A343618 -id:A343618 - cp's OEIS frontend

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

Offset: 0

M. F. Hasler, Apr 25 2021

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.

			0.0019536374331587137208046015123929176069335003912220646291626134042468494...

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.

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).

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 *)

A343619_upto(N=100)={localprec(N+5); digits((PrimeZeta32(9)+1)\.1^N)[^1]} \\ see A343612 for the function PrimeZeta32

P_{3,2}(9) = Sum_{p in A003627} 1/p^9 = P(9) - 1/3^9 - P_{3,1}(9).

A086038 Decimal expansion of the prime zeta modulo function at 8 for primes of the form 4k+1.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 5, 6, 1, 3, 7, 1, 6, 8, 0, 3, 9, 6, 4, 6, 9, 8, 0, 8, 2, 4, 8, 4, 3, 2, 3, 1, 2, 4, 7, 3, 9, 3, 6, 4, 4, 7, 2, 6, 0, 6, 0, 1, 8, 0, 7, 2, 9, 8, 8, 7, 0, 6, 6, 6, 7, 5, 4, 5, 9, 9, 1, 7, 4, 7, 4, 1, 2, 1, 1, 1, 8, 8, 8, 4, 8, 9, 3, 8, 8, 9, 7, 9, 8, 9, 1, 4, 8, 1, 7, 8, 0, 3, 0, 3, 0, 1, 3, 7, 6

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 07 2003

			2.56137168039646980824843231247393644726060180729887066675459917474121... * 10^-6

Jean-François Alcover, Table of n, a(n) for n = 0..1004
X. Gourdon and P. Sebah, Some Constants from Number theory.
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=4, n=1, s=8), page 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A085997 (same for primes 4k+3), A343628 (for primes 3k+1), A343618 (for primes 3k+2), A086032 - A086039 (for 1/p^2, ..., 1/p^9), A085968 (PrimeZeta(8)), A002144 (primes of the form 4k+1).

digits = 1000; m0 = 50; dm = 10; dd = 10; Clear[f, g];
b[s_] := (1+2^-s)^-1*DirichletBeta[s] Zeta[s]/Zeta[2s] // N[#, digits+dd]&;
f[n_] := f[n] = (1/2) MoebiusMu[2n + 1] Log[b[8(2n + 1)]]/(2n + 1);
g[m_] := g[m] = Sum[f[n], {n, 0, m}] ; g[m = m0]; g[m += dm];
While[Abs[g[m] - g[m - dm]] < 10^(-digits - dd), Print[m]; m += dm];
Join[{0, 0, 0, 0, 0}, RealDigits[g[m], 10, digits][[1]]] (* Jean-François Alcover, Jun 24 2011, after X. Gourdon and P. Sebah, updated May 08 2021 *)

A086038_upto(N=100)={localprec(N+3); digits((PrimeZeta41(8)+1)\.1^N)[^1]} \\ see A086032 for the PrimeZeta41 function. - M. F. Hasler, Apr 26 2021

Zeta_Q(8) = Sum_{p in A002144} 1/p^8, where A002144 = {primes p == 1 (mod 4)};

= Sum_{odd m > 0} mu(m)/2m * log(DirichletBeta(8m)*zeta(8m)/zeta(16m)/(1+2^(-8m))) [using Gourdon & Sebah, Theorem 11]. - M. F. Hasler, Apr 26 2021

Edited by M. F. Hasler, Apr 26 2021

cp's OEIS Frontend

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

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