A343614 -id:A343614 - cp's OEIS frontend

A343624 Decimal expansion of the Prime Zeta modulo function P_{3,1}(4) = Sum 1/p^4 over primes p == 1 (mod 3).

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

Offset: 0

M. F. Hasler, Apr 23 2021

The Prime Zeta modulo function at 4 for primes of the form 3k+1 is Sum_{primes in A002476} 1/p^4 = 1/7^4 + 1/13^4 + 1/19^4 + 1/31^4 + ...

The complementary Sum_{primes in A003627} 1/p^4 is given by P_{3,2}(4) = A085964 - 1/3^4 - (this value here) = 0.064186145696557778990099... = A343614.

			P_{3,1}(4) = 0.000461315055343386940174530333409454339939018353816870...

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, p.21
OEIS index to entries related to the (prime) zeta function.

Cf. A175645, A343625 - A343629 (P_{3,1}(3..9): same for 1/p^s, s=3, 5,..., 9).

Cf. A343614 (P_{3,2}(4): same for p==2 (mod 3)), A086034 (P_{4,1}(4): same for p==1 (mod 4)), A085964 (PrimeZeta(4)).

With[{s=4}, Do[Print[N[1/2 * Sum[(MoebiusMu[2*n + 1]/(2*n + 1)) * Log[(Zeta[s + 2*n*s]*(Zeta[s + 2*n*s, 1/6] - Zeta[s + 2*n*s, 5/6])) / ((1 + 2^(s + 2*n*s))*(1 + 3^(s + 2*n*s)) * Zeta[2*(1 + 2*n)*s])], {n, 0, m}], 120]], {m, 100, 500, 100}]] (* adopted from Vaclav Kotesovec's code in A175645 *)

s=0; forprimestep(p=1, 1e8, 3, s+=1./p^4); s \\ Naïve, for illustration: primes up to 10^N give about 3N+2 (= 26 for N=8) correct digits.

A343606_upto(N=100)={localprec(N+5);digits((PrimeZeta31(4)+1)\.1^N)[^1]} \\ cf. A175644 for PrimeZeta31

A086034 Decimal expansion of the prime zeta modulo function at 4 for primes of the form 4k+1.

Original entry on oeis.org

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

Offset: 0

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

			0.0016495841540292915989967613136388518274879099438347321478115258388...

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=4), page 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A085993 (same for primes 4k+3), A343624 (for primes 3k+1), A343614 (for primes 3k+2), A086032 - A086039 (for 1/p^2, ..., 1/p^9), A085541 (PrimeZeta(3)), A002144 (primes of the form 4k+1).

a[s_] = (1 + 2^-s)^-1* DirichletBeta[s] Zeta[s]/Zeta[2 s]; m = 120; $MaxExtraPrecision = 680; Join[{0, 0}, RealDigits[(1/2)* NSum[MoebiusMu[2n + 1]*Log[a[(2n + 1)*4]]/(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 *)

A086034_upto(N=100)={localprec(N+3); digits((PrimeZeta41(4)+1)\.1^N)[^1]} \\ see A086032 for the PrimeZeta41 function. - M. F. Hasler, Apr 26 2021

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

= Sum_{odd m > 0} mu(m)/2m * log(DirichletBeta(4m)*zeta(4m)/zeta(8m)/(1+2^(-4m)))[using Gourdon & Sebah, Theorem 11] - M. F. Hasler, Apr 26 2021.

Edited by M. F. Hasler, Apr 26 2021

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A343624 Decimal expansion of the Prime Zeta modulo function P_{3,1}(4) = Sum 1/p^4 over primes p == 1 (mod 3).

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A086034 Decimal expansion of the prime zeta modulo function at 4 for primes of the form 4k+1.

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