A085998 -id:A085998 - cp's OEIS frontend

A085991 Decimal expansion of the prime zeta modulo function at 2 for primes of the form 4k+3.

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

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003

			0.14843365646700782822586507749... = 1/3^2 + 1/7^2 + 1/11^2 + 1/19^2 + 1/23^2 + ...

Jean-François Alcover, Table of n, a(n) for n = 0..999
P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.
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, section 3.2 constant P(m=4,n=3,s=2).

Cf. A086032 (analog for primes 4k+1), A085548 (PrimeZeta(2)), A002145.

Cf. A085992 .. A085998 (Zeta_R(3..9)).

digits = 1000; nmax0 = 500; dnmax = 10;
Clear[PrimeZeta43];
PrimeZeta43[s_, nmax_] := PrimeZeta43[s, nmax] = (1/2) Sum[(MoebiusMu[2n + 1] ((4n + 2) Log[2] + Log[((-1 + 2^(4n + 2)) Zeta[4n + 2])/(Zeta[4 n + 2, 1/4] - Zeta[4n + 2, 3/4])]))/(2n + 1), {n, 0, nmax}] // N[#, digits+5]&;
PrimeZeta43[2, nmax = nmax0];
PrimeZeta43[2, nmax += dnmax];
While[Abs[PrimeZeta43[2, nmax] - PrimeZeta43[2, nmax - dnmax]] > 10^-(digits+5), Print["nmax = ", nmax]; nmax += dnmax];
PrimeZeta43[2] = PrimeZeta43[2, nmax];
RealDigits[PrimeZeta43[2], 10, digits][[1]] (* Jean-François Alcover, Jun 21 2011, updated May 06 2021 *)

PrimeZeta43(s)={suminf(n=0, my(t=s+s*n*2); moebius(n*2+1)*log(zeta(t)/(zetahurwitz(t, 1/4)-zetahurwitz(t, 3/4))*(4^t-2^t))/(n*2+1))/2}
A085991_upto(N=100)={localprec(N+3); digits((PrimeZeta43(2)+1)\.1^N)[^1]} \\  M. F. Hasler, Apr 25 2021

Zeta_R(2) = Sum_{primes p == 3 (mod 4)} 1/p^2
  = (1/2)*Sum_{n>=0} mobius(2*n+1)*log(b((2*n+1)*2))/(2*n+1),
  where b(x)=(1-2^(-x))*zeta(x)/L(x) and L(x) is the Dirichlet Beta function.

A086039 Decimal expansion of the prime zeta modulo function at 9 for primes of the form 4k+1.

Original entry on oeis.org

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

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 07 2003

			5.121028122527738383259898597063472005396598569391504803757141806973300...* 10^-7

Jean-François Alcover, Table of n, a(n) for n = 0..1009
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, value P(m=4, n=1, s=9), page 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A085998 (same for primes 4k+3), A343629 (for primes 3k+1), A343619 (for primes 3k+2), A086032 - A086038 (for 1/p^2, ..., 1/p^8), A085969 (PrimeZeta(9)), A002144 (primes of the form 4k+1).

digits = 1004;
nmax0 = 50; (* initial number of sum terms *)
dnmax = 10; (* nmax increment *)
dd = 10; (* precision excess *)
Clear[PrimeZeta41];
f[s_] := (1 + 2^-s)^-1 * DirichletBeta[s] Zeta[s]/Zeta[2 s];
PrimeZeta41[s_, nmax_] := PrimeZeta41[s, nmax] = (1/2) Sum[MoebiusMu[2 n + 1]*Log[f[(2 n + 1)*9]]/(2 n + 1), {n, 0, nmax}] // N[#, digits + dd]&;
PrimeZeta41[9, nmax = nmax0];
PrimeZeta41[9, nmax += dnmax];
While[Abs[PrimeZeta41[9, nmax] - PrimeZeta41[9, nmax - dnmax]] > 10^-(digits + dd), Print["nmax = ", nmax]; nmax += dnmax];
PrimeZeta41[9] = PrimeZeta41[9, nmax];
Join[{0, 0, 0, 0, 0, 0}, RealDigits[PrimeZeta41[9], 10, digits][[1]]] (* Jean-François Alcover, Jun 24 2011, after X. Gourdon and P. Sebah, updated May 07 2021 *)

A086039_upto(N=100)={localprec(N+3); digits((PrimeZeta41(9)+1)\.1^N)[^1]} \\ see A086032 for the PrimeZeta41 function. - M. F. Hasler, Apr 26 2021

Zeta_Q(9) = Sum_{p in A002144} 1/p^9, where A002144 = {primes p == 1 mod 4};

= Sum_{odd m > 0} mu(m)/2m * log(DirichletBeta(9m)*zeta(9m)/zeta(18m)/(1 + 2^(-9m))) [using Gourdon & Sebah, Theorem 11]. - M. F. Hasler, Apr 26 2021

Edited by M. F. Hasler, Apr 26 2021

A343619 Decimal expansion of P_{3,2}(9) = Sum 1/p^9 over primes == 2 (mod 3).

