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

Offset: 0

Author

M. F. Hasler, Apr 22 2021

Keywords

• nonn
• cons

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.

Examples

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

Links

• 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.

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).

Programs

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.

Formula

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