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

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

Author

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

Keywords

• cons
• nonn

Examples

			2.56137168039646980824843231247393644726060180729887066675459917474121... * 10^-6
		

Links

• 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.

Crossrefs

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).

Programs

Mathematica

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 *)

PARI

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

Formula

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

Extensions

Edited by M. F. Hasler, Apr 26 2021