A086032 -id:A086032 - cp's OEIS frontend

A085548 Decimal expansion of the prime zeta function at 2: Sum_{p prime} 1/p^2.

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

Offset: 0

Cino Hilliard, Jul 03 2003

Mathar's Table 1 (cited below) lists expansions of the prime zeta function at integers s in 10..39. - Jason Kimberley, Jan 05 2017

			0.4522474200410654985065... = 1/2^2 + 1/3^2 + 1/5^2 +1/7^2 + 1/11^2 + 1/13^2 + ...

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, pp. 94-98.

Jason Kimberley, Table of n, a(n) for n = 0..1093
Henri Cohen, High Precision Computation of Hardy-Littlewood Constants, Preprint, 1998.
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
Persi Diaconis, Frederick Mosteller, and Hironari Onishi, Second-order terms for the variances and covariances of the number of prime factors-including the square free case, J. Number Theory 9 (1977), no. 2, 187--202. MR0434991 (55 #7953).
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 171 and 190.
J. W. L. Glaisher, On the Sums of Inverse Powers of the Prime Numbers, Quart. J. Math. 25, 347-362, 1891.
Xavier Gourdon and Pascal Sebah, Some Constants from Number theory.
Lajos Hajdu and Rob Tijdeman, Integers represented by Lucas sequences, arXiv:2408.04982 [math.NT], 2024. See p. 16.
Shanta Laishram and Florian Luca, Rectangles Of Nonvisible Lattice Points, J. Int. Seq. 18 (2015), Article 15.10.8, Theorem 1.
Jon Lee, Joseph Paat, Ingo Stallknecht, and Luze Xu, Polynomial upper bounds on the number of differing columns of Delta-modular integer programs, arXiv:2105.08160 [math.OC], 2021, see page 23.
R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], 2008-2009. Table 1.
Gerhard Niklasch and Pieter Moree, Some number-theoretical constants. [Cached copy]
Michael Ian Shamos, Property Enumerators and a Partial Sum Theorem, 2011; alternative link.
Hanson Smith, Ramification in the Division Fields of Elliptic Curves and an Application to Sporadic Points on Modular Curves, arXiv:1810.04809 [math.NT], 2018.
Hanson Smith, Ramification in Division Fields and Sporadic Points on Modular Curves, U. Conn. (2020).
Eric Weisstein's World of Mathematics, Distinct Prime Factors.
Eric Weisstein's World of Mathematics, Prime Sums.
Eric Weisstein's World of Mathematics, Prime Zeta Function.
Wikipedia, Prime Zeta Function.
Index to constants which are prime zeta sums {2,0,0}

Decimal expansion of the prime zeta function: this sequence (at 2), A085541 (at 3), A085964 (at 4) to A085969 (at 9).
Cf. A136271 (derivative), A117543 (semiprimes), A222056, A209329, A124012.
Cf. A000720, A001248, A013661, A078437, A242301.

R := RealField(106);
PrimeZeta := func;
Reverse(IntegerToSequence(Floor(PrimeZeta(2,173)*10^105)));
// Jason Kimberley, Dec 30 2016

RealDigits[PrimeZetaP[2], 10, 105][[1]]  (* Jean-François Alcover, Jun 24 2011, updated May 06 2021 *)

recip2(n) = { v=0; p=1; forprime(y=2,n, v=v+1./y^2; ); print(v) }

eps()=my(p=default(realprecision)); precision(2.>>(32*ceil(p*38539962/371253907)),9)
lm=lambertw(log(4)/eps())\log(4);
sum(k=1,lm, moebius(k)/k*log(abs(zeta(2*k)))) \\ Charles R Greathouse IV, Jul 19 2013

sumeulerrat(1/p,2) \\ Hugo Pfoertner, Feb 03 2020

P(2) = Sum_{p prime} 1/p^2 = Sum_{n>=1} mobius(n)*log(zeta(2*n))/n. - Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003
Equals A085991 + A086032 + 1/4. - R. J. Mathar, Jul 22 2010
Equals Sum_{k>=1} 1/A001248(k). - Amiram Eldar, Jul 27 2020
Equals Sum_{k>=2} pi(k)*(2*k+1)/(k^2*(k+1)^2), where pi(k) = A000720(k) (Shamos, 2011, p. 9). - Amiram Eldar, Mar 12 2024

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

Offset corrected by R. J. Mathar, Feb 05 2009

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

Original entry on oeis.org

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.

A175644 Decimal expansion of the sum 1/p^2 over primes p == 1 (mod 3).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Aug 01 2010

The prime zeta modulo function at 2 for primes of the form 3k+1, which is P_{3,2}(2) = Sum_{p in A002476} 1/p^2 = 1/7^2 + 1/13^2 + 1/19^2 + 1/31^2 + ...

The complementary Sum_{p in A003627} 1/p^2 is given by P_{3,2}(2) = A085548 - 1/3^2 - (this value here) = 0.307920758607736436842505075940... = A343612.

			P_{3,1}(2) = 0.03321555032221795055292717778013809648108756665...

