cp's OEIS Frontend

A340617 Decimal expansion of Product_{p prime, p == 3 (mod 4)} (1 - 2/p^2).

%I A340617 #21 Aug 25 2021 15:47:10
%S A340617 7,2,1,0,9,7,9,7,8,2,4,0,7,5,2,4,1,5,8,3,2,4,3,1,1,7,7,5,0,3,5,0,6,4,
%T A340617 1,9,3,3,2,3,8,0,0,9,4,8,8,2,2,7,0,9,0,4,4,8,6,4,2,7,7,4,6,9,5,1,2,7,
%U A340617 0,9,1,2,6,0,3,6,6,0,3,9,4,7,1,7,2,0,6,5,0,1,7,3,7,9,8,4,9,3,6,2,2,8,8,7,6,5
%N A340617 Decimal expansion of Product_{p prime, p == 3 (mod 4)} (1 - 2/p^2).
%H A340617 X. Gourdon and P. Sebah, <a href="http://numbers.computation.free.fr/Constants/Miscellaneous/constantsNumTheory.html">Some Constants from Number theory</a>
%H A340617 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, page 38 (case 4 3 2).
%F A340617 Equals 2*A065474/A335963.
%e A340617 0.7210979782407524158324311775035064193323800948822709044864277469512...
%p A340617 Digits := 150;
%p A340617 with(NumberTheory);
%p A340617 DirichletBeta := proc(s) (Zeta(0, s, 1/4) - Zeta(0, s, 3/4))/4^s; end proc;
%p A340617 alfa := proc(s) DirichletBeta(s)*Zeta(s)/((1 + 1/2^s)*Zeta(2*s)); end proc;
%p A340617 beta := proc(s) (1 - 1/2^s)*Zeta(s)/DirichletBeta(s); end proc;
%p A340617 pzetamod43 := proc(s, terms) 1/2*Sum(Moebius(2*j + 1)*log(beta((2*j + 1)*s))/(2*j + 1), j = 0..terms); end proc;
%p A340617 evalf(exp(-Sum(2^t*pzetamod43(2*t, 70)/t, t = 1..200)));
%t A340617 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 A340617 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 A340617 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 A340617 $MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Z2[4, 3, 2], digits]], 10, digits-1][[1]]
%Y A340617 Cf. A002145, A065474, A085991, A243379, A335963.
%K A340617 nonn,cons
%O A340617 0,1
%A A340617 _Vaclav Kotesovec_, Jan 13 2021