A343609 -id:A343609 - cp's OEIS frontend

A343629 Decimal expansion of the Prime Zeta modulo function P_{3,1}(9) = Sum 1/p^9 over primes p == 1 (mod 3).

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

Offset: 0

M. F. Hasler, Apr 23 2021

The Prime Zeta modulo function at 9 for primes of the form 3k+1 is Sum_{primes in A002476} 1/p^9 = 1/7^9 + 1/13^9 + 1/19^9 + 1/31^9 + ...

The complementary Sum_{primes in A003627} 1/p^8 is given by P_{3,2}(8) = A085969 - 1/3^9 - (this value here) = 0.0039088148233885949714061... = A343609.

			P_{3,1}(9) = 2.4878378446082135873838215937876340672308259947340815...*10^-8

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.

Cf. A086039 (P_{4,1}(9): same for p==1 (mod 4)), A175645, A343624 - A343628 (P_{3,1}(3..8): same for 1/p^n, n = 3..8), A343609 (P_{3,2}(9): same for p==2 (mod 3)).

Cf. A085969 (PrimeZeta(9)), A002476 (primes of the form 3k+1).

With[{s=9}, 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^9); s \\ For illustration: primes up to 10^N give ~ 8N+2 (= 66 for N=8) correct digits.

A343629_upto(N=100)={localprec(N+5);digits((PrimeZeta31(9)+1)\.1^N)[^1]} \\ cf. A175644 for PrimeZeta31

cp's OEIS Frontend

A343629 Decimal expansion of the Prime Zeta modulo function P_{3,1}(9) = Sum 1/p^9 over primes p == 1 (mod 3).

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