Jean-François Alcover, Table of n, a(n) for n = 0..1003
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.
OEIS index to entries related to the (prime) zeta function.

Cf. A086032 (P_{4,1}(2): same for p==1 (mod 4)), A175645 (P_{3,1}(3): same for 1/p^3), A343612 (P_{3,2}(2): same for p==2 (mod 3)), A085548 (PrimeZeta(2)).

With[{s=2}, 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}]] (* Vaclav Kotesovec, Jan 13 2021 *)
digits = 1003;
m = 100; (* initial value of n beyond which summand is considered negligible *)
dm = 100; (* increment of m *)
P[s_, m_] (* "P" short name for "PrimeZeta31" *):= P[s, m] = Sum[Module[{t}, t = s + 2 n*s; MoebiusMu[2n + 1]* ((1/(4n + 2)) (-Log[1 + 2^t] - Log[1 + 3^t] + Log[Zeta[t]] - Log[Zeta[2t]] + Log[Zeta[t, 1/6] - Zeta[t, 5/6]]))], {n, 0, m}] // N[#, digits+10]&;
P[2, m]; P[2, m += dm];
While[ RealDigits[P[2,    m]][[1]][[1;;digits]] !=
       RealDigits[P[2, m-dm]][[1]][[1;;digits]], Print["m = ", m]; m+=dm];
Join[{0}, RealDigits[P[2, m]][[1]][[1;;digits]]] (* Jean-François Alcover, Jun 24 2022, after Vaclav Kotesovec *)

my(s=0); forprimestep(p=1, 1e8, 3, s+=1./p^2); s \\ For illustration: primes up to 10^N give only about 2N+2 (= 18 for N=8) correct digits. - M. F. Hasler, Apr 23 2021

PrimeZeta31(s)=suminf(n=0,my(t=s+2*n*s); moebius(2*n+1)*log((zeta(t)*(zetahurwitz(t,1/6)-zetahurwitz(t,5/6)))/((1+2^t)*(1+3^t)*zeta(2*t)))/(4*n+2)) \\ Inspired from Kotesovec's Mmca code
A175644_upto(N=100)={localprec(N+5);digits((PrimeZeta31(2)+1)\.1^N)[^1]} \\ M. F. Hasler, Apr 23 2021

More digits from Vaclav Kotesovec, Jun 27 2020

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

A343612 Decimal expansion of P_{3,2}(2) = Sum 1/p^2 over primes == 2 (mod 3).

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 22 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.

			0.30792075860773643684250507594099872658103266547551448005201925299378554901...

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.
OEIS index to entries related to the (prime) zeta function.

Cf. A003627 (primes 3k-1), A085548 (PrimeZeta(2)), A021031 (1/27).

Cf. A175644 (same for primes 3k+1), A086032 (for primes 4k+1), A085991 (for primes 4k+3), A343613 - A343619 (P_{3,2}(s): same with 1/p^s, s = 3, ..., 9).

digits = 105; nmax0 = 20; dnmax = 5;
Clear[PrimeZeta31];
PrimeZeta31[s_, nmax_] := PrimeZeta31[s, nmax] = Sum[Module[{t}, t = s + 2 n*s; MoebiusMu[2n + 1] ((1/(4n + 2)) (-Log[1 + 2^t] - Log[1 + 3^t] + Log[Zeta[t]] - Log[Zeta[2t]] + Log[Zeta[t, 1/6] - Zeta[t, 5/6]]))], {n, 0, nmax}] // N[#, digits+5]&;
PrimeZeta31[2, nmax = nmax0];
PrimeZeta31[2, nmax += dnmax];
While[Abs[PrimeZeta31[2, nmax] - PrimeZeta31[2, nmax-dnmax]] > 10^-(digits+5), Print["nmax = ", nmax]; nmax += dnmax];
PrimeZeta32[2] = PrimeZetaP[2] - 1/3^2 - PrimeZeta31[2, nmax];
RealDigits[PrimeZeta32[2], 10, digits][[1]] (* Jean-François Alcover, May 06 2021, after M. F. Hasler's PARI code *)

s=0; forprimestep(p=2,1e8,3,s+=1./p^2);s \\ For illustration: using primes up to 10^N gives about 2N+2 (= 18 for N=8) correct digits.
PrimeZeta32(s)={sumeulerrat(1/p^s)-1/3^s-suminf(n=0, my(t=s+2*n*s); moebius(2*n+1)*log((zeta(t)*(zetahurwitz(t, 1/6)-zetahurwitz(t, 5/6)))/((1+2^t)*(1+3^t)*zeta(2*t)))/(4*n+2))}
A343612_upto(N=100)={localprec(N+5); digits(PrimeZeta32(2)\.1^N)}

P_{3,2}(2) = P(2) - 1/3^2 - P_{3,1}(2) = A085548 - A000012 - A175644.

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

cp's OEIS Frontend

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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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A343612 Decimal expansion of P_{3,2}(2) = Sum 1/p^2 over primes == 2 (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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