A343616 -id:A343616 - cp's OEIS frontend

A086036 Decimal expansion of the prime zeta modulo function at 6 for primes of the form 4k+1.

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

Offset: 0

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

			6.4250963664773791101819138043576598984545546978815052898566258438984520...*10^-5

Jean-François Alcover, Table of n, a(n) for n = 0..1006
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=6), page 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A085995 (same for primes 4k+3), A343626 (for primes 3k+1), A343616 (for primes 3k+2), A086032, ..., A086039 (for 1/p^2, ..., 1/p^9), A085966 (PrimeZeta(6)), A002144 (primes of the form 4k+1).

digits = 1003; 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[(2n + 1)*6]]/(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}, RealDigits[g[m], 10, digits][[1]]] (* Jean-François Alcover, Jun 24 2011, after X. Gourdon and P. Sebah, updated May 08 2021 *)

A086036_upto(N=100)={localprec(N+3); digits((PrimeZeta41(6)+1)\.1^N)[^1]} \\ see A086032 for the PrimeZeta41 function. - M. F. Hasler, Apr 26 2021

Zeta_Q(6) = Sum_{p in A002144} 1/p^6 where A002144 = {primes p == 1 mod 4};

= Sum_{odd m > 0} mu(m)/2m * log(DirichletBeta(6m)*zeta(6m)/zeta(12m)/(1+2^(-6m))) [using Gourdon & Sebah, Theorem 11]. - M. F. Hasler, Apr 26 2021

Edited by M. F. Hasler, Apr 26 2021

A085995 Decimal expansion of the prime zeta modulo function at 6 for primes of the form 4k+3.

Original entry on oeis.org

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

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003

			0.0013808358869717391630318541280158226106013963275654296802648025785307522...

P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.
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=3, s=6), page 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A002145 (primes 4k+3), A001014 (n^6), A085966 (PrimeZeta(6)).

Cf. A085991 - A085998 (Zeta_R(2..9): same for 1/p^2, ..., 1/p^9), A086036 (same for primes 4k+1), A343626 (for primes 3k+1), A343616 (for primes 3k+2).

b[x_] = (1 - 2^(-x))*(Zeta[x]/DirichletBeta[x]); $MaxExtraPrecision = 250; m = 40; Join[{0, 0}, RealDigits[(1/2)*NSum[MoebiusMu[2n + 1]* Log[b[(2n + 1)*6]]/(2n + 1), {n, 0, m}, AccuracyGoal -> 120, NSumTerms -> m, PrecisionGoal -> 120, WorkingPrecision -> 120] ][[1]]][[1 ;; 105]] (* Jean-François Alcover, Jun 22 2011, updated Mar 14 2018 *)

A085995_upto(N=100)={localprec(N+3); digits((PrimeZeta43(6)+1)\.1^N)[^1]} \\ see A085991 for the PrimeZeta43 function. - M. F. Hasler, Apr 25 2021

Zeta_R(6) = Sum_{p in A002145} 1/p^6 where A002145 = {primes p == 3 (mod 4)},
  = (1/2)*Sum_{n >= 0} möbius(2*n+1)*log(b((2*n+1)*6))/(2*n+1),
  where b(x) = (1-2^(-x))*zeta(x)/L(x) and L(x) is the Dirichlet Beta function.

Edited by M. F. Hasler, Apr 25 2021

cp's OEIS Frontend

A086036 Decimal expansion of the prime zeta modulo function at 6 for primes of the form 4k+1.

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