A086039 -id:A086039 - cp's OEIS frontend

A086032 Decimal expansion of the prime zeta modulo function at 2 for primes of the form 4k+1.

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

Offset: 0

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

			0.053813763574057670280678287341536562285675501495085532293911422295866827...

Jean-François Alcover, Table of n, a(n) for n = 0..1004
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=1,s=2).
X. Gourdon and P. Sebah, Some Constants from Number theory.
OEIS index to entries related to the (prime) zeta function.

Cf. A085991 (same for primes 4k+3), A175644 (for primes 3k+1), A343612 (for primes 3k+2), A086033 - A086039 (for 1/p^3, ..., 1/p^9), A085548 (PrimeZeta(2)), A002144 (primes 4k+1).

digits = 1004; nmax0 = 100; dnmax = 10;
Clear[PrimeZeta41];
f[s_] := (1 + 2^-s)^-1*DirichletBeta[s] Zeta[s]/Zeta[2s];
PrimeZeta41[s_, nmax_] := PrimeZeta41[s, nmax] = (1/2) Sum[MoebiusMu[2n + 1]* Log[f[(2n + 1)*2]]/(2n + 1), {n, 0, nmax}] // N[#, digits+5]&;
PrimeZeta41[2, nmax = nmax0];
PrimeZeta41[2, nmax += dnmax];
While[Abs[PrimeZeta41[2, nmax] - PrimeZeta41[2, nmax - dnmax]] > 10^-(digits + 5), Print["nmax = ", nmax]; nmax += dnmax];
PrimeZeta41[2] = PrimeZeta41[2, nmax];
Join[{0}, RealDigits[PrimeZeta41[2], 10, digits][[1]]] (* Jean-François Alcover, Jun 24 2011, after X. Gourdon and P. Sebah, updated May 06 2021 *)

PrimeZeta41(s)={suminf(n=0, my(t=s+s*n*2); moebius(2*n+1)*log(zeta(t)/zeta(2*t)*(zetahurwitz(t,1/4)-zetahurwitz(t,3/4))/(4^t+2^t))/(4*n+2))}
A086032_upto(N=100)={localprec(N+3);digits((PrimeZeta41(2)+1)\.1^N)[^1]} \\  M. F. Hasler, Apr 24 2021

Zeta_Q(2) = Sum_{p in A002144} 1/p^2, where A002144 = {primes p == 1 (mod 4)}.
Equals A085548 - 1/4 - A085991. - R. J. Mathar, Apr 03 2011
Zeta_Q(2) = Sum_{odd m > 0} mu(m)/2m * log(DirichletBeta(2m)*zeta(2m)/zeta(4m)/(1 + 4^-m)) [using Gourdon & Sebah, Theorem 11]. - M. F. Hasler, Apr 26 2021

A085998 Decimal expansion of the prime zeta modulo function at 9 for primes of the form 4k+3.

Original entry on oeis.org

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

Offset: 0

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

			0.000050830472150197889235259150923411189622380689881639399795... ~ 5.08...*10^-5

Jean-François Alcover, Table of n, a(n) for n = 0..1006
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=9), page 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A085991 .. A085997 (Zeta_R(2..8)).

Cf. A086039 (analog for primes 4k+1), A085969 (PrimeZeta(9)), A002145 (primes 4k+3).

digits = 1003;
nmax0 = 100;(* initial number of sum terms *)
dnmax = 10;(* nmax increment *)
dd = 10;(* precision excess *)
Clear[PrimeZeta43];
f[s_] := (1 - 2^(-s))*(Zeta[s]/DirichletBeta[s]);
PrimeZeta43[s_, nmax_] := PrimeZeta43[s, nmax] = (1/2) Sum[MoebiusMu[2 n + 1]*Log[f[(2 n + 1)*9]]/(2 n + 1), {n, 0, nmax}] // N[#, digits + dd] &;
PrimeZeta43[9, nmax = nmax0];
PrimeZeta43[9, nmax += dnmax];
While[Abs[PrimeZeta43[9, nmax] - PrimeZeta43[9, nmax - dnmax]] > 10^-(digits + dd), Print["nmax = ", nmax]; nmax += dnmax];
PrimeZeta43[9] = PrimeZeta43[9, nmax];
Join[{0, 0, 0, 0}, RealDigits[PrimeZeta43[9], 10, digits][[1]]] (* Jean-François Alcover, Jun 22 2011, updated May 07 2021 *)

A085998_upto(N=100)={localprec(N+3); digits((PrimeZeta43(9)+1)\.1^N)[^1]} \\ see A085991 for the PrimeZeta43 function. - M. F. Hasler, Apr 25 2021

Zeta_R(9) = Sum_{primes p == 3 (mod 4)} 1/p^9
  = (1/2)*Sum_{n>=0} mobius(2*n+1) *log(b((2*n+1)*9))/(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

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).

