A343616 - OEIS

%I #8 Apr 26 2021 03:42:04

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

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

%U 7,2,3,3,5,6,2,3,7,3,0,4,0,7,0,0,5,5,2,4,2,2,1,0,3,3,6,8,4,3,5,2

%N Decimal expansion of P_{3,2}(6) = Sum 1/p^6 over primes == 2 (mod 3).

%C The prime zeta modulo function P_{m,r}(s) = Sum_{primes p == r (mod m)} 1/p^s generalizes the prime zeta function P(s) = Sum_{primes p} 1/p^s.

%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, value P(m=3, n=2, s=6), p. 21.

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

%F P_{3,2}(6) = Sum_{p in A003627} 1/p^6 = P(6) - 1/3^6 - P_{3,1}(6).

%e 0.015689614727130461563527666152209091814208675553077763366153188676457...

%o (PARI) A343616_upto(N=100)={localprec(N+5); digits((PrimeZeta32(6)+1)\.1^N)[^1]} \\ see A343612 for the function PrimeZeta32

%Y Cf. A003627 (primes 3k-1), A001014 (n^6), A085966 (PrimeZeta(6)), A021733 (1/3^6).

%Y Cf. A343612 - A343619 (P_{3,2}(s): analog for 1/p^s, s = 2 .. 9).

%Y Cf. A343626 (for primes 3k+1), A086036 (for primes 4k+1), A085995 (for primes 4k+3).

%K nonn,cons

%O 0,3

%A _M. F. Hasler_, Apr 25 2021