cp's OEIS Frontend

A343615 Decimal expansion of P_{3,2}(5) = Sum 1/p^5 over primes == 2 (mod 3).

%I A343615 #8 Apr 26 2021 03:58:16
%S A343615 0,3,1,5,7,7,1,3,5,7,1,9,0,0,3,9,4,1,9,5,6,0,3,3,7,8,0,3,4,3,7,1,6,3,
%T A343615 9,6,3,4,7,7,7,2,9,9,6,3,8,3,2,4,8,6,1,4,5,7,9,0,2,5,8,3,4,1,2,2,8,2,
%U A343615 9,7,5,5,7,1,9,8,1,1,7,3,0,3,9,1,5,9,6,1,1,0,7,5,2,9,7,6,2,6,2
%N A343615 Decimal expansion of P_{3,2}(5) = Sum 1/p^5 over primes == 2 (mod 3).
%C A343615 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 A343615 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=5) on p. 21.
%H A343615 <a href="/index/Z#zeta_function">OEIS index to entries related to the (prime) zeta function</a>.
%F A343615 P_{3,2}(5) = P(5) - 1/3^5 - P_{3,1}(5).
%e A343615 0.0315771357190039419560337803437163963477729963832486145790258341228297557...
%o A343615 (PARI) s=0;forprimestep(p=2,1e8,3,s+=1./p^4);s \\ For illustration: using primes up to 10^N gives about 3N+2 (= 26 for N=8) correct digits.
%o A343615 (PARI) A343615_upto(N=100, s=5)={localprec(N+5); digits((sumeulerrat(1/p^s)-1/3^s-PrimeZeta31(s)+1)\.1^N)[^1]} \\ see A175644 for the function PrimeZeta31, A343612 for a function PrimeZeta32
%Y A343615 Cf. A003627 (primes 3k-1), A000584 (n^5), A085965 (PrimeZeta(5)), A021247 (1/3^5).
%Y A343615 Cf. A343625 (same for primes 3k+1), A086035 (for primes 4k+1), A085994 (for primes 4k+3), A343612 - A343619 (P_{3,2}(2..9): same for 1/p^2, ..., 1/p^9).
%K A343615 nonn,cons
%O A343615 0,2
%A A343615 _M. F. Hasler_, Apr 22 2021