A343629 Decimal expansion of the Prime Zeta modulo function P_{3,1}(9) = Sum 1/p^9 over primes p == 1 (mod 3).

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 23 2021

The Prime Zeta modulo function at 9 for primes of the form 3k+1 is Sum_{primes in A002476} 1/p^9 = 1/7^9 + 1/13^9 + 1/19^9 + 1/31^9 + ...

The complementary Sum_{primes in A003627} 1/p^8 is given by P_{3,2}(8) = A085969 - 1/3^9 - (this value here) = 0.0039088148233885949714061... = A343609.

			P_{3,1}(9) = 2.4878378446082135873838215937876340672308259947340815...*10^-8

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, p.21.

Cf. A086039 (P_{4,1}(9): same for p==1 (mod 4)), A175645, A343624 - A343628 (P_{3,1}(3..8): same for 1/p^n, n = 3..8), A343609 (P_{3,2}(9): same for p==2 (mod 3)).

Cf. A085969 (PrimeZeta(9)), A002476 (primes of the form 3k+1).

With[{s=9}, Do[Print[N[1/2 * Sum[(MoebiusMu[2*n + 1]/(2*n + 1)) * Log[(Zeta[s + 2*n*s]*(Zeta[s + 2*n*s, 1/6] - Zeta[s + 2*n*s, 5/6])) / ((1 + 2^(s + 2*n*s))*(1 + 3^(s + 2*n*s)) * Zeta[2*(1 + 2*n)*s])], {n, 0, m}], 120]], {m, 100, 500, 100}]] (* adopted from Vaclav Kotesovec's code in A175645 *)

s=0; forprimestep(p=1, 1e8, 3, s+=1./p^9); s \\ For illustration: primes up to 10^N give ~ 8N+2 (= 66 for N=8) correct digits.

A343629_upto(N=100)={localprec(N+5);digits((PrimeZeta31(9)+1)\.1^N)[^1]} \\ cf. A175644 for PrimeZeta31

A086033 Decimal expansion of the prime zeta modulo function at 3 for primes of the form 4k+1.

Original entry on oeis.org

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

Offset: 0

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

			0.008755082732970504494226765813746675051112061220425472440026374989908715100...

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=1, s=3).
OEIS index to entries related to the (prime) zeta function.

Cf. A085992 (same for primes 4k+3), A175645 (for primes 3k+1), A343613 (for primes 3k+2), A086032, ..., A086039 (for 1/p^2, ..., 1/p^9), A085541 (PrimeZeta(3)), A002144 (primes of the form 4k+1).

a[s_] = (1 + 2^-s)^-1* DirichletBeta[s] Zeta[s]/Zeta[2 s]; m = 110; $MaxExtraPrecision = 470; Join[{0, 0}, RealDigits[(1/2)* NSum[MoebiusMu[2n + 1]*Log[a[(2n + 1)*3]]/(2n + 1), {n, 0, m}, AccuracyGoal -> m, NSumTerms -> m, PrecisionGoal -> m, WorkingPrecision -> m]][[1]]][[1 ;; 105]] (* Jean-François Alcover, Jun 24 2011, after X. Gourdon and P. Sebah, updated Mar 14 2018 *)

A086033_upto(N=100)={localprec(N+3);digits((PrimeZeta41(3)+1)\.1^N)[^1]} \\ See A086032 for the function PrimeZeta41. - M. F. Hasler, Apr 24 2021

Zeta_Q(3) = Sum_{p in A002144} 1/p^3  where  A002144 = {primes p == 1 (mod 4)};
  = Sum_{odd m > 0} mu(m)/2m * log(DirichletBeta(3m)*zeta(3m)/zeta(6m)/(1+8^-m))) [using Gourdon & Sebah, Theorem 11]. - M. F. Hasler, Apr 26 2021
Equals A085541 - 1/2^3 - A085992. - R. J. Mathar, Apr 03 2011

A086035 Decimal expansion of the prime zeta modulo function at 5 for primes of the form 4k+1.

Original entry on oeis.org

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

Offset: 0

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

			0.0003234740342217975185119081860410839774427337057991473366959335726302601...

