%I #8 Apr 24 2021 03:12:52

%S 0,0,0,0,0,1,2,3,1,3,7,2,2,5,5,4,8,1,9,1,9,6,7,4,4,4,8,9,4,7,1,2,4,4,

%T 4,4,0,0,3,9,3,6,6,6,9,0,5,7,8,6,6,2,6,3,7,0,7,2,8,1,9,6,3,7,0,6,2,0,

%U 2,1,0,5,7,4,1,2,0,6,7,2,6,0,0,6,9,5,5,9,2,2,1,2,7,4,9,2,4,8,2,5

%N Decimal expansion of the Prime Zeta modulo function P_{3,1}(7) = Sum 1/p^7 over primes p == 1 (mod 3).

%C The Prime Zeta modulo function at 7 for primes of the form 3k+1 is Sum_{primes in A002476} 1/p^7 = 1/7^7 + 1/13^7 + 1/19^7 + 1/31^7 + ...

%C The complementary Sum_{primes in A003627} 1/p^7 is given by P_{3,2}(7) = A085967 - 1/3^7 - (this value here) = 0.0078253541130504928742517... = A343607.

%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, p.21.

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

%e P_{3,1}(7) = 1.231372255481919674448947124444003936669057866...*10^-6

%t With[{s=7}, 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 (PARI) s=0; forprimestep(p=1, 1e8, 3, s+=1./p^7); s \\ For illustration: primes up to 10^N give 6N+2 (= 50 for N=8) correct digits.

%o (PARI) A343627_upto(N=100)={localprec(N+5);digits((PrimeZeta31(7)+1)\.1^N)[^1]} \\ cf. A175644 for PrimeZeta31

%Y Cf. A175645, A343624 - A343629 (P_{3,1}(3..9): same for 1/p^n, n = 3..9), A343607 (P_{3,2}(7): same for p==2 (mod 3)), A086037 (P_{4,1}(7): same for p==1 (mod 4)).

%Y Cf. A085967 (PrimeZeta(7)), A002476 (primes of the form 3k+1).

%K cons,nonn

%O 0,7

%A _M. F. Hasler_, Apr 23 2021