,1,0,8,0,0,3,3,2,2
%N Decimal expansion of Product_{p prime, p == 1 (mod 4)} (1 - 2/p^2).
%C The asymptotic density of the numbers k such that k^2+1 is squarefree (A049533) (Estermann, 1931).
%C The constant c in Sum_{k=0..n} phi(k^2 + 1) = A333170(n) ~ (1/4)*c*n^3 (Finch, 2018).
%C The constant c in Sum_{k=0..n} phi(k^2 + 1)/(k^2 + 1) = (3/4)*c*n + O(log(n)^2) (Postnikov, 1988).
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 101.
%D Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 166.
%D A. G. Postnikov, Introduction to Analytic Number Theory, Amer. Math. Soc., 1988, pp. 192-195.
%H Theodor Estermann, <a href="https://doi.org/10.1007/BF01455836">Einige Sätze über quadratfreie Zahlen</a>, Mathematische Annalen, Vol. 105 (1931), pp. 653-662, <a href="https://eudml.org/doc/159528">alternative link</a>.
%H Steven R. Finch, <a href="https://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, arXiv:2001.00578 [math.HO], 2020, p. 14.
%H D. R. Heath-Brown, <a href="https://doi.org/10.4064/aa155-1-1">Square-free values of n^2 + 1</a>, Acta Arithmetica, Vol. 155, No. 1 (2012), pp. 1-13.
%H 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], variable F(m=4,n=1,s=2), p. 38.
%H Wolfgang Schwarz, <a href="https://doi.org/10.1007/BF01418877">Über die Summe Sigma_{n <= x} phi(f)(n) und verwandte Probleme</a>, Monatshefte für Mathematik, Vol. 66, No. 1 (1962), pp. 43-54, <a href="https://eudml.org/doc/177151">alternative link</a>.
%H Radoslav Tsvetkov, <a href="http://nntdm.net/volume-25-2019/number-3/207-222/">On the distribution of k-free numbers and r-tuples of k-free numbers. A survey</a>, Notes on Number Theory and Discrete Mathematics, Vol. 25, No. 3 (2019), pp. 207-222.
%F Equals 2*A065474/A340617.
%e 0.89484122456248817072566150690843732198754780892071...
%p Digits := 150;
%p with(NumberTheory);
%p DirichletBeta := proc(s) (Zeta(0, s, 1/4) - Zeta(0, s, 3/4))/4^s; end proc;
%p alfa := proc(s) DirichletBeta(s)*Zeta(s)/((1 + 1/2^s)*Zeta(2*s)); end proc;
%p beta := proc(s) (1 - 1/2^s)*Zeta(s)/DirichletBeta(s); end proc;
%p pzetamod41 := proc(s, terms) 1/2*Sum(Moebius(2*j + 1)*log(alfa((2*j + 1)*s))/(2*j + 1), j = 0..terms); end proc;
%p evalf(exp(-Sum(2^t*pzetamod41(2*t, 50)/t, t = 1..200))); # _Vaclav Kotesovec_, Jan 13 2021
%t f[p_] := If[Mod[p, 4] == 1, 1 - 2/p^2, 1]; RealDigits[N[Product[f[Prime[i]], {i, 1, 10^6}], 10], 10, 8][[1]] (* for calculating only the first few terms *)
%t (* -------------------------------------------------------------------------- *)
%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 Z2[m_, n_, s_] := (w = 1; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = 2^w * P[m, n, s*w]/w; sumz = sumz + difz; w++]; Exp[-sumz]);
%t $MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Z2[4, 1, 2], digits]], 10, digits-1][[1]] (* _Vaclav Kotesovec_, Jan 15 2021 *)
%o (PARI) f(lim,poly=1-'x-'x^2/2)=prodeulerrat(subst(poly,'x,1/'x^2))*prodeuler(p=2,lim, my(pm2=1./p^2); if(p%4==1,1.-2*pm2,1.)/subst(poly,'x,pm2)) \\ Gets 14 digits at lim=1e9; _Charles R Greathouse IV_, Aug 10 2022
%Y Cf. A002144, A049533, A065474, A069987, A088539, A333169, A333170, A340617.
%K nonn,cons
%O 0,1
%A _Amiram Eldar_, Jul 01 2020
%E More digits (from the paper by _R. J. Mathar_) added by _Jon E. Schoenfield_, Jan 12 2021
%E More digits from _Vaclav Kotesovec_, Jan 13 2021
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