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

Author

R. J. Mathar, Aug 01 2010

Keywords

• cons
• nonn

Comments

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...

Examples

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

Links

• 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.

Crossrefs

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)).

Programs

Mathematica

(* 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 *)

PARI

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

Extensions

More digits from Vaclav Kotesovec, Jun 27 2020