cp's OEIS Frontend

A343629 Decimal expansion of the Prime Zeta modulo function P_{3,1}(9) = Sum 1/p^9 over primes p == 1 (mod 3).

%I A343629 #7 Apr 24 2021 03:13:00
%S A343629 0,0,0,0,0,0,0,2,4,8,7,8,3,7,8,4,4,6,0,8,2,1,3,5,8,7,3,8,3,8,2,1,5,9,
%T A343629 3,7,8,7,6,3,4,0,6,7,2,3,0,8,2,5,9,9,4,7,3,4,0,8,1,5,2,5,9,4,9,1,8,7,
%U A343629 4,6,7,2,3,8,2,1,9,0,9,2,0,8,9,0,0,5,0,1,9,8,4,2,1,9,4,7,7,0,1,4
%N A343629 Decimal expansion of the Prime Zeta modulo function P_{3,1}(9) = Sum 1/p^9 over primes p == 1 (mod 3).
%C A343629 The Prime Zeta modulo function at 9 for primes of the form 3k+1 is Sum_{primes in A002476} 1/p^9 = 1/7^9 + 1/13^9 + 1/19^9 + 1/31^9 + ...
%C A343629 The complementary Sum_{primes in A003627} 1/p^8 is given by P_{3,2}(8) = A085969 - 1/3^9 - (this value here) = 0.0039088148233885949714061... = A343609.
%H A343629 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, p.21.
%e A343629 P_{3,1}(9) = 2.4878378446082135873838215937876340672308259947340815...*10^-8
%t A343629 With[{s=9}, 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}]] (* adopted from _Vaclav Kotesovec_'s code in A175645 *)
%o A343629 (PARI) s=0; forprimestep(p=1, 1e8, 3, s+=1./p^9); s \\ For illustration: primes up to 10^N give ~ 8N+2 (= 66 for N=8) correct digits.
%o A343629 (PARI) A343629_upto(N=100)={localprec(N+5);digits((PrimeZeta31(9)+1)\.1^N)[^1]} \\ cf. A175644 for PrimeZeta31
%Y A343629 Cf. A086039 (P_{4,1}(9): same for p==1 (mod 4)), A175645, A343624 - A343628 (P_{3,1}(3..8): same for 1/p^n, n = 3..8), A343609 (P_{3,2}(9): same for p==2 (mod 3)).
%Y A343629 Cf. A085969 (PrimeZeta(9)), A002476 (primes of the form 3k+1).
%K A343629 cons,nonn
%O A343629 0,8
%A A343629 _M. F. Hasler_, Apr 23 2021