A085991 -id:A085991 - cp's OEIS frontend

A360095 Decimal expansion of Sum_{p primes, p == 3 (mod 4)} log(p)/p^2.

2, 1, 2, 4, 4, 4, 7, 6, 8, 9, 3, 1, 6, 6, 5, 0, 5, 7, 7, 0, 5, 0, 6, 7, 7, 9, 2, 6, 8, 2, 8, 2, 5, 2, 1, 4, 8, 7, 0, 3, 7, 3, 6, 9, 5, 8, 4, 3, 7, 6, 6, 6, 9, 7, 8, 1, 0, 4, 9, 7, 5, 3, 7, 1, 6, 7, 7, 0, 9, 5, 9, 7, 6, 0, 2, 0, 8, 1, 1, 5, 3, 5, 8, 9, 6, 1, 3, 7, 0, 5, 9, 6, 1, 4, 0, 7, 4, 3, 8, 3, 3, 7, 4, 4, 7, 3

Offset: 0

Vaclav Kotesovec, Jan 25 2023

			0.212444768931665057705067792682825214870373695843766697810497537167709...

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

Cf. A085548, A085991, A086239, A136271, A358789, A360094.

beta[s_]:= (1 - 1/2^s) * Zeta[s] / DirichletBeta[s]; Do[Print[N[-1/2*Sum[MoebiusMu[2*n + 1]/(2*n + 1) * D[Log[beta[(2*n + 1)*s]], s] /. s->2, {n, 0, m}], 120]], {m, 10, 100, 10}]

Equals A136271 - A360094 - log(2)/4.

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

Original entry on oeis.org

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

Vaclav Kotesovec, Jan 13 2021

			0.7210979782407524158324311775035064193323800948822709044864277469512...

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).

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

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)));

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]]

Equals 2*A065474/A335963.

A086240 Decimal expansion of Sum_{k>=2} (p mod 4 - 2)/p^2 where p=prime(k).

Original entry on oeis.org

0, 9, 4, 6, 1, 9, 8, 9, 2, 8, 9, 2, 9, 5, 0, 1, 5, 7, 9, 4, 5, 1, 8, 6, 7, 9, 0, 1, 4, 9, 1, 7, 4, 8, 0, 9, 6, 0, 1, 8, 8, 0, 3, 4, 0, 2, 4, 9, 7, 2, 1, 3, 5, 7, 1, 4, 8, 5, 9, 6, 0, 8, 5, 7, 5, 9, 4, 3, 1, 3, 7, 3, 2, 7, 5, 6, 2, 5, 5, 8, 4, 1, 6, 3, 9, 0, 4, 4, 2, 9, 3, 4, 6, 4, 3, 8, 0, 7, 9, 9, 6, 4, 2, 0, 8, 7

Offset: 0

Eric W. Weisstein, Jul 13 2003

			0.094619892892950157945186790149174809601880340249721357148596...

S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 94-98

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, variable S(m=4,s=2,r=2).
Eric Weisstein's World of Mathematics, Prime Sums

Equals A085991 - A086032. - Vaclav Kotesovec, Jun 19 2020

A missing digit inserted and more digits added by R. J. Mathar, Jul 28 2010

More digits from Vaclav Kotesovec, Jun 19 2020

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A360095 Decimal expansion of Sum_{p primes, p == 3 (mod 4)} log(p)/p^2.

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

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A086240 Decimal expansion of Sum_{k>=2} (p mod 4 - 2)/p^2 where p=prime(k).

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