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A175645 Decimal expansion of the sum 1/p^3 over primes == 1 (mod 3). 12
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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...

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]

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

EXAMPLE

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

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 *)

PROG

(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

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)).

Sequence in context: A068635 A156695 A330251 * A178514 A154924 A071105

Adjacent sequences:  A175642 A175643 A175644 * A175646 A175647 A175648

KEYWORD

cons,nonn

AUTHOR

R. J. Mathar, Aug 01 2010

EXTENSIONS

More digits from Vaclav Kotesovec, Jun 27 2020

STATUS

approved

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Last modified July 29 07:26 EDT 2021. Contains 346340 sequences. (Running on oeis4.)