A343615 - OEIS

A343615

Decimal expansion of P_{3,2}(5) = Sum 1/p^5 over primes == 2 (mod 3).

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

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OFFSET

0,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..98.

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, value P(m=3, n=2, s=5) on p. 21.

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

FORMULA

P_{3,2}(5) = P(5) - 1/3^5 - P_{3,1}(5).

EXAMPLE

0.0315771357190039419560337803437163963477729963832486145790258341228297557...

PROG

(PARI) s=0; forprimestep(p=2, 1e8, 3, s+=1./p^4); s \\ For illustration: using primes up to 10^N gives about 3N+2 (= 26 for N=8) correct digits.

(PARI) A343615_upto(N=100, s=5)={localprec(N+5); digits((sumeulerrat(1/p^s)-1/3^s-PrimeZeta31(s)+1)\.1^N)[^1]} \\ see A175644 for the function PrimeZeta31, A343612 for a function PrimeZeta32

CROSSREFS

Cf. A003627 (primes 3k-1), A000584 (n^5), A085965 (PrimeZeta(5)), A021247 (1/3^5).

Cf. A343625 (same for primes 3k+1), A086035 (for primes 4k+1), A085994 (for primes 4k+3), A343612 - A343619 (P_{3,2}(2..9): same for 1/p^2, ..., 1/p^9).

Sequence in context: A169998 A326729 A171998 * A159285 A021080 A354576

Adjacent sequences: A343612 A343613 A343614 * A343616 A343617 A343618

KEYWORD

nonn,cons

AUTHOR

M. F. Hasler, Apr 22 2021

STATUS

approved