A085991 -id:A085991 - 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

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

Original entry on oeis.org

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

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.

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

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

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