A085996 -id:A085996 - cp's OEIS frontend

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

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

			1.2818448599795268251026582166507935820606749563344794362656914682... * 10^-5

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.

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).

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 *)

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

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

Edited by M. F. Hasler, Apr 26 2021

A343617 Decimal expansion of P_{3,2}(7) = Sum 1/p^7 over primes == 2 (mod 3).

Original entry on oeis.org

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

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.

			0.0078253541130504928742517016707559206033079309751324433146804883394...

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=7), p. 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A003627 (primes 3k-1), A001015 (n^7), A085967 (PrimeZeta(7)).
Cf. A343612 - A343619 (P_{3,2}(s): analog for 1/p^s, s = 2 .. 9).
Cf. A343627 (for primes 3k+1), A086037 (for primes 4k+1), A085996 (for primes 4k+3).

A343617_upto(N=100)={localprec(N+5); digits((PrimeZeta32(7)+1)\.1^N)[^1]} \\ see A343612 for the function PrimeZeta32

P_{3,2}(7) = Sum_{p in A003627} 1/p^7 = P(7) - 1/3^7 - P_{3,1}(7).

cp's OEIS Frontend

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

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