A343613 -id:A343613 - cp's OEIS frontend

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

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

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

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A175645 Decimal expansion of the sum 1/p^3 over primes == 1 (mod 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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A086033 Decimal expansion of the prime zeta modulo function at 3 for primes of the form 4k+1.

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