A343619 - OEIS

A343619

Decimal expansion of P_{3,2}(9) = Sum 1/p^9 over primes == 2 (mod 3).

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

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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.

OEIS index to entries related to the (prime) zeta function.

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

Adjacent sequences: A343616 A343617 A343618 * A343620 A343621 A343622

KEYWORD

nonn,cons

AUTHOR

M. F. Hasler, Apr 25 2021

STATUS

approved