cp's OEIS Frontend

A343628 Decimal expansion of the Prime Zeta modulo function P_{3,1}(8) = Sum 1/p^8 over primes p == 1 (mod 3).

%I A343628 #8 Apr 24 2021 03:12:56
%S A343628 0,0,0,0,0,0,1,7,4,7,5,2,8,5,3,3,6,3,0,0,8,7,1,7,9,9,4,1,0,9,0,8,7,9,
%T A343628 7,0,3,8,1,1,0,4,7,4,0,4,9,1,9,7,7,3,4,6,2,8,1,7,7,9,6,6,7,9,6,1,3,7,
%U A343628 9,8,3,7,4,9,9,6,3,5,3,6,4,5,7,9,2,3,2,5,8,3,2,9,9,5,9,9,0,2,0,8
%N A343628 Decimal expansion of the Prime Zeta modulo function P_{3,1}(8) = Sum 1/p^8 over primes p == 1 (mod 3).
%C A343628 The Prime Zeta modulo function at 8 for primes of the form 3k+1 is Sum_{primes in A002476} 1/p^8 = 1/7^8 + 1/13^8 + 1/19^8 + 1/31^8 + ...
%C A343628 The complementary Sum_{primes in A003627} 1/p^8 is given by P_{3,2}(8) = A085968 - 1/3^8 - (this value here) = 0.0039088148233885949714061... = A343608.
%H A343628 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 A343628 P_{3,1}(8) = 1.7475285336300871799410908797038110474049197734628...*10^-7
%t A343628 With[{s=8}, 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 A343628 (PARI) s=0; forprimestep(p=1, 1e8, 3, s+=1./p^8); s \\ For illustration: primes up to 10^N give ~ 7N+2 (= 58 for N=8) correct digits.
%o A343628 (PARI) A343628_upto(N=100)={localprec(N+5);digits((PrimeZeta31(8)+1)\.1^N)[^1]} \\ cf. A175644 for PrimeZeta31
%Y A343628 Cf. A175645, A343624 - A343629 (P_{3,1}(3..9): same for 1/p^n, n = 3..9), A343608 (P_{3,2}(8): same for p==2 (mod 3)), A086038 (P_{4,1}(8): same for p==1 (mod 4)).
%Y A343628 Cf. A085968 (PrimeZeta(8)), A002476 (primes of the form 3k+1).
%K A343628 cons,nonn
%O A343628 0,8
%A A343628 _M. F. Hasler_, Apr 23 2021