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

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

Author

M. F. Hasler, Apr 22 2021

Keywords

• nonn
• cons

Comments

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.

Examples

			0.30792075860773643684250507594099872658103266547551448005201925299378554901...
		

Links

• 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.

Crossrefs

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).

Programs

Mathematica

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 *)

PARI

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)}

Formula

P_{3,2}(2) = P(2) - 1/3^2 - P_{3,1}(2) = A085548 - A000012 - A175644.