A343628 Decimal expansion of the Prime Zeta modulo function P_{3,1}(8) = Sum 1/p^8 over primes p == 1 (mod 3).

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

Offset

0,8

Comments

The Prime Zeta modulo function at 8 for primes of the form 3k+1 is Sum_{primes in A002476} 1/p^8 = 1/7^8 + 1/13^8 + 1/19^8 + 1/31^8 + ...

The complementary Sum_{primes in A003627} 1/p^8 is given by P_{3,2}(8) = A085968 - 1/3^8 - (this value here) = 0.0039088148233885949714061... = A343608.

Links

Table of n, a(n) for n=0..99.

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.

Example

P_{3,1}(8) = 1.7475285336300871799410908797038110474049197734628...*10^-7

Mathematica

With[{s=8}, 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 *)

Prog

(PARI) s=0; forprimestep(p=1, 1e8, 3, s+=1./p^8); s \\ For illustration: primes up to 10^N give ~ 7N+2 (= 58 for N=8) correct digits.
(PARI) A343628_upto(N=100)={localprec(N+5); digits((PrimeZeta31(8)+1)\.1^N)[^1]} \\ cf. A175644 for PrimeZeta31

Crossrefs

Cf. A175645, A343624 - A343629 (P_{3,1}(3..9): same for 1/p^n, n = 3..9), A343608 (P_{3,2}(8): same for p==2 (mod 3)), A086038 (P_{4,1}(8): same for p==1 (mod 4)).

Cf. A085968 (PrimeZeta(8)), A002476 (primes of the form 3k+1).

Sequence in context: A153586 A319883 A153186 * A085469 A050996 A085541

Adjacent sequences: A343625 A343626 A343627 * A343629 A343630 A343631

Keyword

cons,nonn

Author

M. F. Hasler, Apr 23 2021

Status

approved