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A343627
Decimal expansion of the Prime Zeta modulo function P_{3,1}(7) = Sum 1/p^7 over primes p == 1 (mod 3).
2
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, 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, 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
OFFSET
0,7
COMMENTS
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 + ...
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.
EXAMPLE
P_{3,1}(7) = 1.231372255481919674448947124444003936669057866...*10^-6
MATHEMATICA
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 *)
PROG
(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.
(PARI) A343627_upto(N=100)={localprec(N+5); digits((PrimeZeta31(7)+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), A343607 (P_{3,2}(7): same for p==2 (mod 3)), A086037 (P_{4,1}(7): same for p==1 (mod 4)).
Cf. A085967 (PrimeZeta(7)), A002476 (primes of the form 3k+1).
Sequence in context: A292915 A271700 A136555 * A188107 A174014 A236376
KEYWORD
cons,nonn
AUTHOR
M. F. Hasler, Apr 23 2021
STATUS
approved