A343626 - OEIS

%I #8 Apr 24 2021 03:12:48

%S 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,

%T 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,

%U 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

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

%C 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 + ...

%C 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.

%H R. J. Mathar, <a href="http://arxiv.org/abs/1008.2547">Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli</a>, arXiv:1008.2547 [math.NT], 2010-2015, p.21.

%H <a href="/index/Z#zeta_function">OEIS index to entries related to the (prime) zeta function</a>.

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

%t 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 *)

%o (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.

%o (PARI) A343626_upto(N=100)={localprec(N+5);digits((PrimeZeta31(6)+1)\.1^N)[^1]} \\ cf. A175644 for PrimeZeta31

%Y 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)).

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

%K cons,nonn

%O 0,6

%A _M. F. Hasler_, Apr 23 2021