A343617 - OEIS

%I #13 Aug 25 2021 15:47:01

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

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

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

%N Decimal expansion of P_{3,2}(7) = Sum 1/p^7 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=7), p. 21.

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

%F P_{3,2}(7) = Sum_{p in A003627} 1/p^7 = P(7) - 1/3^7 - P_{3,1}(7).

%e 0.0078253541130504928742517016707559206033079309751324433146804883394...

%o (PARI) A343617_upto(N=100)={localprec(N+5); digits((PrimeZeta32(7)+1)\.1^N)[^1]} \\ see A343612 for the function PrimeZeta32

%Y Cf. A003627 (primes 3k-1), A001015 (n^7), A085967 (PrimeZeta(7)).

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

%Y Cf. A343627 (for primes 3k+1), A086037 (for primes 4k+1), A085996 (for primes 4k+3).

%K nonn,cons

%O 0,3

%A _M. F. Hasler_, Apr 25 2021