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 25 2021

The prime zeta modulo function P_{m,r}(s) = Sum_{primes p == r (mod m)} 1/p^s generalizes the prime zeta function P(s) = Sum_{primes p} 1/p^s.

			0.0019536374331587137208046015123929176069335003912220646291626134042468494...

Jean-François Alcover, Table of n, a(n) for n = 0..1005
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, value P(m=3, n=2, s=9), p. 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A003627 (primes 3k-1), A001017 (n^9), A085969 (PrimeZeta(9)).
Cf. A343612 - A343618 (P_{3,2}(s): analog for 1/p^s, s = 2 .. 8).
Cf. A343629 (for primes 3k+1), A086039 (for primes 4k+1), A085998 (for primes 4k+3).

digits = 1004; nmax0 = 50; dnmax = 10;
Clear[PrimeZeta31];
PrimeZeta31[s_, nmax_] := PrimeZeta31[s, nmax] = Sum[Module[{t}, t = s + 2 n*s; MoebiusMu[2 n + 1] ((1/(4 n + 2)) (-Log[1 + 2^t] - Log[1 + 3^t] + Log[Zeta[t]] - Log[Zeta[2 t]] + Log[Zeta[t, 1/6] - Zeta[t, 5/6]]))], {n, 0, nmax}] // N[#, digits + 5] &;
PrimeZeta31[9, nmax = nmax0];
PrimeZeta31[9, nmax += dnmax];
While[Abs[PrimeZeta31[9, nmax] - PrimeZeta31[9, nmax - dnmax]] > 10^-(digits + 5), Print["nmax = ", nmax]; nmax += dnmax];
PrimeZeta32[9] = PrimeZetaP[9] - 1/3^9 - PrimeZeta31[9, nmax];
Join[{0, 0}, RealDigits[PrimeZeta32[9], 10, digits][[1]] ] (* Jean-François Alcover, May 07 2021, after M. F. Hasler's PARI code *)

A343619_upto(N=100)={localprec(N+5); digits((PrimeZeta32(9)+1)\.1^N)[^1]} \\ see A343612 for the function PrimeZeta32

P_{3,2}(9) = Sum_{p in A003627} 1/p^9 = P(9) - 1/3^9 - P_{3,1}(9).

A085992 Decimal expansion of the prime zeta modulo function at 3 for primes of the form 4k+3.

Original entry on oeis.org

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

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003

			0.04100755656647303192888654885196002592430006070572381744864564171...

P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.
P. Fortuny Ayuso, J. M. Grau, and A. Oller-Marcen, A von Staudt-type formula for sum_{z in Z_n[i]} z^k, arXiv:1402.0333 [math.NT], 2014.
X. Gourdon and P. Sebah, Some Constants from Number theory.
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo functions..., arXiv:1008.2547 [math.NT], 2010-2015, value P(m=4, s=3, n=3), page 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A085991 .. A085998 (Zeta_R(2..9)).

Cf. A086033 (analog for primes 4k+1), A085541 (PrimeZeta(3)), A002145 (primes 4k+3).

b[x_] = (1 - 2^(-x))*(Zeta[x] / DirichletBeta[x]); $MaxExtraPrecision = 200; m = 40; Prepend[ RealDigits[(1/2)* NSum[MoebiusMu[2n+1]* Log[b[(2n+1)*3]]/(2n+1), {n, 0, m}, AccuracyGoal -> 120, NSumTerms -> m, PrecisionGoal -> 120, WorkingPrecision -> 120] ][[1]], 0][[1 ;; 105]] (* Jean-François Alcover, Jun 21 2011, updated Mar 14 2018 *)

A085992_upto(N=100)={localprec(N+3); digits((PrimeZeta43(3)+1)\.1^N)[^1]} \\ see A085991 for the PrimeZeta43 function. - M. F. Hasler, Apr 25 2021

Zeta_R(3) = Sum_{primes p == 3 (mod 4)} 1/p^3
  = (1/2)*Sum_{n>=0} mobius(2*n+1)*log(b((2*n+1)*3))/(2*n+1),
  where b(x) = (1-2^(-x))*zeta(x)/L(x) and L(x) is the Dirichlet Beta function.

Edited by M. F. Hasler, Apr 25 2021

A085993 Decimal expansion of the prime zeta modulo function at 4 for primes of the form 4k+3.

