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

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, 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, 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

Offset

0,1

Links

Table of n, a(n) for n=0..105.

X. Gourdon and P. Sebah, Some Constants from Number theory

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, page 38 (case 4 3 2).

Formula

Equals 2*A065474/A335963.

Example

0.7210979782407524158324311775035064193323800948822709044864277469512...

Maple

Digits := 150;
with(NumberTheory);
DirichletBeta := proc(s) (Zeta(0, s, 1/4) - Zeta(0, s, 3/4))/4^s; end proc;
alfa := proc(s) DirichletBeta(s)*Zeta(s)/((1 + 1/2^s)*Zeta(2*s)); end proc;
beta := proc(s) (1 - 1/2^s)*Zeta(s)/DirichletBeta(s); end proc;
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;
evalf(exp(-Sum(2^t*pzetamod43(2*t, 70)/t, t = 1..200)));

Mathematica

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);
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}];
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]);
$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Z2[4, 3, 2], digits]], 10, digits-1][[1]]

Crossrefs

Cf. A002145, A065474, A085991, A243379, A335963.

Sequence in context: A351480 A378699 A153589 * A010505 A363360 A020844

Adjacent sequences: A340614 A340615 A340616 * A340618 A340619 A340620

Keyword

nonn,cons

Author

Vaclav Kotesovec, Jan 13 2021

Status

approved