cp's OEIS Frontend

A343619 Decimal expansion of P_{3,2}(9) = Sum 1/p^9 over primes == 2 (mod 3).

%I A343619 #17 May 07 2021 20:34:38
%S A343619 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,
%T A343619 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,
%U A343619 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
%N A343619 Decimal expansion of P_{3,2}(9) = Sum 1/p^9 over primes == 2 (mod 3).
%C A343619 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.
%H A343619 Jean-François Alcover, <a href="/A343619/b343619.txt">Table of n, a(n) for n = 0..1005</a>
%H A343619 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, value P(m=3, n=2, s=9), p. 21.
%H A343619 <a href="/index/Z#zeta_function">OEIS index to entries related to the (prime) zeta function</a>.
%F A343619 P_{3,2}(9) = Sum_{p in A003627} 1/p^9 = P(9) - 1/3^9 - P_{3,1}(9).
%e A343619 0.0019536374331587137208046015123929176069335003912220646291626134042468494...
%t A343619 digits = 1004; nmax0 = 50; dnmax = 10;
%t A343619 Clear[PrimeZeta31];
%t A343619 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] &;
%t A343619 PrimeZeta31[9, nmax = nmax0];
%t A343619 PrimeZeta31[9, nmax += dnmax];
%t A343619 While[Abs[PrimeZeta31[9, nmax] - PrimeZeta31[9, nmax - dnmax]] > 10^-(digits + 5), Print["nmax = ", nmax]; nmax += dnmax];
%t A343619 PrimeZeta32[9] = PrimeZetaP[9] - 1/3^9 - PrimeZeta31[9, nmax];
%t A343619 Join[{0, 0}, RealDigits[PrimeZeta32[9], 10, digits][[1]] ] (* _Jean-François Alcover_, May 07 2021, after _M. F. Hasler_'s PARI code *)
%o A343619 (PARI) A343619_upto(N=100)={localprec(N+5); digits((PrimeZeta32(9)+1)\.1^N)[^1]} \\ see A343612 for the function PrimeZeta32
%Y A343619 Cf. A003627 (primes 3k-1), A001017 (n^9), A085969 (PrimeZeta(9)).
%Y A343619 Cf. A343612 - A343618 (P_{3,2}(s): analog for 1/p^s, s = 2 .. 8).
%Y A343619 Cf. A343629 (for primes 3k+1), A086039 (for primes 4k+1), A085998 (for primes 4k+3).
%K A343619 nonn,cons
%O A343619 0,4
%A A343619 _M. F. Hasler_, Apr 25 2021