A086033 -id:A086033 - cp's OEIS frontend

A085541 Decimal expansion of the prime zeta function at 3.

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

Offset: 0

Cino Hilliard, Jul 02 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

			0.1747626392994435364231...

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
J. W. L. Glaisher, On the Sums of Inverse Powers of the Prime Numbers, Quart. J. Math. 25, 347-362, 1891.

Jason Kimberley, Table of n, a(n) for n = 0..1497
Henri Cohen, High Precision Computation of Hardy-Littlewood Constants, Preprint, 1998.
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
X. Gourdon and P. Sebah, Some Constants from Number theory
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]
Eric Weisstein's World of Mathematics, Prime Zeta Function
Index to constants which are prime zeta sums {3,0,0}

Decimal expansion of the prime zeta function: A085548 (at 2), this sequence (at 3), A085964 (at 4) to A085969 (at 9).

Cf. A002117, A030078, A242302.

R := RealField(106);
PrimeZeta := func;
Reverse(IntegerToSequence(Floor(PrimeZeta(3,117)*10^105)));
// Jason Kimberley, Dec 30 2016

(* If Mathematica version >= 7.0 then RealDigits[PrimeZetaP[3]//N[#,105]&][[1]] else : *) m = 200; $MaxExtraPrecision = 200; PrimeZetaP[s_] := NSum[MoebiusMu[k]*Log[Zeta[k*s]]/k, {k, 1, m}, AccuracyGoal -> m, NSumTerms -> m, PrecisionGoal -> m, WorkingPrecision -> m]; RealDigits[PrimeZetaP[3]][[1]][[1 ;; 105]] (* Jean-François Alcover, Jun 24 2011 *)

recip3(n) = { v=0; p=1; forprime(y=2,n, v=v+1./y^3; ); print(v) }

sumeulerrat(1/p,3) \\ Hugo Pfoertner, Feb 03 2020

P(3) = Sum_{p prime} 1/p^3 = Sum_{n>=1} mobius(n)*log(zeta(3*n))/n. - Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003
Equals A086033 + A085992 + 1/8. - R. J. Mathar, Jul 22 2010
Equals Sum_{k>=1} 1/A030078(k). - Amiram Eldar, Jul 27 2020

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

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

A175645 Decimal expansion of the sum 1/p^3 over primes == 1 (mod 3).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Aug 01 2010

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

The complementary sum, Sum_{prime p in A003627} 1/p^3 is given by P_{3,2}(3) = A085541 - 1/3^3 - (this value here) = 0.13412517891546354042859932999943119899...

			P_{3,1}(3) = 0.00360042334694295895747694762923846494249516...

Jean-François Alcover, Table of n, a(n) for n = 0..1008
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. A086033 (P_{4,1}(3): same for p==1 (mod 4)), A175644 (P_{3,1}(2): same for 1/p^2), A343613 (P_{3,2}(3): same for p==2 (mod 3)), A085541 (PrimeZeta(3)).

(* A naive solution yielding 12 correct digits: *) s1 = s2 = 0.; Do[Switch[Mod[n, 3], 1, If[PrimeQ[n], s1 += 1/n^3], 2, If[PrimeQ[n], s2 += 1/n^3]], {n, 10^7}]; Join[{0, 0}, RealDigits[(PrimeZetaP[3] + s1 - s2 - 1/27)/2, 10, 12][[1]]] (* Jean-François Alcover, Mar 15 2018 *)
With[{s=3}, 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 *)
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}];
$MaxExtraPrecision = 1000; digits = 121; Join[{0,0}, RealDigits[Chop[N[P[3, 1, 3], digits]], 10, digits-1][[1]]] (* Vaclav Kotesovec, Jan 22 2021 *)

s=0; forprimestep(p=1,1e8,3,s+=1./p^3);s \\ for illustration only: primes up to 10^N give about 2N+2 correct digits. - M. F. Hasler, Apr 22 2021
A175645_upto(N=100)=localprec(N+5);digits((PrimeZeta31(3)+1)\.1^N)[^1] \\ Cf. A175644 for PrimeZeta31. - M. F. Hasler, Apr 23 2021

More digits from Vaclav Kotesovec, Jun 27 2020

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

A343613 Decimal expansion of P_{3,2}(3) = Sum 1/p^3 over primes == 2 (mod 3).

Original entry on oeis.org

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

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.

			0.134125178915463540428599329999431198995879919752168337370599106153853349956...

Jean-François Alcover, Table of n, a(n) for n = 0..1006
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=3) on p. 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A003627 (primes 3k-1), A085541 (PrimeZeta(3)), A021031 (1/27).

Cf. A175645 (same for p==1 (mod 3)), A086033 (for primes 4k+1), A085992 (for primes 4k+3), A343612 - A343619 (P_{3,2}(2..9): same for 1/p^2, ..., 1/p^9).

s=0;forprimestep(p=2,1e8,3,s+=1./p^3);s \\ For illustration: using primes up to 10^N gives about 2N+2 (= 18 for N=8) correct digits.

A343613_upto(N=100)={localprec(N+5); digits((PrimeZeta32(3)+1)\.1^N)[^1]} \\ see A343612 for the function PrimeZeta32.

P_{3,2}(3) = P(3) - 1/3^3 - P_{3,1}(3) = A085541 - A021031 - A175645.

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A085541 Decimal expansion of the prime zeta function at 3.

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

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A175645 Decimal expansion of the sum 1/p^3 over primes == 1 (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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A343613 Decimal expansion of P_{3,2}(3) = Sum 1/p^3 over primes == 2 (mod 3).

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