Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #16 May 07 2021 08:07:55
%S 1,3,4,1,2,5,1,7,8,9,1,5,4,6,3,5,4,0,4,2,8,5,9,9,3,2,9,9,9,9,4,3,1,1,
%T 9,8,9,9,5,8,7,9,9,1,9,7,5,2,1,6,8,3,3,7,3,7,0,5,9,9,1,0,6,1,5,3,8,5,
%U 3,3,4,9,9,5,6,0,4,7,9,3,7,6,7,1,5,2,8,6,5,3,7,4,0,4,0,3,4,4,4,3,3,6,7,8,6
%N Decimal expansion of P_{3,2}(3) = Sum 1/p^3 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="/A343613/b343613.txt">Table of n, a(n) for n = 0..1006</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, value P(m=3, n=2, s=3) on p. 21.
%H <a href="/index/Z#zeta_function">OEIS index to entries related to the (prime) zeta function</a>.
%F P_{3,2}(3) = P(3) - 1/3^3 - P_{3,1}(3) = A085541 - A021031 - A175645.
%e 0.134125178915463540428599329999431198995879919752168337370599106153853349956...
%o (PARI) s=0;forprimestep(p=2,1e8,3,s+=1./p^3);s \\ For illustration: using primes up to 10^N gives about 2N+2 (= 18 for N=8) correct digits.
%o (PARI) A343613_upto(N=100)={localprec(N+5); digits((PrimeZeta32(3)+1)\.1^N)[^1]} \\ see A343612 for the function PrimeZeta32.
%Y Cf. A003627 (primes 3k-1), A085541 (PrimeZeta(3)), A021031 (1/27).
%Y Cf. A175645 (same for p==1 (mod 3)), A086033 (for primes 4k+1), A085992 (for primes 4k+3), A343612 - A343619 (P_{3,2}(2..9): same for 1/p^2, ..., 1/p^9).
%K nonn,cons
%O 0,2
%A _M. F. Hasler_, Apr 22 2021