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A343619 Decimal expansion of P_{3,2}(9) = Sum 1/p^9 over primes == 2 (mod 3). 9
0, 0, 1, 9, 5, 3, 6, 3, 7, 4, 3, 3, 1, 5, 8, 7, 1, 3, 7, 2, 0, 8, 0, 4, 6, 0, 1, 5, 1, 2, 3, 9, 2, 9, 1, 7, 6, 0, 6, 9, 3, 3, 5, 0, 0, 3, 9, 1, 2, 2, 2, 0, 6, 4, 6, 2, 9, 1, 6, 2, 6, 1, 3, 4, 0, 4, 2, 4, 6, 8, 4, 9, 4, 2, 8, 9, 4, 9, 0, 3, 2, 9, 3, 0, 3, 4, 2, 1, 7, 9, 3, 7, 8, 2, 6, 9, 0, 7, 9, 1, 2, 4, 6, 3, 8 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
The prime zeta modulo function P_{m,r}(s) = Sum_{primes p == r (mod m)} 1/p^s generalizes the prime zeta function P(s) = Sum_{primes p} 1/p^s.
LINKS
Jean-François Alcover, Table of n, a(n) for n = 0..1005
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, value P(m=3, n=2, s=9), p. 21.
FORMULA
P_{3,2}(9) = Sum_{p in A003627} 1/p^9 = P(9) - 1/3^9 - P_{3,1}(9).
EXAMPLE
0.0019536374331587137208046015123929176069335003912220646291626134042468494...
MATHEMATICA
digits = 1004; nmax0 = 50; dnmax = 10;
Clear[PrimeZeta31];
PrimeZeta31[s_, nmax_] := PrimeZeta31[s, nmax] = Sum[Module[{t}, t = s + 2 n*s; MoebiusMu[2 n + 1] ((1/(4 n + 2)) (-Log[1 + 2^t] - Log[1 + 3^t] + Log[Zeta[t]] - Log[Zeta[2 t]] + Log[Zeta[t, 1/6] - Zeta[t, 5/6]]))], {n, 0, nmax}] // N[#, digits + 5] &;
PrimeZeta31[9, nmax = nmax0];
PrimeZeta31[9, nmax += dnmax];
While[Abs[PrimeZeta31[9, nmax] - PrimeZeta31[9, nmax - dnmax]] > 10^-(digits + 5), Print["nmax = ", nmax]; nmax += dnmax];
PrimeZeta32[9] = PrimeZetaP[9] - 1/3^9 - PrimeZeta31[9, nmax];
Join[{0, 0}, RealDigits[PrimeZeta32[9], 10, digits][[1]] ] (* Jean-François Alcover, May 07 2021, after M. F. Hasler's PARI code *)
PROG
(PARI) A343619_upto(N=100)={localprec(N+5); digits((PrimeZeta32(9)+1)\.1^N)[^1]} \\ see A343612 for the function PrimeZeta32
CROSSREFS
Cf. A003627 (primes 3k-1), A001017 (n^9), A085969 (PrimeZeta(9)).
Cf. A343612 - A343618 (P_{3,2}(s): analog for 1/p^s, s = 2 .. 8).
Cf. A343629 (for primes 3k+1), A086039 (for primes 4k+1), A085998 (for primes 4k+3).
Sequence in context: A154543 A203132 A201515 * A259982 A200483 A057888
KEYWORD
nonn,cons
AUTHOR
M. F. Hasler, Apr 25 2021
STATUS
approved

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Last modified March 29 05:43 EDT 2024. Contains 371264 sequences. (Running on oeis4.)