Original entry on oeis.org

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

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003

			0.012843555610217553343622534619519018334553149771008458117126483020416...

P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.
X. Gourdon and P. Sebah, Some Constants from Number theory.
R. J. Mathar, Table of ... Prime Zeta Modulo functions..., arXiv:1008.2547 [math.NT], 2010-2015, value P(m=4, n=3, s=4), page 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A085991 .. A085998 (Zeta_R(2..9)).

Cf. A086034 (analog for primes 4k+1), A085964 (PrimeZeta(4)), A002145 (primes 4k+3).

b[x_] = (1 - 2^(-x))*(Zeta[x]/DirichletBeta[x]); $MaxExtraPrecision = 200; m = 40; Prepend[ RealDigits[ (1/2)*NSum[ MoebiusMu[2n+1]* Log[b[(2n+1)*4]]/(2n+1), {n, 0, m}, AccuracyGoal -> 120, NSumTerms -> m, PrecisionGoal -> 120, WorkingPrecision -> 120] ][[1]], 0][[1 ;; 105]] (* Jean-François Alcover, Jun 22 2011, updated Mar 14 2018 *)

A085993_upto(N=100)={localprec(N+3); digits((PrimeZeta43(4)+1)\.1^N)[^1]} \\ see A085991 for the PrimeZeta43 function. - M. F. Hasler, Apr 25 2021

Zeta_R(4) = Sum_{primes p == 3 mod 4} 1/p^4
  = (1/2)*Sum_{n >= 0} mobius(2*n+1)*log(b((2*n+1)*4))/(2*n+1),
  where b(x) = (1-2^(-x))*zeta(x)/L(x) and L(x) is the Dirichlet Beta function.

Edited by M. F. Hasler, Apr 25 2021

A085994 Decimal expansion of the prime zeta modulo function at 5 for primes of the form 4k+3.

Original entry on oeis.org

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

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003

			0.004181543449702459614306334352814627154254543020852184353396741251345574...

P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.
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, value P(m=4, n=3, s=5), page 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A085991 .. A085998 (Zeta_R(2..9)).

Cf. A086035 (analog for primes 4k+1), A085965 (PrimeZeta(5)), A002145 (primes 4k+3).

b[x_] = (1 - 2^(-x))*(Zeta[x]/DirichletBeta[x]); $MaxExtraPrecision = 200; m = 40; Join[{0, 0}, RealDigits[(1/2)*NSum[MoebiusMu[2n + 1]* Log[b[(2n + 1)*5]]/(2n + 1), {n, 0, m}, AccuracyGoal -> 120, NSumTerms -> m, PrecisionGoal -> 120, WorkingPrecision -> 120] ][[1]]][[1 ;; 105]] (* Jean-François Alcover, Jun 22 2011, updated Mar 14 2018 *)

A085994_upto(N=100)={localprec(N+3); digits((PrimeZeta43(5)+1)\.1^N)[^1]} \\ see A085991 for the PrimeZeta43 function. - M. F. Hasler, Apr 25 2021

Zeta_R(5) = Sum_{primes r == 3 mod 4} 1/p^5
  = (1/2)*Sum_{n=0..inf} mobius(2*n+1)*log(b((2*n+1)*5))/(2*n+1),
  where b(x) = (1-2^(-x))*zeta(x)/L(x) and L(x) is the Dirichlet Beta function.

A085995 Decimal expansion of the prime zeta modulo function at 6 for primes of the form 4k+3.

Original entry on oeis.org

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

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003

			0.0013808358869717391630318541280158226106013963275654296802648025785307522...

P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.
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, value P(m=4, n=3, s=6), page 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A002145 (primes 4k+3), A001014 (n^6), A085966 (PrimeZeta(6)).

Cf. A085991 - A085998 (Zeta_R(2..9): same for 1/p^2, ..., 1/p^9), A086036 (same for primes 4k+1), A343626 (for primes 3k+1), A343616 (for primes 3k+2).

b[x_] = (1 - 2^(-x))*(Zeta[x]/DirichletBeta[x]); $MaxExtraPrecision = 250; m = 40; Join[{0, 0}, RealDigits[(1/2)*NSum[MoebiusMu[2n + 1]* Log[b[(2n + 1)*6]]/(2n + 1), {n, 0, m}, AccuracyGoal -> 120, NSumTerms -> m, PrecisionGoal -> 120, WorkingPrecision -> 120] ][[1]]][[1 ;; 105]] (* Jean-François Alcover, Jun 22 2011, updated Mar 14 2018 *)

A085995_upto(N=100)={localprec(N+3); digits((PrimeZeta43(6)+1)\.1^N)[^1]} \\ see A085991 for the PrimeZeta43 function. - M. F. Hasler, Apr 25 2021

Zeta_R(6) = Sum_{p in A002145} 1/p^6 where A002145 = {primes p == 3 (mod 4)},
  = (1/2)*Sum_{n >= 0} möbius(2*n+1)*log(b((2*n+1)*6))/(2*n+1),
  where b(x) = (1-2^(-x))*zeta(x)/L(x) and L(x) is the Dirichlet Beta function.

