A343614 - OEIS

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A343614

Decimal expansion of P_{3,2}(4) = Sum 1/p^4 over primes == 2 (mod 3).

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

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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..99.

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

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

FORMULA

P_{3,2}(4) = P(4) - 1/3^4 - P_{3,1}(4) = A085964 - A021085 - A343624.

EXAMPLE

P_{3,2}(4) = 0.06418614569655777899009908658740273681...

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) A343614_upto(N=100)={localprec(N+5); digits((PrimeZeta32(4)+1)\.1^N)[^1]} \\ see for the function PrimeZeta32.

CROSSREFS

Cf. A003627 (primes 3k-1), A085964 (PrimeZeta(4)), A021085 (1/3^4).

Cf. A343624 (same for primes 3k+1), A086034 (for primes 4k+1), A085993 (for primes 4k+3), A343612 - A343619 (P_{3,2}(2..9): same for 1/p^2, ..., 1/p^9).

Sequence in context: A309222 A324034 A365319 * A086241 A378412 A204023

Adjacent sequences: A343611 A343612 A343613 * A343615 A343616 A343617

KEYWORD

nonn,cons

AUTHOR

M. F. Hasler, Apr 22 2021

STATUS

approved