A343626 Decimal expansion of the Prime Zeta modulo function P_{3,1}(6) = Sum 1/p^6 over primes p == 1 (mod 3).

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

Offset

0,6

Comments

The Prime Zeta modulo function at 6 for primes of the form 3k+1 is Sum_{primes in A002476} 1/p^6 = 1/7^6 + 1/13^6 + 1/19^6 + 1/31^6 + ...

The complementary Sum_{primes in A003627} 1/p^6 is given by P_{3,2}(6) = A085966 - 1/3^6 - (this value here) = 0.015689614727130461563527666... = A343606.

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.

OEIS index to entries related to the (prime) zeta function.

Example

P_{3,1}(6) = 8.7300110231981670120427791452319495610797645391837...*10^-8

Mathematica

With[{s=6}, 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^6); s \\ For illustration: primes up to 10^N give 5N+2 (= 42 for N=8) correct digits.
(PARI) A343626_upto(N=100)={localprec(N+5); digits((PrimeZeta31(6)+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), A343606 (P_{3,2}(6): same for p==2 (mod 3)), A086036 (P_{4,1}(6): same for p==1 (mod 4)).

Cf. A085966 (PrimeZeta(6)), A002476 (primes of the form 3k+1).

Sequence in context: A244839 A329450 A203069 * A272531 A244684 A201742

Adjacent sequences: A343623 A343624 A343625 * A343627 A343628 A343629

Keyword

cons,nonn

Author

M. F. Hasler, Apr 23 2021

Status

approved