Edited by M. F. Hasler, Apr 25 2021

A085997 Decimal expansion of the prime zeta modulo function at 8 for primes of the form 4k+3.

Original entry on oeis.org

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

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003

			0.000152593994837434090715190710370606586529883910264442130365934082553889...

P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.
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, value P(m=4, n=3, s=8), page 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A086038 (analog for primes 4k+1), A085968 (PrimeZeta(8)), A002145 (primes 4k+3).

Cf. A085991 .. A085998 (Zeta_R(2..9)).

b[x_] = (1 - 2^(-x))*(Zeta[x]/DirichletBeta[x]); $MaxExtraPrecision = 320; m = 40; Join[{0, 0, 0}, RealDigits[(1/2)* NSum[MoebiusMu[2n + 1]* Log[b[(2n + 1)*8]]/(2n + 1), {n, 0, m}, AccuracyGoal -> 120, NSumTerms -> m, PrecisionGoal -> 120, WorkingPrecision -> 120] ][[1]]][[1 ;; 105]] (* Jean-François Alcover, Jun 22 2011, updated Mar 14 2018 *)

A085997_upto(N=100)={localprec(N+3); digits((PrimeZeta43(8)+1)\.1^N)[^1]} \\ see A085991 for the PrimeZeta43 function. - M. F. Hasler, Apr 25 2021

Zeta_R(8) = Sum_{primes p == 3 mod 4} 1/p^8
  = (1/2)*Sum_{n=0..inf} mobius(2*n+1)*log(b((2*n+1)*8))/(2*n+1),
  where b(x) = (1-2^(-x))*zeta(x)/L(x) and L(x) is the Dirichlet Beta function.

Edited by M. F. Hasler, Apr 25 2021

A085996 Decimal expansion of the prime zeta modulo function at 7 for primes of the form 4k+3.

Original entry on oeis.org

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

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003

			0.0004585144075337972668731121472822151533627221357444614502792647239732950115...

P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.
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, value P(m=4, n=3, s=7), page 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A086037 (analog for primes 4k+1), A085967 (PrimeZeta(7)), A002145 (primes 4k+3).

Cf. A085991 .. A085998 (Zeta_R(2..9)).

b[x_] = (1 - 2^(-x))*(Zeta[x]/DirichletBeta[x]); $MaxExtraPrecision = 275; m = 40; Join[{0, 0, 0}, RealDigits[(1/2)* NSum[MoebiusMu[2n + 1]* Log[b[(2n + 1)*7]]/(2n + 1), {n, 0, m}, AccuracyGoal -> 120, NSumTerms -> m, PrecisionGoal -> 120, WorkingPrecision -> 120] ][[1]]][[1 ;; 105]] (* Jean-François Alcover, Jun 22 2011, updated Mar 14 2018 *)

A085996_upto(N=100)={localprec(N+3); digits((PrimeZeta43(7)+1)\.1^N)[^1]} \\ see A085991 for the PrimeZeta43 function. - M. F. Hasler, Apr 25 2021

Zeta_R(7) = Sum_{primes p == 3 mod 4} 1/p^7
  = (1/2)*Sum_{n=0..inf} mobius(2*n+1)*log(b((2*n+1)*7))/(2*n+1),
  where b(x) = (1-2^(-x))*zeta(x)/L(x) and L(x) is the Dirichlet Beta function.

Edited by M. F. Hasler, Apr 25 2021

cp's OEIS Frontend

A085991 Decimal expansion of the prime zeta modulo function at 2 for primes of the form 4k+3.

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A086039 Decimal expansion of the prime zeta modulo function at 9 for primes of the form 4k+1.

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A343619 Decimal expansion of P_{3,2}(9) = Sum 1/p^9 over primes == 2 (mod 3).

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A085992 Decimal expansion of the prime zeta modulo function at 3 for primes of the form 4k+3.

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A085993 Decimal expansion of the prime zeta modulo function at 4 for primes of the form 4k+3.

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A085994 Decimal expansion of the prime zeta modulo function at 5 for primes of the form 4k+3.

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A085995 Decimal expansion of the prime zeta modulo function at 6 for primes of the form 4k+3.

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