Jean-François Alcover, Table of n, a(n) for n = 0..1006
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=5), page 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A085994 (same for primes 4k+3), A343625 (for primes 3k+1), A343615 (for primes 3k+2), A086032, ..., A086039 (for 1/p^2, ..., 1/p^9), A085965 (PrimeZeta(5)), A002144 (primes of the form 4k+1).

digits = 1004; m0 = 50; dm = 10; dd = 10; Clear[f, g];
b[s_] := (1+2^-s)^-1 * DirichletBeta[s] Zeta[s]/Zeta[2s] // N[#, digits+dd]&;
f[n_] := f[n] = (1/2) MoebiusMu[2n + 1] Log[b[5(2n + 1)]]/(2n + 1);
g[m_] := g[m] = Sum[f[n], {n, 0, m}]; g[m = m0]; g[m += dm];
While[Abs[g[m] - g[m-dm]] < 10^(-digits-dd), Print[m]; m += dm];
Join[{0, 0, 0}, RealDigits[g[m], 10, digits][[1]]] (* Jean-François Alcover, Jun 24 2011, after X. Gourdon and P. Sebah, updated May 08 2021 *)

A086035_upto(N=100)={localprec(N+3); digits((PrimeZeta41(5)+1)\.1^N)[^1]} \\ see A086032 for the PrimeZeta41 function. - M. F. Hasler, Apr 26 2021

Zeta_Q(5) = Sum_{p in A002144} 1/p^5 where A002144 = {primes p == 1 (mod 4)};

= Sum_{odd m > 0} mu(m)/2m *log(DirichletBeta(5m)*zeta(5m)/zeta(10m)/(1+2^(-5m))) [using Gourdon & Sebah, Theorem 11]. - M. F. Hasler, Apr 26 2021

Edited by M. F. Hasler, Apr 26 2021

A086036 Decimal expansion of the prime zeta modulo function at 6 for primes of the form 4k+1.

Original entry on oeis.org

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

Offset: 0

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

			6.4250963664773791101819138043576598984545546978815052898566258438984520...*10^-5

Jean-François Alcover, Table of n, a(n) for n = 0..1006
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=6), page 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A085995 (same for primes 4k+3), A343626 (for primes 3k+1), A343616 (for primes 3k+2), A086032, ..., A086039 (for 1/p^2, ..., 1/p^9), A085966 (PrimeZeta(6)), A002144 (primes of the form 4k+1).

digits = 1003; m0 = 50; dm = 10; dd = 10; Clear[f, g];
b[s_] := (1 + 2^-s)^-1 DirichletBeta[s] Zeta[s]/Zeta[2s] // N[#, digits + dd]&;
f[n_] := f[n] = (1/2) MoebiusMu[2n + 1]*Log[b[(2n + 1)*6]]/(2n + 1);
g[m_] := g[m] = Sum[f[n], {n, 0, m}]; g[m = m0]; g[m += dm];
While[Abs[g[m] - g[m - dm]] < 10^(-digits - dd), Print[m]; m += dm];
Join[{0, 0, 0, 0}, RealDigits[g[m], 10, digits][[1]]] (* Jean-François Alcover, Jun 24 2011, after X. Gourdon and P. Sebah, updated May 08 2021 *)

A086036_upto(N=100)={localprec(N+3); digits((PrimeZeta41(6)+1)\.1^N)[^1]} \\ see A086032 for the PrimeZeta41 function. - M. F. Hasler, Apr 26 2021

Zeta_Q(6) = Sum_{p in A002144} 1/p^6 where A002144 = {primes p == 1 mod 4};

= Sum_{odd m > 0} mu(m)/2m * log(DirichletBeta(6m)*zeta(6m)/zeta(12m)/(1+2^(-6m))) [using Gourdon & Sebah, Theorem 11]. - M. F. Hasler, Apr 26 2021

Edited by M. F. Hasler, Apr 26 2021

A086038 Decimal expansion of the prime zeta modulo function at 8 for primes of the form 4k+1.

Original entry on oeis.org

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

Offset: 0

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

			2.56137168039646980824843231247393644726060180729887066675459917474121... * 10^-6

Jean-François Alcover, Table of n, a(n) for n = 0..1004
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=8), page 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A085997 (same for primes 4k+3), A343628 (for primes 3k+1), A343618 (for primes 3k+2), A086032 - A086039 (for 1/p^2, ..., 1/p^9), A085968 (PrimeZeta(8)), A002144 (primes of the form 4k+1).

digits = 1000; m0 = 50; dm = 10; dd = 10; Clear[f, g];
b[s_] := (1+2^-s)^-1*DirichletBeta[s] Zeta[s]/Zeta[2s] // N[#, digits+dd]&;
f[n_] := f[n] = (1/2) MoebiusMu[2n + 1] Log[b[8(2n + 1)]]/(2n + 1);
g[m_] := g[m] = Sum[f[n], {n, 0, m}] ; g[m = m0]; g[m += dm];
While[Abs[g[m] - g[m - dm]] < 10^(-digits - dd), Print[m]; m += dm];
Join[{0, 0, 0, 0, 0}, RealDigits[g[m], 10, digits][[1]]] (* Jean-François Alcover, Jun 24 2011, after X. Gourdon and P. Sebah, updated May 08 2021 *)

A086038_upto(N=100)={localprec(N+3); digits((PrimeZeta41(8)+1)\.1^N)[^1]} \\ see A086032 for the PrimeZeta41 function. - M. F. Hasler, Apr 26 2021

Zeta_Q(8) = Sum_{p in A002144} 1/p^8, where A002144 = {primes p == 1 (mod 4)};

= Sum_{odd m > 0} mu(m)/2m * log(DirichletBeta(8m)*zeta(8m)/zeta(16m)/(1+2^(-8m))) [using Gourdon & Sebah, Theorem 11]. - M. F. Hasler, Apr 26 2021

Edited by M. F. Hasler, Apr 26 2021

A086034 Decimal expansion of the prime zeta modulo function at 4 for primes of the form 4k+1.

