A086037 Decimal expansion of the prime zeta modulo function at 7 for primes of the form 4k+1.

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

Offset

0,6

Links

Table of n, a(n) for n=0..104.

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=7), page 21.

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

Formula

Zeta_Q(7) = Sum_{p in A002144} 1/p^7 where A002144 = {primes p == 1 mod 4};

= Sum_{odd m > 0} mu(m)/2m*log(DirichletBeta(7m)*zeta(7m)/zeta(14m)/(1+2^(-7m))) [using Gourdon & Sebah, Theorem 11]. - M. F. Hasler, Apr 26 2021

Example

1.2818448599795268251026582166507935820606749563344794362656914682... * 10^-5

Mathematica

a[s_] = (1 + 2^-s)^-1* DirichletBeta[s] Zeta[s]/Zeta[2 s]; m = 120; $MaxExtraPrecision = 1200; Join[{0, 0, 0, 0}, RealDigits[(1/2)* NSum[MoebiusMu[2n + 1]*Log[a[(2n + 1)*7]]/(2n + 1), {n, 0, m}, AccuracyGoal -> m, NSumTerms -> m, PrecisionGoal -> m, WorkingPrecision -> m]][[1]]][[1 ;; 105]] (* Jean-François Alcover, Jun 24 2011, after X. Gourdon and P. Sebah, updated Mar 14 2018 *)

Prog

(PARI) A086037_upto(N=100)={localprec(N+3); digits((PrimeZeta41(7)+1)\.1^N)[^1]} \\ see A086032 for the PrimeZeta41 function. - M. F. Hasler, Apr 26 2021

Crossrefs

Cf. A085996 (same for primes 4k+3), A343627 (for primes 3k+1), A343617 (for primes 3k+2), A086032, ..., A086039 (for 1/p^2, ..., 1/p^9), A085967 (PrimeZeta(7)), A002144 (primes of the form 4k+1).

Sequence in context: A201763 A254277 A244688 * A343487 A199787 A165274

Adjacent sequences: A086034 A086035 A086036 * A086038 A086039 A086040

Keyword

cons,nonn

Author

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

Extensions

Edited by M. F. Hasler, Apr 26 2021

Status

approved