A343612 - OEIS

%I #26 May 07 2021 08:07:47

%S 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,

%T 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,

%U 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

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

%C 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.

%H Jean-François Alcover, <a href="/A343612/b343612.txt">Table of n, a(n) for n = 0..1005</a>

%H R. J. Mathar, <a href="http://arxiv.org/abs/1008.2547">Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli</a>, arXiv:1008.2547 [math.NT], 2010-2015.

%H <a href="/index/Z#zeta_function">OEIS index to entries related to the (prime) zeta function</a>.

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

%e 0.30792075860773643684250507594099872658103266547551448005201925299378554901...

%t digits = 105; nmax0 = 20; dnmax = 5;

%t Clear[PrimeZeta31];

%t 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]&;

%t PrimeZeta31[2, nmax = nmax0];

%t PrimeZeta31[2, nmax += dnmax];

%t While[Abs[PrimeZeta31[2, nmax] - PrimeZeta31[2, nmax-dnmax]] > 10^-(digits+5), Print["nmax = ", nmax]; nmax += dnmax];

%t PrimeZeta32[2] = PrimeZetaP[2] - 1/3^2 - PrimeZeta31[2, nmax];

%t RealDigits[PrimeZeta32[2], 10, digits][[1]] (* _Jean-François Alcover_, May 06 2021, after _M. F. Hasler_'s PARI code *)

%o (PARI)

%o 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.

%o 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))}

%o A343612_upto(N=100)={localprec(N+5); digits(PrimeZeta32(2)\.1^N)}

%Y Cf. A003627 (primes 3k-1), A085548 (PrimeZeta(2)), A021031 (1/27).

%Y 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).

%K nonn,cons

%O 0,1

%A _M. F. Hasler_, Apr 22 2021