Original entry on oeis.org

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

Offset: 0

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

			0.0016495841540292915989967613136388518274879099438347321478115258388...

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=4), page 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A085993 (same for primes 4k+3), A343624 (for primes 3k+1), A343614 (for primes 3k+2), A086032 - A086039 (for 1/p^2, ..., 1/p^9), A085541 (PrimeZeta(3)), A002144 (primes of the form 4k+1).

a[s_] = (1 + 2^-s)^-1* DirichletBeta[s] Zeta[s]/Zeta[2 s]; m = 120; $MaxExtraPrecision = 680; Join[{0, 0}, RealDigits[(1/2)* NSum[MoebiusMu[2n + 1]*Log[a[(2n + 1)*4]]/(2n + 1), {n, 0, m}, AccuracyGoal -> m, NSumTerms -> m, PrecisionGoal -> m, WorkingPrecision -> m]][[1]]][[1 ;; 105]] (* Jean-François Alcover, Jun 24 2011, after X. Gourdon and P. Sebah, updated Mar 14 2018 *)

A086034_upto(N=100)={localprec(N+3); digits((PrimeZeta41(4)+1)\.1^N)[^1]} \\ see A086032 for the PrimeZeta41 function. - M. F. Hasler, Apr 26 2021

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

= Sum_{odd m > 0} mu(m)/2m * log(DirichletBeta(4m)*zeta(4m)/zeta(8m)/(1+2^(-4m)))[using Gourdon & Sebah, Theorem 11] - M. F. Hasler, Apr 26 2021.

Edited by M. F. Hasler, Apr 26 2021

A086037 Decimal expansion of the prime zeta modulo function at 7 for primes of the form 4k+1.

Original entry on oeis.org

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

Offset: 0

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

			1.2818448599795268251026582166507935820606749563344794362656914682... * 10^-5

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=7), page 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A085996 (same for primes 4k+3), A343627 (for primes 3k+1), A343617 (for primes 3k+2), A086032, ..., A086039 (for 1/p^2, ..., 1/p^9), A085967 (PrimeZeta(7)), A002144 (primes of the form 4k+1).

a[s_] = (1 + 2^-s)^-1* DirichletBeta[s] Zeta[s]/Zeta[2 s]; m = 120; $MaxExtraPrecision = 1200; Join[{0, 0, 0, 0}, RealDigits[(1/2)* NSum[MoebiusMu[2n + 1]*Log[a[(2n + 1)*7]]/(2n + 1), {n, 0, m}, AccuracyGoal -> m, NSumTerms -> m, PrecisionGoal -> m, WorkingPrecision -> m]][[1]]][[1 ;; 105]] (* Jean-François Alcover, Jun 24 2011, after X. Gourdon and P. Sebah, updated Mar 14 2018 *)

A086037_upto(N=100)={localprec(N+3); digits((PrimeZeta41(7)+1)\.1^N)[^1]} \\ see A086032 for the PrimeZeta41 function. - M. F. Hasler, Apr 26 2021

Zeta_Q(7) = Sum_{p in A002144} 1/p^7 where A002144 = {primes p == 1 mod 4};

= Sum_{odd m > 0} mu(m)/2m*log(DirichletBeta(7m)*zeta(7m)/zeta(14m)/(1+2^(-7m))) [using Gourdon & Sebah, Theorem 11]. - M. F. Hasler, Apr 26 2021

Edited by M. F. Hasler, Apr 26 2021

cp's OEIS Frontend

A086032 Decimal expansion of the prime zeta modulo function at 2 for primes of the form 4k+1.

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A085998 Decimal expansion of the prime zeta modulo function at 9 for primes of the form 4k+3.

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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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A343629 Decimal expansion of the Prime Zeta modulo function P_{3,1}(9) = Sum 1/p^9 over primes p == 1 (mod 3).

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A086033 Decimal expansion of the prime zeta modulo function at 3 for primes of the form 4k+1.

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A086035 Decimal expansion of the prime zeta modulo function at 5 for primes of the form 4k+1.

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A086036 Decimal expansion of the prime zeta modulo function at 6 for primes of the form 4k+1.

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