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A343628
Decimal expansion of the Prime Zeta modulo function P_{3,1}(8) = Sum 1/p^8 over primes p == 1 (mod 3).
3
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, 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, 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
OFFSET
0,8
COMMENTS
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 + ...
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.
EXAMPLE
P_{3,1}(8) = 1.7475285336300871799410908797038110474049197734628...*10^-7
MATHEMATICA
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 *)
PROG
(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.
(PARI) A343628_upto(N=100)={localprec(N+5); digits((PrimeZeta31(8)+1)\.1^N)[^1]} \\ cf. A175644 for PrimeZeta31
CROSSREFS
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)).
Cf. A085968 (PrimeZeta(8)), A002476 (primes of the form 3k+1).
Sequence in context: A153586 A319883 A153186 * A085469 A050996 A085541
KEYWORD
cons,nonn
AUTHOR
M. F. Hasler, Apr 23 2021
STATUS
approved